Download introduction to financial mathematics (map 5601)

(Spring) 2016
TABLE OF CONTENTS
Summary of the lecture
I NTRODUCTION TO F INANCIAL
M ATHEMATICS (MAP 5601)
A RASH FAHIM
T HIS LECTURE NOT IS CREATED UNDER THE SUPPORT OF NSF DMS-1209519 ‚ (S PRING ) 2016 ‚
F LORIDA S TATE U NIVERSITY
Last Revision: December 8, 2016
Table of Contents
I
Introduction
1 Basic financial derivatives
1.1 Futures and forward contracts
1.2 Vanilla call and put options . .
1.3 American options . . . . . . .
1.4 Bond . . . . . . . . . . . . . .
1.5 Credit derivatives . . . . . . .
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. 3
. 6
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. 10
2 No dominance principle and model-independent arbitrage
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3 Arrow-Debreu model
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3.1 One-period binomial model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 One-period trinomial model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Replication and complete market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
II Modeling financial assets in discrete-time
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1 Arbitrage and trading in discrete-time markets
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Arbitrage strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Binomial model
2.1 No arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Basic properties of binomial tree . . . . . . . . . . . . . . .
Time homogeneity . . . . . . . . . . . . . . . . . . . . . .
Martingale property . . . . . . . . . . . . . . . . . . . . . .
Markovian property . . . . . . . . . . . . . . . . . . . . . .
2.3 Pricing and replicating contingent claims in binomial model
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(Spring) 2016
2.4
TABLE OF CONTENTS
Summary of the lecture
Markovian property of the option price and the hedging components . . . . . . . . . . . . . 37
Martingale property of discounted option price and hedging strategy components . . . . . . 39
Dividend paying stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Calibrating the parameters of the model by the market date: Binomial model
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3.1 Time-varying return and volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
III
Modeling financial assets in continuous-time
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1 Trading and arbitrage in continuous-time markets
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2 Continuous-time market of Bachelier
2.1 Pricing and replicating contingent claims in Bachelier model
Price of contingent claim at time zero . . . . . . . . . . . .
Markovian property of the option price . . . . . . . . . . . .
Time homogeneity of the option price . . . . . . . . . . . .
Martingale property of the option price . . . . . . . . . . . .
Replication in Bachelier model: Delta Hedging . . . . . . .
2.2 Numerical methods for option pricing in Bachelier model . .
Finite-difference scheme for heat equation . . . . . . . . . .
Monte Carlo approximation for Bachelier model . . . . . . .
Quadrature approximation for Bachelier model . . . . . . .
2.3 Discussion: drawbacks of Bachelier model . . . . . . . . . .
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3 Continuous-time market of Black-Scholes
3.1 Black-Scholes model: limit of binomial under risk neutral probability . . . . . . . . . .
Asymptotics of parameters u, l, and p . . . . . . . . . . . . . . . . . . . . . . . . . . .
Arithmetic return versus log return . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Weak convergence of binomial model to geometric Brownian motion . . . . . . . . . . .
Calibrating binomial model: revised . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Pricing contingent claims in Black-Scholes model . . . . . . . . . . . . . . . . . . . . .
3.3 Delta Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Completeness of Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Error of discrete hedging in Black-Scholes model and Greeks . . . . . . . . . . . . . . .
Discrete hedging without money market account . . . . . . . . . . . . . . . . . . . . . .
Other derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Time-varying B-S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Black-Scholes with yield curve and forward interest rate . . . . . . . . . . . . . . . . .
3.8 Black-Scholes model and Brownian motion . . . . . . . . . . . . . . . . . . . . . . . .
Markovian property of Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . .
Martingale property of Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Physical versus risk neutral in Black-Scholes model . . . . . . . . . . . . . . . . . . . .
Volatility estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Black-Scholes equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Heat equation and Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . .
Solving Black-Scholes equation via finite-difference scheme for Black-Scholes equation
Binomial tree scheme for Black-Scholes . . . . . . . . . . . . . . . . . . . . . . . . . .
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(Spring) 2016
TABLE OF CONTENTS
Summary of the lecture
Monte Carlo scheme for Black-Scholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.11 Stock price with dividend in Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . 95
Black-Scholes equation with dividend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4 Term structure of the volatility and volatility smile
4.1 Local volatility models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Constant elasticity of variance model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calibration of local volatility models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 American options
5.1 Pricing American option in binomial model . . . . . .
Hedging American option in binomial model . . . . .
5.2 Pricing American option in Black-Scholes model . . .
Finding the edges of exercise boundary . . . . . . . .
Smooth fit . . . . . . . . . . . . . . . . . . . . . . . .
American option with finite maturity . . . . . . . . . .
American call option on discrete dividend paying asset
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Appendices
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Appendix A Convex functions
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Appendix B A review of probability theory
B.1 Basic concepts and definitions of discrete probability
B.2 General probability spaces . . . . . . . . . . . . . .
B.3 Martingales . . . . . . . . . . . . . . . . . . . . . .
B.4 Continuous random variables . . . . . . . . . . . . .
B.5 Characteristic function . . . . . . . . . . . . . . . .
B.6 Weak convergence . . . . . . . . . . . . . . . . . .
Weak convergence of probabilities . . . . . . . . . .
B.7 Donsker Invariance Principle and Brownian motion .
Sample space for Brownian motion . . . . . . . . . .
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Appendix C Stochastic analysis
C.1 Stochastic integral with respect to Brownian motion and Itô formula . . .
Martingale property of stochastic integral . . . . . . . . . . . . . . . . .
C.2 Itô formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3 Martingale property of stochastic integral and partial differential equations
C.4 Stochastic integral and Stochastic differential equation . . . . . . . . . .
C.5 Itô calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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(Spring) 2016
Summary of the lecture
1 FINANCIAL DERIVATIVES
Part I
Introduction
Risky (financial) asset or simple risky asset is a product which has an uncertain price pattern over time. For
example, stocks of a company in an exchange market behaves randomly over time. Unlike a risky asset, a
bank account with a fixed interest rate which has an absolutely predictable value, is called a risk-free asset.
Risky assets are classified in many different categories. Among them financial securities, which the largest
body of risky assets, are traded papers in the exchange market, and are divided into three subcategories;
equity, debt, and derivatives.
Equity is a claim of ownership of a company. If it is issued by a corporation, it is called common stock,
stock or share. Debt which are sometimes referred to as fixed-income instrument promises a fixed cash flow
until a time called maturity. The cash flow of some fixed-income securities is contingent on the default of the
issuer of the asset. Derivatives are asset with price depending on certain events. For example, a derivative
can promise a payment (payoff) dependent on the price of a stock, price of a fixed-income instrument, on
the event of default of a company, or on a climate event.
An important class of assets which are no financial securities is (securities) commodities. Commodities include crops, energy, metals, etc. The commodities are in particular important because our daily life depends
on them.
There are other assets which are not usually part of any of the above classes; e.g. real estate. If the asset is
easily traded in an exchange market it is called liquid. Equities are the most liquid; fixed-income instruments
and derivatives are less liquid. Commodities are now very liquid partly due to the introduction of emerging
economics. Real estate is one of the most illiquid assets.
1
Basic financial derivatives
In this section, we cover the basics of some simple financial derivatives, bonds and some credit derivatives.
1.1
Futures and forward contracts
Forward and futures contracts are the same in principle, but they differ in operational aspects. In both
contracts, the two sides are obliged to exchange a specified underlying asset at a specific date T in the future,
so-called maturity, at a fair price which both agree upon. In other words, futures and forward contracts lock
the price of a deal in the future.
Forward contracts are simpler than futures. They are non-tradable contracts between two specific parties;
one of which is the buyer of the underlying or the long position and the other is the seller of the underlying
or the short position. The buyer (seller) is obliged to buy (sell) the determined units of the underlying asset
from the seller (to the buyer) at a price specified in the forward contract, called forward price. Forward
price is usually agreed between two parties of the contract so that no money is exchanged between them
for entering the contract. The forward contract price is not universal and depend upon what the two parties
agree upon. Usually one party is the issuer of the forward and quotes the forward price to the other part
who is facing a decision to agree or decline to enter the deal. Generally, the issuer is a financial firm and the
buyer is a financial or industrial firm.
Unlike forward contracts, futures contracts are tradable in specialized markets. The price listed in a futures
contract is called futures price of the underlying asset for delivery at date T , which reflects the attitude of
the investors toward the price of underlying asset at maturity T . The futures price at time t for maturity T is
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(Spring) 2016
1 FINANCIAL DERIVATIVES
Summary of the lecture
denoted by Ft pT q. The futures price and the spot price St of the underlying asset are related through
lim Ft pT q “ ST .
tÑT
Either parties in a futures are supposed to open a marginal account managed by the market regulator. The
function of the marginal account is to prevent the counterparty risk, i.e. the risk that either of the parties
will be unable to meet their obligation on the futures. A holder of a futures contract is supposed to keep
the amount of money in the marginal account as the futures price changes. To understand the operation of
marginal account consider the case of the long position, the party who is obliged to buy the underlying at
time T at price F0 pT q. Her obligation at time T is ST ´ F0 pT q, i.e. she is supposed to buy the underlying
at price F0 pT q while the market price is ST . While F0 pT q is specified in the futures, the future price of the
underlying, ST , is unknown. If at time T , F0 pT q ď ST then she gains, but otherwise she loses F0 pT q ´ ST .
Based on this discussion, the regulator of futures market required the long position to hold at least F0 pT q´St
at any day t to cover the possible future loss. An example of marginal account rebalancing is shown in Table
1.
For example, the long position, who entered the futures contract at time 0, has to keep at least St ´ F0 pT q
in his marginal account at any time t till maturity. At the maturity, the value of the marginal account is
St ´ F0 pT q which is the same as the obligation of the long position. This allows market to clear and settle
all the futures contracts and reduces the counterparty risk, and is in contrast to the forward contract that has
the risk of one party not being able to meet its obligation. For more information of the mechanism of futures
markets, see [17, Chapter 2].
Time t
Underlying asset price St
Changes to the marginal account
1
+.05
-.05
2
-.04
+.04
3
-.6
+.6
4
+.02
-.02
Table 1: Rebalancing marginal account of long position in futures.
Remark 1.1. The marginal account can be subject to several regulations including minimum cash holding.
Remark 1.2. Unlike the forward contract, futures contract is not subject to counterparty risk. However, the
existence of marginal account creates an opportunity cost; the fund in the marginal account can alternatively
be invested somewhere else for profit; at least in a risk-free account with interest rate.
Example 1.1 (Futures opportunity cost). Consider a futures contract with maturity T “ 2 days and futures
price equal to $99.95 and a forward contract with the same maturity and forward price $100; both are
written on a risky asset with price S0 “ $99.94. The marginal account for the futures contract needs at
least $10 upon entering the contract and should by rebalanced thereafter according to the closing price at
the end of each day. We denote the day-end price by S1 and S2 for day one and day two respectively. Given
the risk free daily compounded interest rate is 0.2%, we what to know for which values of the spot price of
the underlying asset, pS1 , S2 q, the forward contract is more interesting than the futures contract for long
position.
The payoff of the forward contract for the long position is ST ´ 100, while the same quantity for the futures
is ST ´ 99.95. Therefore, the payoff of futures is worth .05 more than the payoff of forward on the maturity
date T “ 2. Now, let’s calculate the opportunity cost. On day one, the marginal account must have $10.01;
the cost of holding $10.01 in the marginal account for day one is
p10.01qp1 ` .002q ´ 10.01 “ p10.01qp.002q “ .02002,
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(Spring) 2016
Summary of the lecture
1 FINANCIAL DERIVATIVES
which is equivalent to p.02002qp1.002q “ .02006004 at the end of day two. On day to we have to keep $10
plus p99.95´S1 q` 1 in the marginal account which creates an opportunity cost of p.002qp10`p99.95´S1 q` q.
Therefore, the total gain/loss of futures minus forward is
f pS1 q :“ .02993996 ´ p.002qp99.95 ´ S1 q` .
As shown in Figure 1.1,
f pS1 q
´.16996004
84.98002
99.95
S1
Figure 1.1: The difference between the gain of the futures and forward in Example 1.1.
Exercise 1.1. Consider a futures contract with maturity T “ 2 days and futures price equal to $100 and a
forward contract with the same maturity and forward price $99; both are written on a risky asset with price
S0 “ $99. The marginal account for the futures contract needs at least $20 upon entering the contract and
should by rebalanced thereafter according to the spot price at the beginning of the day. Given the risk free
daily compounded interest rate is .2%, for which values of the spot price of the underlying asset, pS1 , S2 q,
the forward contract is more interesting than the futures contract for short position?
A futures market provides easy access to the futures contracts for a variety of products and for different
maturities. In addition, it makes termination of a contract possible. A long position in a futures can even his
position out by entering a short position of the same contract.
One of the practices in the futures market is rolling over. Imagine a trader who needs to have a contract
on a product at a maturity T far in the future, such that there is no contract in the futures market with such
maturities. However, there is one with maturity T1 ă T . The trader can enter the futures with maturity T1
and later close it when longer maturities become available. Then, he can continue thereafter until he reaches
certain maturity or even forever.
It is well-known result, see Proposition 2.2, that in an ideal world, the discounted projected price of an
asset in the future is greater or equal to the forward (futures) price; aka backwardation. In reality this may
be violated, i.e. the projected price of an asset is less than the futures price. This situation is called contango.
1.2
Vanilla call and put options
Call option gives the holder right but not obligation to buy a certain asset in a specified time in the future
at a pre-determined price. The specified time is called maturity and often is denoted by T and the predetermined price is called strike price and is denoted by K. The asset price at current time t is denoted by
St and at maturity by ST . Call options are available in the option markets at a price which depends, among
other factors, on T , K and St . We denote the price of the call option by CpT, K, S, tq2 or simply C when
appropriate. Thus, call option protects its owner against increase in the price of the underlying asset in the
future.
Another type of vanilla option, put option, protects its owner against decrease in the price, i.e. it promises
the seller of the underlying asset at least the price at maturity. The price of put option is denoted by
P pT, K, S, tq or simply P when appropriate.
1
2
pxq` :“ maxtx, 0u
We will see later that C only depends on T ´ t in many models.
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The payoff of an option is the gain of the owner in dollar amount. For instance, the payoff of a call option is
pST ´ Kq` . This is because when the market price at maturity is ST and the strike price is K, the holder of
the option is buying the underlying asset at lower price K and gains ST ´ K provided ST ą K. Otherwise
when ST ď K, the holder do not exercise the option and buy the asset from the market directly. Similarly,
the payoff of a put option is pK ´ ST q` .
K
ST
K
ST
Figure 1.2: Left: the payoff pST ´ Kq` of a call option with strike K. Right: the payoff pK ´ ST q` of a
put option with strike K.
Figure 1.3: Call and put option quotes on Tesla Motors Inc. stocks on January 11, 2016. The first column
is the price of the underlying asset (NASDAQ:TSLA). The bid price is the price that trades are willing to
buy the options and the ask price is the price that trades are willing to sell. The spot price at the time was
$206.11. Source: google finance.
Similar to futures, options are also traded in specialized markets. You can see option chain for Tesla Motors
Inc. in Figure 1.3. The columns“bid” and “ask” are showing the best buy and sell price in the outstanding
orders and column“Open Int” (open interest) show the total volume of outstanding orders.
When the price ST of the underlying asset is greater than K, we say that the call options are in-the-money
and the puts are out-of-the-money. Otherwise, when ST ă K, puts are in-the-money and the calls are
out-of-the-money. If the strike price K is (approximately) the same as asset price ST , we call the option
at-the-money (or ATM).
Far in-the-money call or put options are behaving like forward contracts but with a wrong forward price!
Similarly, far out-of-the-money call or put options are worth little.
The holder of an option is called long position and the issuer of the option is called short position. While
the holder has the privilege of exercising the option when profitable, the issuer has the obligation to pay the
holder the amount of payoff upon exercise.
A European option is an option whose payoff is a function gpST q of the asset price at maturity ST .
The function g is called a payoff function. Call and put options are in particular important because any
piecewise linear continuous European payoff can be written as a linear combination (possibly infinite!)
of call options and put options with possibly different strikes but the same maturity and a constant cash
4
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Summary of the lecture
Figure 1.4: Call and put option quotes on IBM. stocks on January 13, 2016. I the money options are
highlighted. Source: google finance.
amount; or equivalently
a linear combination of call options and put options. Therefore, the price of payoff
ř
gpST q “ a0 ` i ai pST ´ Ki q` is given by
ÿ
ai Cpτ, Ki , Sq.
Bt pT qa0 `
i
Remark 1.3. The underlying asset ST at time T is like a payoff of a call option with strike K “ 0.
Example 1.2. Put-call parity suggests that the payoff pK ´ ST q` of a put option can be written as long in
K amount of cash, long in one call option pST ´ Kq` , and short in one underlying asset (by Remark 1.3 a
call option with strike 0). See Figure 1.5.
“
K
ST
`
K
ST
`
K
ST
ST
Figure 1.5: pK ´ ST q` “ K ` pST ´ Kq` ´ ST .
Example 1.3. An option which promises the payoff gpST q :“ |ST ´ K| is made of a long position in a
call option with strike K, long position in a put with strike K both with the same maturity. Equivalently,
this payoff can be written as K amount of cash, a short position in call option with strike 03 , and two long
positions in a call option with strike K all with the same maturity.
Example 1.4. A put option with payoff pK ´ ST q` can be written as K amount of cash, a short position in
a call option with strike 0, and a long position on a call option with strike K.
Exercise 1.2. Consider the payoff gpST q shown in Figure 1.7.
3
Call option with strike 0 is the asset itself; pST ´ 0q` “ S0 .
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gpST q “ |ST ´ K|
K
ST
K
Figure 1.6: Payoff of Example 1.3
gpST q
K2 ´K1
2
K1
K1 `K2
2
K2
ST
Figure 1.7: Payoff
a) Write this payoff as a linear combination of the payoffs of some call options and put options with
different strikes and same maturity.
b) Repeat part (a) with call options and cash. (No put option is allowed.)
1.3
American options
The vanilla call and put options considered above are European options, i.e. they can only be exercised at
a maturity. American options are a boosted version of European options that give their owner the right but
not obligation to exercise any date before or at the maturity. Therefore, at the time of exercise τ P r0, T s,
an American call option has value equal to pSτ ´ Kq` . The exercise time is determined by comparing the
continuation value to the exercise value pSτ ´ Kq` . We will discuss this in details in the future. We use
CAm pT, K, S, tq and PAm pT, K, S, tq to denote the price of american call and american put, respectively.
The exercise time τ is not a deterministic time. More precisely, it can be a random time which depends
upon occurrence of certain events. Some of the famous exercise time are those of threshold type; the option
is exercised before maturity if the market price of option becomes equal to the exercise value. Since both of
these quantities behave randomly over time, we can not predict when this time exactly occurs.
1.4
Bond
Zero-coupon bond (or simply zero-coupon bond) is a fixed-income security which promises a fixed amount
of cash in a specified currency, called principle, face value or par value at a maturity T , e.g. a $100 at Jan.
30. A zero-coupon bond is traded in specialized market with a price often lower than the principle4 .
4
There has been instances when this has been violates, e.g. the financial crisis of 2007.
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For simplicity in the sequel, we always consider a zero-coupon bond with principle $1 and denote its price
at time t with maturity T by Bt pT q. The interest rate can have several meanings depending on the time
frame and frequency of discounting. The annual interest rate (APR) calculated yearly is simply given by
Rmo 12
q “ Bt1pT q , or
1 ` APRyr “ Bt1pT q , while monthly calculated APR is given by p1 ` AP12
1 ` APRyr “ p1 ` APRmo q12 .
1
n
n-times compounded APR satisfies p1 ` APR
n q “ Bt pT q . When the frequency of compounding n approaches infinity, we obtain the continuous compounding instantaneous (a.k.a. spot or short) interest rate,
denoted by r,ş and Bt pT q “ e´rpT ´tq . If the short interest rate is time dependent, i.e. rs with s P rt, T s, then
T
Bt pT q “ e´ t rs ds .
Similarly if R is an interest rate for the time period rt, T s, and Ri is respectively the interest rate for periods
rti´1 , ti s, where t0 “ t ă t1 ă t2 ă ¨ ¨ ¨ ă tn “ T , we have
p1 ` Rq “ p1 ` R1 q ¨ ¨ ¨ p1 ` Rn q.
The short rate r or rs is an abstract concept; it exists hypothetically. In practice, the interest rate is usually
given by APR with specified frequency of compounding, e.g. APR=2% with compounded quarterly, which
means a quarterly interest rate of .5%, or more commonly it is given by the notion of yield curve. The yield
Rt pT q of the zero-coupon bond Bt pT q; it is defined by
Bt pT q “ e´pT ´tqRt pT q or Rt pT q :“ ´
1
ln Bt pT q.
T ´t
(1.1)
The yield is a bivariate function R : D Ñ Rě0 where D is given by tpt, T q : T ą 0 and t ă T u. Therefore.
it is natural to call is yield surface. A model for R is often referred to as term structure. Then, the short rate
can be defined as rt :“ limT Ót Rt pT q.
Beside zero-coupon bonds, there are other bonds which pay coupons in regular basis, for example a bond
which pays $100 as principle in 12 months and $20 every quarter before maturity. A coupon-carrying bond
with coupons payments of $c at later dates T1 ă ... ă Tn and principle payment P at maturity T with
Tn ă T is worth
n
ÿ
c
e´pTi ´tqRt pTi q ` P e´pT ´tqRt pT q .
i“1
This is because each coupon is the same as a zero-coupon bond with principle c and maturity Ti , for i “
1, ..., n. Then, one write the price of coupon carrying option as
c
n
ÿ
Bt pTi q ` P Bt pT q.
i“1
Therefore, zero-coupon bond is the building block of all bonds.
Exercise 1.3. Let Bt pT q be the price of a zero-coupon bond at time t with principle $1 maturing at time T .
Recall that the yield of this bond is defined by Rt pT q which satisfies
Bt pT q “ e´Rt pT qpT ´tq .
a) If a risk-free 1-year zero-coupon bond with $100 principle is priced B0 p1q “ $96 and a risk-free
2-year zero-coupon bond with $100 principle is priced B0 p2q “ $96, find the price of a risk-free zero
coupon bond B1 p2q and the yield curve R1 p2q.
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b) What is the price of a bond which pays a $30 coupon at the end of this year and $100 as the principle
in two years?
The above discussion makes sense only when the issuer of the bond is not subject to default on payment of
coupons or principle. These kind of bonds are called sovereign bond and often are issued by federal reserve
or central bank of country in their own currency. For example, the sovereign bonds in US are T-bill, T-note,
and T-bond. T-bills are bonds have maturity less than a year, T-notes have maturity more than a year up to
10 years, and T-bonds have maturity more than 10 years. Bonds issued by other entities or governments in
a foreign currency is usually called corporate bond.
The zero-coupon bond price Bt pT q can be used to discount a payment or cashflow at time T without
appealing to a specific interest rate model. For example, a cashflow of $10 at time T “ 1 is worth $10B0 p1q
10
now. While, in the short rate model we write 10e´r , or in the one period interest rate model we write 1`r
.
Similar to yield curve, the forward rate Ft pT q of a zero-coupon bond in defined by
Bt pT q “ e´
şT
t
Ft puqdu
or Ft pT q :“ ´
B ln Bt pT q
.
BT
The interpretation of forward rate is to charge different interest rates Ft ptq, Ft pt ` δq, ..., Ft pT ´ δq on
different periods of time rt, t ` δs, rt ` δ, t ` 2δs, ..., rT ´ δ, T s, respectively, while all the rates are agreed
upon at time t. The rates are all determined at the beginning of the lending time by the forward contacts in
money market.
Unlike the forward rate and short rate rt , yield curve Rt pT q is accessible through market data. For example,
LIBOR5 is the rate at which banks agree to lend at each other and is considered more or less a benchmark for
international trades. Or the US treasury yield curve is considered a risk free rate for inside US transactions.
The quotes of yield curve Rt pT q for LIBOR and US treasury for different maturities are given in Tables 2
and 3, respectively.
Maturity
USD LIBOR - overnight
USD LIBOR - 1 week
USD LIBOR - 1 month
USD LIBOR - 2 months
USD LIBOR - 3 months
USD LIBOR - 6 months
USD LIBOR - 12 months
02-19-2016
0.37090 %
0.39305 %
0.43350 %
0.51720 %
0.61820 %
0.86790 %
1.13975 %
02-18-2016
0.37140 %
0.39200 %
0.43200 %
0.51895 %
0.61820 %
0.87040 %
1.14200 %
02-17-2016
0.37000%
0.39160%
0.43005 %
0.51675 %
0.61940 %
0.86660 %
1.13465 %
02-16-2016
0.37100 %
0.39050 %
0.42950 %
0.51605 %
0.61820 %
0.86585 %
1.13215 %
02-15-2016
0.39340 %
0.42925 %
0.51580 %
0.61820 %
0.86360 %
1.12825 %
Table 2: LIBOR yield curve for US dollar. Source www.global-rates.com.
Date
02/22/16
02/23/16
02/24/16
1 Mo
0.28%
0.28%
0.28%
3 Mo
0.33%
0.32%
0.33%
6 Mo
0.46%
0.47%
0.46%
1 Yr
0.55%
0.55%
0.55%
2 Yr
0.78%
0.76%
0.75%
3 Yr
0.92%
0.90%
0.90%
5 Yr
1.25%
1.23%
1.21%
7 Yr
1.54%
1.51%
1.52%
10 Yr
1.77%
1.74%
1.75%
20 Yr
2.18%
2.16%
2.16%
30 Yr
2.62%
2.60%
2.61%
Table 3: Treasury yield curve for US dollar. Source https://www.treasury.gov.
The forward rates can also be obtained from the data on forward rate agreements6 , which is not publicly
5
London Inter Bank Offered Rate
Forward rate agreements are agreements between two parties to set an interest rate for a future loan. It is more or less in the
same spirit of when you lock your mortgage rate with a bank for a fixed period of time.
6
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1 FINANCIAL DERIVATIVES
Summary of the lecture
available.
We should clarify that there is a slight difference between the yield curve defined by (1.1) and the yield
curve data in Tables 2 and 3. The recorded data on the yield curve follows the formula
ˆ
R̂t pT q :“
1
Bt pT q
˙
1
T ´t
´ 1,
where the time-to-maturity T ´ t is recorded in year. Therefore,
´
´
¯¯ 1
´
¯
T ´t
R̂t pT q “ exp Rt pT qpT ´ tq
´ 1 “ exp Rt pT q ´ 1 « Rt pT q,
when Rt pT q is small. For example, in Table 3, the yield of zero-coupon bond which expires in one month is
given by .28% “ .0028 and the price of such bond is equal to p1.0028q´12 “ 0.9670. Similarly, the yield of
zero-coupon bond which expires in 30 years is given by 2.62% “ .0028 and the price of such bond is equal
1
to p1.262q´ 30 “ 0.9923.
It also important to know that some interpolation method is used to create yield curve data. This is because
a bond which expires exactly in one month, three months, six months or etc. does not necessarily exists. A
bond which expires in a month will be a three-week maturity bond after a week. Therefore, after calculating
the yield for the available maturities, an interpolation gives us the interpolated yields of the standard maturities in the yield charts. For example in Table 3, Treasury department uses cubic Hermite spline method is
used to generate daily yield curve quotes.
Knowing the bond price Bt pT q and and knowing yield Rt pT q are equivalent. Since the data on the yield is
sparse, they do not give us full knowledge on the short rate rt or forward rate Ft pT q. But, even these sparse
data impose some conditions of the forward rate or equivalently short rate rt . By definition we have:
Rt pT q “
1
T ´t
żT
Ft puqdu.
t
Therefore, one can get the spot interest rate rt from forward rate by
rt “ lim Ft pT q.
T Ót
As an approximate formula, for small δ we have
rt « Ft pt ` δq « Rt pt ` δq.
Normally, we do not have maturities as close to use the above formula. Therefore, the task of finding rt is
a modeling task. Consider two maturities T1 ă T2 and two yield curves rates Rt pT1 q and Rt pT2 q. A dollar
now is worth $epT1 ´tqRt pT1 q at 1 and is worth $epT2 ´tqRt pT2 q at T2 . Therefore, $epT1 ´tqRt pT1 q at time T1 is
worth $epT2 ´tqRt pT2 q at time T2 , i.e.
BT1 pT2 q “
epT1 ´tqRt pT1 q
“ epT1 ´tqRt pT1 q´pT2 ´tqRt pT2 q
epT2 ´tqRt pT2 q
From the definition of forward rate, we have BT1 pT2 q “ e
´
şT2
T1
FT1 puqdu
and
ż T2
FT1 puqdu “ pT2 ´ tqRt pT2 q ´ pT1 ´ tqRt pT1 q.
T1
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1 FINANCIAL DERIVATIVES
Summary of the lecture
In particular if we assume that the interest rate is not changing. i.e.FT1 puq “ FT1 pT2 q for u P rT1 , T2 s, we
deduce that
pT2 ´ tqRt pT2 q ´ pT1 ´ tqRt pT1 q
FT1 pT2 q “
.
T2 ´ T1
1.5
Credit derivatives
Financial instruments are issued by financial companies such as banks. There is always a risk that the issuer
goes bankrupt or at least defaults on some payments and cannot make its obligation. The same situation
holds when a debt such as mortgage is issued. In such cases, the beneficiary of the issued security is
exposed to credit risk. There are two major credit derivatives in the market: credit default swap (CDS) and
collateralized debt obligation (CDO).
A CDO is a quite complicated financial instrument. Here we present a rather simplified structure of CDO.
One leg of CDO is a special purpose entity (SPE) which holds a portfolio of defaultable assets such as
mortgage-backed securities, commercial real estate bonds and corporate loans. These assets serve as collateral. Then, it issues bonds which pays the cashflow of the assets to the investors. The holders of these special
bonds do not uniformly receive the cashflow. There are four type of bonds in three trenches: senior, mezzanine, junior and equity. The cashflow is distributed among investors first to the holders of senior bonds, then
junior bond holders and at the end equity bond holders. In the case of default of some of the collateral assets
in the portfolio, the equity holders are the first to lose income. Therefore, a senior trench bond is the most
expensive and equity trench bond is the cheapest. CDO is traded in specialized debt markets, derivative
markets, or over-the-counter (OTC).
# of defaults
100
senior
75
mezzanine
50
junior
25
equity
0
%100
% of loss
Figure 1.8: Simplification of CDO structure
Example 1.5. Consider a collateral portfolio of 100 different zero coupon bonds with the same maturity. Let
trenchize the CDO as four equally sized trenches shown in Figure 1.8. If none of the bonds in the portfolio
in the collateral defaults, the total of $100 payoff will evenly be distributed among CDO holders. However,
if 10 bonds default, then $90 received cashflow will be split into $75 to be evenly distribution among the
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1 FINANCIAL DERIVATIVES
junior, mazzanine and senior holders and the remaining $15 dollars will be evenly distributed among equity
holders. On the other hand, in the case of 50 defaults, equity and junior holders receive nothing. The senior
trench receives full payment unless more than 75 defaults occur.
We learned from 2007 financial crisis that even the CDO senior trench bond can yield to lower cashflow
caused by systemic default in the portfolio, especially when the structure of the portfolio of defaultable
assets creates a systemic risk; see Figure 1.9. For example, a portfolio made of mortgages and mortgagebased securities are linked through several risk factors. Knowing the correlation of the losses is not sufficient
to determine the credit risk behind complex derivatives such as CDOs. A more detailed structure of the risk
must be known to enable us to understand the loss distribution better.
0.15
Uncorrelated
Correlated
Correlated with systemic risk
0.1
5 · 10−2
Spike

y
Fat
 tail
y
0
0
20
40
60
80 100
Severity of loss
120
140
Figure 1.9: Distribution of loss: Correlation increases the tail chance. Systemic risk adds an spike to the
loss distribution. All distributions have the same mean. The fat-tailed loss distribution and the systemic risk
loss distribution have the same correlation of default.
Exercise 1.4. Consider a portfolio of 100 standard zero-coupon defaultable bonds with the same maturity
and principle of $1.
a) Plot the loss pdf (probability density function), given the defaults of bonds are independent events
with probability .01.
b) Now assume that the defaults are correlated in the following way. Given the number of defaults N ,
the defaulted bonds can be any combination of N out of 100 bonds, and N is distributed as negative
binomial with parameters pr, pq “ p9, .1q conditional on that N ď 100.
c) In the same framework as in part (b) assume that there is a random variable Y takes values 0 and
1 with probabilities .995 and .005, respectively, such that given Y “ 0, the defaults are independent
with probability 1/99, and given Y “ 1, all bonds default at the same time. See Example B.9
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1 FINANCIAL DERIVATIVES
CDS is a swap which protects the holder of a defaultable asset against the default before a certain maturity
time T by recovering a percentage R of the nominal value specified in the contract in case that the default
happens before maturity. In return, the holder makes regular constant premium payments κ until the default
event or the maturity, whichever happens first. The maturity of CDS is often the same as maturity of the
asset if there is any. For example, a CDS on a bond with maturity T also expires at time T . If we denote the
unknown default time by τ the discounted payoff of CDS starting at time t is given by
ÿ
p1 ´ RqP Bt pT ^ τ q1tτ ďT u ´ κ
Bt pTi q
(1.2)
tďTi ăτ ^T
The recovery rate is normally a percentage of the nominal of the defaultable asset which can be evaluated
priori to the time of issue. There is no initial premium for CDS; therefore the regularly paid premium should
be determined in a way to make the expected value of (1.2) zero. The time of default is a random time with
possibly a known distribution. In this case, κ can be determined by the taking the expectation.
«
ff
ÿ
“
‰
p1 ´ RqP E Bt pT ^ τ q1tτ ďT u “ κE
Bt pTi q .
(1.3)
tďTi ăτ ^T
Exercise 1.5. Consider a CDS on a defaultable bond with maturity T “ 1 year and recovery rate R be 90%.
Let the default time τ be Poisson random variable with mean 6 month. Assume the yield of a zero-coupon
risk free bond is constant 1 for all maturities within a year. Find the monthly premiums of the CDS in terms
of the principle of the defaultable bond P “ $1.
12
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2
Summary of the lecture
2 MODEL-FREE EVALUATION
No dominance principle and model-independent arbitrage
In this section, we focus on the logical implications in the market which holds true regardless of the choice
of the model. Therefore, here we do make any assumption of the dynamics of the assets or the interest rate
and the yield curve. Instead, we on;y impose the following basic assumption. We also ignore any friction
in the market such as transaction cost, liquidity restriction, etc. Some of the content of this section can be
found in [27, Section 1.2].
We consider a sample space Ω and let χ be given set of (measurable)7 mappings form Ω to R. The members
of the set χ are called payoff. We assume that there exists a pricing function Π : χ Ñ R; i.e. the price of
the payoff P is given by ΠpP q.
Assumption 2.1 (No dominance). If the payoff X P χ of a financial instrument is nonnegative, then the
price ΠpXq of the financial instrument is nonnegative.
Remark 2.1. No dominance principle implies that two financial instruments with the same payoff has the
same price.
One of the most important consequence of no dominance is the linearity of the pricing function Π, explained
in the following proposition.
Proposition 2.1. Assume that no-dominance holds. Then,
ΠpX1 ` X2 q “ ΠpX1 q ` ΠpX2 q.
Let’s first fix some terminology which will will be used by being in a long position in asset, bond, etc, we
mean that we hold asset, bond, etc. Similarly, by being in a short position in asset, bond, etc, we mean
that we owe asset, bond, etc. Short selling is a common practice in the market which involves in borrowing
an asset, bond, etc and then selling it for cash or keeping it for other reasons. In practice, the borrower is
obliged to pay the short sold security back upon the request of the lender. All the material on this section is
build on the no dominance principle, Assumption 2.1.
Proposition 2.2. The forward price Kt satisfies
Kt “
St
.
Bt pT q
Proof. Consider a portfolio made of a long position in asset St and a short position in Kt zero coupon bonds
of value Bt pT q. The value of this portfolio at time T is ST ´ Kt . This value is the same as the payoff of a
forward contract. Therefore, by dominance principle (Remark 2.1), we obtain that that Ft “ St ´ Kt Bt pT q.
On the other hand, the price of forward is zero and therefore we have the result.
Exercise 2.1. Let Ft pT q be a forward price for the zero coupon bond B0 pT q for delivery time t. For
0 ă t ă T , use no dominance to show that if B0 ptqFt pT q “ B0 pT q.
Exercise 2.2. Consider a zero-coupon bond B0d pT q on the domestic currency and another zero-coupon bond
B0f pT q on a foreign currency. The domestic-to-foreign exchange rate is denoted by E and and the forward
exchange rate8 for time T is denoted by F . Show
B0d pT qE “ B0f pT qF.
7
8
We ignore measurability concerns here. However, we briefly address them them Section B.2.
Forward exchange rate is an exchange rate which is guaranteed for the maturity.
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2 MODEL-FREE EVALUATION
Proposition 2.3. The price of an American option is always greater then or equal to the price of a European
option with the same payoff.
Proof. An American option can always be exercised not necessarily optimally at the maturity an generates
the same payoff as the European.
Proposition 2.4. The price of vanilla options satisfy
CpT, K1 , S, tq ď CpT, K2 , S, tq and P pT, K1 , S, tq ě P pT, K2 , S, tq
where K1 ě K2 .
Proof. Consider a portfolio which consists of a long position in call option with strike price K2 and a short
position in a call option with strike K1 , both maturing at T . Then, the terminal value of the portfolio is
pST ´ K2 q` ´ pST ´ K1 q` which is nonnegative. By no dominance condition, we have CpT, K2 , S, tq ´
CpT, K1 , S, tq ě 0. For put option similar argument works.
Exercise 2.3. Show that American call and put prices are increasing in maturity T .
Exercise 2.4. Let λ P p0, 1q. Then,
CpT, λK1 ` p1 ´ λqK2 , S, tq ď λCpT, K1 , S, tq ` p1 ´ λqCpT, K2 , S, tq.
In other words, the price of call/put option is convex in K.
Show the same claim for the price of put option, American call option, and American put option.
Exercise 2.5. It is well known that a convex function has right and left derivatives at all points. From the
above exercise it follows that BK˘ CpT, K, S, tq exists. Use no dominance to show that
´Bt pT q ď BK˘ CpT, K, S, tq ď 0
Hint: Consider a portfolio made of a long position in a call with strike K2 a long position in K2 ´ K1 bonds
and short position in a call option with strike K1 .
Proposition 2.5 (Put-call parity). The price of a call option and the price of the put option with the same
strike and maturity satisfy
CpT, K, S, tq ` KBt pT q “ S ` P pT, K, S, tq.
Proof. Since pST ´ Kq` ` K “ ST ` pK ´ ST q` , a portfolio consisting of a call option and a K units
of zero coupon bond Bt pT q is worth as a portfolio made of a put option and one unit of underlying asset at
maturity of the options T . Thus, the result follows by the no dominance.
Exercise 2.6. A portfolio of long positions in call options with the same maturity and strikes on different
assets is worth more than a call option on a portfolio of the same assets with the same weight; i.e.
n
ÿ
λi CpT, Ki , S piq , tq ě CpT, K̂, Ŝ, tq,
(2.1)
i“1
ř
ř
where λi ě 0, Ki ě 0, S piq for i “ 1, ..., n are current asset prices, K̂ “ ni“1 λi Ki and Ŝ “ ni“1 λi S piq
is value of a portfolio which has λi units of asset S piq for each i “ 1, ..., n.
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2 MODEL-FREE EVALUATION
Remark 2.2. Exercise 2.6 has an important implication about risk of a portfolio. A portfolio made of
different assets can be insured against the risk of price increase in two ways, by purchasing a call option for
each unit of each asset or by purchasing a call option on the whole portfolio. It follows from (2.1) that the
latter choice is cheaper than the former. An option of a portfolio is called a basket option.
Exercise 2.7 (Arbitrage bounds for the price of a call option). Show that price of call option should satisfy
pS ´ Bt pT qKq` ď CpT, K, S, tq ď S.
The notion of model specific arbitrage will later be explored in Section 3. However, in this section, we
present model-independent arbitrage which is in the same context as no dominance principle.
Definition 2.1. A positive payoff with zero price is called a model-independent arbitrage.
As a result of Exercise 2.7, if the price of the option is lower that pS ´ Bt pT qKq` or upper than the asset
price S, then there is an immediate risk free opportunity of making extra money, i.e. model-independent
arbitrage.
Proposition 2.6. If there is no model-independent arbitrage, no dominance principle holds.
Proof. Assume that no dominance does not hold, i.e. there is a non-negative payoff P with a negative price
p. Then, new payoff P 1 :“ P ´ p is positive and has zero price.
Exercise 2.8. For 0 ă t ă T , show that if B0 ptqBt pT q ą B0 pT q (equiv. B0 ptqBt pT q ă B0 pT q), there is a
model-independent arbitrage.
Exercise 2.9. Consider a zero-coupon bond B0d pT q on the domestic currency and another zero-coupon bond
B0f pT q on a foreign currency. The domestic-to-foreign exchange rate is denoted by F0 and FT at time t “ 0
and t “ T , respectively. Show that if
B0d pT qF0 ą B0f pT qFT ,
then there exists a model-independent arbitrage.
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3 ARROW-DEBREU MODEL
Arrow-Debreu model
The content of this section adopted from [2]. In Arrow-Debreu theory, an asset has a given price and a set of
possible values. There are N assets with prices arranged in a column vector p “ pp1 , ..., pN qT9 . For each
i “ 1, ..., N , the possible future values (cashflow) of asset i is given by tDi,j : j “ 1, ..., M u. Di,j is the
jth state of future value of asset i and M is a universal number for all assets.
Then, one can encode D into a N -by-M matrix
»
fi
D1,1 ¨ ¨ ¨ D1,M
—
.. ffi .
..
D :“ rDi,j si“1,...,N,j“1,...,M “ – ...
.
. fl
DN,1 ¨ ¨ ¨ DN,M
Each row Di,¨ of the matrix represents an asset in the Arrow-Debreu market, while each column D¨,j represents a future state of the market.
Di,1
Di,2
..
.
pi
Di,M ´1
Di,M
Figure 3.1: Arrow-Debreu model
Example 3.1 (Game of chance). Let N “ 1, M “ 2 and D1,1 “ ´D1,2 “ 1. In other words, there is a fee
p1 to enter a game of chance in which the player is either gains or loses a dollar based on the outcome of
flipping a coin. Notice that for now we do not specify the head-tail probability for the coin. This probability
determines whether the price of the game p1 is a fair price or not.
D1,1 “ 1
p1
D1,2 “ ´1
Figure 3.2: Game of chance described in Arrow-Debreu model
A portfolio is a row vector θ “ pθ1 , ¨ ¨ ¨ , θN q where θi P R which represents the number of units of asset i
in the portfolio. The total price of the portfolio is then given by
θ¨p“
N
ÿ
i“1
9
AT is the transpose of matrix A.
16
θi pi .
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3 ARROW-DEBREU MODEL
Here notation “¨” is the dot product in Euclidean space. Notice that if θi ą 0 the position of the portfolio in
asset i is called long an otherwise if θi ă 0 it is called short.
Arbitrage is a portfolio which costs no money but gives a nonnegative future value and for some states
positive values. More precisely, we have the following definition.
Definition 3.1. θ is called an arbitrage portfolio or arbitrage opportunity if
a) θp “ 0
b) θD¨,j ě 0 for all j “ 1, ..., M
c) θD¨,j ą 0 for at least one j.
Notice that for a given θ “ pθ1 , ¨ ¨ ¨ , θN q, the portfolio represented by θ is itself an asset with value θD¨j
at the state j of the market.
We say that a market model is free of arbitrage or it satisfies no arbitrage condition (NA for short), if
there is no arbitrage opportunity in this model. Besides no arbitrage condition, there is a similar but rather
different notion, i.e. “no free lunch with vanishing risk” (NFLVR), in mathematical finance literature. In
our context, the two notions are the same.
Sometimes arbitrage opportunities are stronger than arbitrage in Definition 3.1 a.k.a. weak arbitrage. For
instance, when we can start with a zero-valued portfolio and end up positive for all states of the market.
Equivalently, strong arbitrage can be defined as follows.
Definition 3.2. θ is called a strong arbitrage portfolio or arbitrage opportunity if
a) θp ă 0
b) θD¨,j ě 0 for all j “ 1, ..., M
Remark 3.1. Notice that if we remove some of the states of the market, then weak arbitrage can disappears.
However, strong arbitrage does not.
Notice that model-independent arbitrage defined in Definition 2.1 is stronger than strong arbitrage.
The following theorem is the most important result in financial mathematics which characterize the notion
of arbitrage in a simple way. basically, it presents a rather simple criterion to see if a market model satisfies
NA.
Theorem 3.1 (Fundamental theorem of asset pricing (FTAP)). There is no arbitrage opportunity iff there
exist a column vector of positive numbers π “ pπ1 , ..., πM qT such that
p “ Dπ.
(3.1)
Proof. Let π be as in the assertion of the theorem and assume that there is an arbitrage portfolio represented
by θ P RN . Then,
M
ÿ
θp “ θDπ “
θD¨,j πj .
j“1
By the definition of arbitrage (Definition 3.1), we have θp “ 0. Also, by the positiveness of entities of π and
Definition 3.1.(b-c), the right hand side above is strictly positive which contradicts with the left hand side
being zero. Thus, there is no arbitrage.
`1
To show the converse, consider the positive closed cone RM
:“ tx P RM `1 : xj ě 0u and defined the
`
linear space L by
L :“ tp´θp, θD¨,1 , ..., θD¨,M q, : θ P RN u Ď RN
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3 ARROW-DEBREU MODEL
`1
Then, NA is equivalent to L X RM
“ t0u. By separating hyperplane theorem (see [8, Section 2.5.1], there
`
exists a hyperplane given by
H “ tx P RM `1 : λ ¨ x ` b “ 0u,
for some λ “ pλ0 , ..., λM q P RM `1 and b P R such that
`1
λ ¨ x ` b ě 0 ě λ ¨ y ` b for all x P RM
and y P L.
`
`1
Since 0 P L X RM
, we obtain b “ 0. Since L is a vector space, both y and ´y belong to L and we obtain
`
`1
λ ¨ y “ 0 for all y P Lzt0u and thus, L Ď H. In addition, for x P RM
zt0u, we have
`
λ¨xąλ¨y
`1
for all y P L; otherwise the NA condition is violated. In particular, for any x P RM
zt0u, we have
`
λ ¨ x ą 0,
which implies that λi ą 0 for all i “ 0, ..., M .
On the other hand, since L Ď H, we have
0 “ λ ¨ p´θp, θD¨,1 , ..., θD¨,M q
“ ´θpλ0 `
M
ÿ
θD¨,j λj .
j“1
Since the above is true for all θ P RN , we conclude that
pλ0 “
M
ÿ
θD¨,j λj .
j“1
If we define πj “
λj
λ0
for all j “ 1, ..., M , π “ pπ1 , ..., πM qT satisfies the desired properties.
Remark 3.2. The weak and strong arbitrage are model specific. If we change the model, the arbitrage
opportunity disappears. However, model-independent arbitrage remain an arbitrage opportunity in any
possible model.
Remark 3.3. Notice that vector π is not necessarily unique and therefore there can be several risk neutral
1
probabilities π̂ such that (3.1) holds. If we intend to add a new asset with price p1 and values D11 , ..., DM
to the current Arrow-Debreu market, the new asset does not create an arbitrage opportunity if and only if at
least one of the existing risk neutral probabilities π̂ satisfies
p1 “
M
ÿ
Dj1 π̂j .
j“1
π
For each j “ 1, ..., M , one can introduce π̂j “ řM j . The vector π̂ “ pπ̂1 , ..., π̂M qT is a probability
k“1 πk
ř
vector, i.e. M
π̂
“
1
and
often
is
referred
to
as
risk
neutral probability or risk adjusted probability.
j
j“1
We next clarify the relation between the vector π of Theorem 3.1 and the interest rate R. Consider an
Arrow-Debreu model with N ` 1 assets, one of which is a risk-free zero-coupon bond. with price given by
p0 . The cashflow of the bond is always $1 at all states of the market. See Figure For no arbitrage condition
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3 ARROW-DEBREU MODEL
D0,1 “ 1
D0,2 “ 1
p0
.
.
.
D0,M ´1 “ 1
D0,M “ 1
Figure 3.3: Risk free asset in Arrow-Debreu model
to hold, (3.1) implies that p0 “
řM
j“1 πj .
Therefore, the risk-free interest rate R satisfies
1
1 ` R “ řM
j“1 πj
.
On the other hand, (3.1) for the price of risky assets, i.e. pj for j “ 1, ..., M , we have
pi “
M
ÿ
Dij πj “
j“1
N
1 ÿ
Dij π̂j .
1 ` R j“1
(3.2)
The RHS10 above has a practical meaning; provided that the no arbitrage condition holds, the discounted
expected future cash flow of an asset with respect to risk neutral evaluation equals the price of the asset.
pi “
1
ÊrDi¨ s.
1`R
ř
Here, ÊrDi¨ s :“ M
j“1 π̂j Dij is the expected cashflow of asset j with respect to the risk neutral probability
1
π̂. Factor 1`R is a discounting which brings the future values to present value.
By rearranging (3.2), one obtains
˙
M ˆ
ÿ
Dij
R“
´ 1 π̂j .
pi
j“1
D
The term piji ´ 1 in RHS is the realized return of asset i if the states j of the market occurs. Therefore, the
interpretation of the above equality is that the expected return of each asset under risk neutral probability π̂
is equal to risk-free interest rate R.
Example 3.2. Consider an Arrow-Debreu model with a risky asset shown below and a risk-free asset with
interest rate R “ .5. To see if there is no arbitrage in this model, we should investigate the solutions of the
1
.3
.5
10
Short for right hand side
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3 ARROW-DEBREU MODEL
system
1 “ π̂1 ` π̂2
1
pπ̂1 ` .5π̂2 q
.3 “
1.5
The first equation accounts for that the π̂ is a probability vector and second equation comes from (3.2).
However, the only solution is π̂ “ p´.1, 1.1qT which is not a probability.
Example 3.3. Consider an Arrow-Debreu model with a risky asset shown below and a risk-free asset with
interest rate R “ .5. To see if there is no arbitrage in this model, we should investigate the solutions of the
2
1
1
.5
system
1 “ π̂1 ` π̂2 ` π̂3
1
1“
p2π̂1 ` π̂2 ` .5π̂3 q
1.5
One of the infinitely many solutions of the above system is π̂ “ p.6, .2, .2qT which implies no arbitrage.
If a second risky asset, shown below, is added to the market, we still do not have arbitrage because vector
π̂ “ p.6, .2, .2qT works for the new market.
1
.4
0
0
Exercise 3.1. Consider an Arrow-Debreu model with two risky assets shown below and a risk-free asset
with interest rate R “ .5. Find all the values for p such that the market is arbitrage free.
2
1
1
p2
1
.5
0
0
Remark 3.4. A zero-coupon bond is a risk-free asset in the currency of reference. For example, a zerocoupon bond which pays $1 is risk-free under the Dollar. However, it is not risk-free if the currency of
reference is Euro. In the latter case, a Euro zero-coupon bond is subject to the risk of exchange rate and is
a risky asset. See Exercise (3.2) below.
Exercise 3.2. Consider an Arrow-Debreu model with two assets and two states, one is a zero-coupon bond
on the domestic currency with interest rate Rd under domestic currency and the other is a zero-coupon bond
on a foreign currency with interest rate Rf under foreign currency.
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3 ARROW-DEBREU MODEL
a) Given the domestic-to-foreign exchange rate at time 0 is F0 11 , and at time 1 takes non-negative values
F1 and F2 , what is the Arrow-Debreu description of foreign bond in the domestic currency?
b) A currency swap is a contract which guarantees a fixed domestic-to-foreign exchange rate, a.k.a.
forward exchange rate for the maturity. The forward exchange rate is agreed upon between two
parties such that the value of the contract is zero. Express the forward exchange rate of a currency
swap maturing at 1 in terms of F0 , Rd , F1 , Rf and F2 .
Remark 3.5. The risk neutral probability π̂ has little to do with the actual probability (frequency) of the
states of a cashflow in the market. The probabilities fj :“ Pp state j ocurresq can be obtained through
statistical techniques from the market data. However, risk neutral probability π̂ depends only on the matrix
D and vector p and not the market data.
One can interpret π̂ as investor’s preference toward the different states of the market. To see this, let rewrite
(3.2) as the following statistical average
pi “
M
”´ π̂ ¯ ı
1 ÿ ´ π̂j ¯
1
¨
fj Dij “
E
Di¨ .
1 ` R j“1 fj
1`R
f¨
(3.3)
Here E is the expectation with respect to the probability obtained from observing the frequency of the states
π̂
in the market. The quotient fjj is the risk preference of the investor toward the state j of the market, which
is referred to as state-price deflator.
One can analyze Arrow-Debreu model by introducing M new elementary securities to the market; for
i “ N ` 1, ..., N ` M , we introduce sN `j which pays $1 when market state j happens. See Figure 3.4.
Then, it is not hard to see that π̂j is the arbitrage free price of asset sN `j . Therefore, the cashflow from asset
i is equivalent to the cashflow of a basket of Di,1 asset sN `1 , ..., Di,M asset sN `M . Recall from (3.2) that
pi “
M
1 ÿ
Dij π̂j .
1 ` R j“1
DN `j,1 “ 0
..
.
DN `j,j “ 1
..
.
DN `j,M “ 0
π̂j
Figure 3.4: Elementary asset sN `j
Exercise 3.3. Consider one period Arrow-Debreu model with N “ 2 and M “ 4 shown in Figure 3.5 and
take R “ 0.
a) Show that any risk neutral probability π̂ “ pπ̂1 , π̂2 , π̂3 , π̂4 q satisfies
$
1
’
&π̂1 ` π̂2 “ 2
π̂3 ` π̂4 “ 12 .
’
%
π̂1 ` π̂3 “ 12
11
1 unit of domestic is worth F0 units of foreign.
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D1
D1
D2
p1 “
3 ARROW-DEBREU MODEL
D1
D1 `D2
2
p2 “
D1 `D2
2
D1
D2
D2
D2
Figure 3.5: Exercise 3.3
b) Recall the notion of independent random variables. Find a risk neutral probability that makes the
random variables of the price of two assets independent.
3.1
One-period binomial model
Let M “ 2 and N “ 2 with one risk free zero-coupon bond and a risky asset with price S0 :“ p1 , and
future cash flow given byD1,1 “ S0 u and D1,2 “ S0 l where S0 and l ă u are all positive real numbers.
By Theorem 3.1, in one-period binomial model no arbitrage condition is equivalent to l ă 1 ` R ă u and
S0 u
S0
S0 l
Figure 3.6: One-period binomial model
1`R´l
pu´lqp1`Rq
u´1´R
u´l .
π “ pπl , πu qT with πu “
π̂u “
1`R´l
u´l
and π̂l “
and πl “
u´1´R
pu´lqp1`Rq .
The risk neutral probability is then given by
Exercise 3.4. Show the above claims.
From FTAP, we know that l ă 1 ` R ă u is equivalent to NA condition. But, it is often insightful to
construct an arbitrage portfolio when l ă 1 ` R ă u is violated. For example, consider the case when
u ď 1 ` R. Then, consider a portfolio with short position in one unit of the asset and long position in S0
units of bonds. To construct this portfolio, no cash is needed and it is worth zero. However, the two possible
future outcomes are either S0 p1 ` Rq ´ S0 u ě 0 or S0 p1 ` Rq ´ S0 l ą 0, which matches with the definition
of (weak) arbitrage in Definition 3.1.
Next, we consider addition of a new asset into the market which pays the cashflow D1 in state 1 and D2 in
state 2. Then, no arbitrage condition implies that the price p of this asset is must be given by
ˆ
˙
1
u´1´R
1`R´l
p“
D1 `
D2 .
1`R
u´l
u´l
For instance, a call option with payoff pS ´ Kq` with lS0 ď K ă uS0 , shown in figure 3.7, has a “noarbitrage price”
1`R´l
C“
puS0 ´ Kq.
pu ´ lqp1 ` Rq
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3 ARROW-DEBREU MODEL
Summary of the lecture
S0 u ´ K
C
0
Figure 3.7: Cashflow of call option in one-period binomial model
0 ´Kq
We shall now see why any price other than puS
p1`Rq π̂u for the call option causes arbitrage in the binomial
market with a zero-coupon bond, a risky asset and a call option on the risky asset. For this reason, we need
to first introduce the notion of replicating portfolio. Consider a portfolio with θ0 investment in cash and θ1
units of risky asset. Then, this portfolio generates the cashflow shown in Figure 3.8. One can choose pθ0 , θ1 q
such that
#
θ1 S0 u ` θ0 p1 ` Rq “ S0 u ´ K
(3.4)
θ1 S0 l ` θ0 p1 ` Rq “ 0
In other words,
θ0 “ ´
S0 lpS0 u ´ Kq
p1 ` Rqpu ´ lq
and
θ1 “
S0 u ´ K
.
S0 pu ´ lq
θ1 S0 u ` θ0 p1 ` Rq “ S0 u ´ K
θ0 ` θ1 S0
θ1 S0 l ` θ0 p1 ` Rq “ 0
Figure 3.8: Replicating portfolio in one-period binomial model
Then, one can see that the value of replicating portfolio is equal to the price of the call option, i,e,
θ0 ` θ1 S0 “ ´
S0 lpS0 u ´ Kq
S0 u ´ K
1
`
“
pS0 ´ Kqπ̂u .
p1 ` Rqpu ´ lq
u´l
1`R
Now, we return on building an arbitrage in the case where the price of call option C is different from
0 ´Kq
We only cover the case C ă puS
p1`Rq π̂u . Consider a portfolio which consists of a long position
in a call option and a short position in a replicating portfolio on the same call option. Shorting a replicating
portfolio is equivalent to ´θ0 position in cash and ´θ1 position in the underlying asset. Then, the value of
0 ´Kq
such portfolio is equal to C ´ puS
p1`Rq π̂u ă 0. This means there is some extra cash in the pocket, while the
payoff of the call option can be used to clear off the shorted replicating portfolio in full. Here the arbitrage
is a strong arbitrage as in Definition 3.2.
puS0 ´Kq
p1`Rq π̂u .
Exercise 3.5. Consider a one-period binomial model with parameters l, u and R and let K P pS0 l, S0 us.
Find a replicating portfolio for a put option with strike K. Verify that the value of the replicating portfolio is
equal to the no-arbitrage price of the put option C “ πl pK ´ S0 lq. Then, find an arbitrage portfolio when
the price of the put option with strike K is less that C.
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3 ARROW-DEBREU MODEL
S1
q has an important
Under the pricing probability π̂, the discounted price of the asset pS0 , Ŝ1 :“ 1`R
property, called martingale property. In one period binomial model, this property is written as
ÊrS̃1 s “
1
ÊrS1 s “ S0 ,
1`R
(3.5)
Here Ê is the expectation under probability π̂ and S1 is a random variable of the price of asset at time t “ 1
which takes values lS0 and uS0 . Under the risk neutral probability S1 takes lS0 and uS0 with probability π̂l
and π̂u , respectively. From introductory probability, one can see that
ˆ
˙
u´1´R
1`R´l
ÊrS1 s “ S0
ˆl`
ˆ u “ S0 p1 ` Rq.
u´l
u´l
Exercise 3.6. Consider an Arrow-Debreu market with M “ 2 which consists of a risk-free bond with
interest rate R “ .01 and a forward contract12 on a non-storable asset13 with forward price K and maturity
of one period. Given the payoff of the forward contract takes values D1,1 “ 4, and D1,2 “ ´2 respective to
the state of the market at maturity, is there any arbitrage?
Now assume that the underlying asset is storable and has price p “ 10. Given there is no arbitrage, find K,
and binomial model parameters u and d for the underlying asset.
3.2
One-period trinomial model
In one-period trinomial model, M “ 3 and N “ 1 and S0 :“ p1 , u :“ D1,1 {S0 , m :“ D1,2 {S0 and
l :“ D1,3 {S0 where S0 , D1,1 , D1,2 and D1,3 are all positive real numbers. By Theorem 3.1, no abitrage
S0 u
S0
S0 m
S0 l
Figure 3.9: One-period trinomial model
condition is equivalent to existence of a positive vector π “ pπl , πm , πu q such that
"
πl l ` πm m ` πu u “
1
1
πl ` πm ` πu “ 1`R
.
(3.6)
It is not hard to see that no arbitrage condition has the same condition as in one-period binomial model, i.e.
l ă 1 ` R ă u.
Exercise 3.7. Derive no arbitrage condition for multinomial model below.
12
13
In the context of this exercise, the forward can be replaced by a futures contract.
E.g. electricity.
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3 ARROW-DEBREU MODEL
S0 u M
S0 uM ´1
..
.
S0 u 2
S0
S0 u 1
Here u1 , ..., uM are positive numbers.
3.3
Replication and complete market
Recall from Section 3.1 that a call option in one period binomial model can be replicated by a portfolio
0 ´Kq
0 ´K
of θ1 “ SuS
in risk-free asset, a.k.a. replicating portfolio.
units of underlying and θ0 “ ´ lpuSu´l
0 pu´lq
A contingent claim on an underlying asset S is a new asset with a payoff equal to a given function of the
price of underlying at a given maturity. Call and put options are examples of contingent claims with payoff
functions pST ´ Kq` and pK ´ ST q` , respectively. For a general contingent claim, a replicating portfolio
is a portfolio with the same future value as the payoff of the contingent claim at all states of the market.
In the binomial model, for any contingent claim with payoff D1 and D2 in states u and l respectively, one
can always solve the following system of equations to obtain a replicating portfolio.
#
θ0 p1 ` Rq ` θ1 S0 u “ D1
,
(3.7)
θ0 p1 ` Rq ` θ1 S0 l “ D2
We leave to find the values of θ0 and θ1 to the reader. Then, the reader can also verify that the no-arbitrage
price of this contingent claim is equal to π̂u D1 ` π̂l D2 to avoid arbitrage.
In trinomial model, a call option with K ą S0 l may or may not be replicable. This is because one has to
solve a system of three equations and two unknowns:
$
’
&θ0 p1 ` Rq ` θ1 S0 u “ pS0 u ´ Kq`
θ0 p1 ` Rq ` θ1 S0 m “ pS0 m ´ Kq` .
’
%
θ0 p1 ` Rq ` θ1 S0 l
“0
Except a narrow range of parameter, this system of equations do not possess a solution. A market model
in which every contingent claim is replicable is called a complete market. A binomial model is a complete
market whereas a trinomial model is not.
For a general Arrow-Debreu model, the condition of completeness is expressed in the following theorem.
Theorem 3.2. Assume that there is no arbitrage, i.e. there exits a risk neutral probability π̂. Then, the
market is complete if and only if there is a unique risk neutral probability, i.e. the system of linear equation
(3.1)
p “ Dπ
has a unique positive solution.
While in the binomial model there is only one risk neutral probability, one can easily verify that in trinomial
model, there are infinitely many risk neutral probabilities provided that l ă 1 ` R ă u. This is because the
system of two equation and three unknowns (3.6) have infinitely many positive solutions when l ă 1 ` R ă
u.
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3 ARROW-DEBREU MODEL
Remark 3.6. Replication is a normal practice for the issuer of the option to hedge the exposed risk of issuing
an option. When the market is not complete, one cannot perfectly replicate all contingent claims. Therefore,
the issuer should take another approach. We often deal with a non-perfect replication, i.e. super-replication
or sub-replication. Super-replication price is defined as the cheapest price of a portfolio which generates a
value at maturity larger than or equal to the payoff of contingent claim for all states of the market. Similarly,
sub-replication price is the most expensive price of a portfolio which generates a value at maturity smaller
than or equal to the the payoff of contingent claim for all states of the market. For instance in trinomial
model, the super-replication price of a call option is defined by
mintθ0 ` θ1 S0 u
over all a, b P R subject to the constraints
$
’
&θ0 p1 ` Rq ` θ1 S0 u
θ0 p1 ` Rq ` θ1 S0 m
’
%
θ0 p1 ` Rq ` θ1 S0 l
ě pS0 u ´ Kq`
ě pS0 m ´ Kq` .
ě0
More precisely, super-replication of a contingent claim is to find the smallest value of a portfolio which has a
payoff equal to or greater than the payoff of the contingent claim at any state of the market. Sub-replication
price can be defined similarly. Sub- or super-replication is a linear programming problem which can be
solved by some standard algorithms.
Exercise 3.8. Show that in any model if there are two distinct risk neutral probabilities, then there are
infinite number of them.
Exercise 3.9. In trinomial model, let S0 “ 1, R “ 0, u “ 2, m “ 1, and l “ 1{2.
a) Find all risk neutral probabilities and the range of prices generated by them for a call option with
strike K “ 1.
b) Find the super-replication price and sub-replication price for this call option and compare them to
the lowest and highest prices in part (a).
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1 DISCRETE-TIME MARKETS
Part II
Modeling financial assets in discrete-time
Section 3 are dealing with a single-period market, i.e. there are only two trading dates, one at the beginning
of the period and one at maturity. In this part, we extend the results of Section 3 to the multi-period market
with a focus of the binomial model. This is also important in our later study of the continuous-time markets,
which can be seen as the limit of discrete-time markets.
Consider a discrete-time market with time horizon T in which trading occurs at only at time t “ 0, ..., T ,
piq
and there are d assets; the price of asset i at time t “ 0, ..., T is denoted by St , for i “ 0, ..., d. The
T
canonical sample space for discrete-time is given by Ω :“ pRd`1
` q equipped with the Borel σ-field. Here
piq
ω P Ω is made of T -tuples of the form pω1 , ...ωT q such that each ωt “ pωt : i “ 0, ..., dq is a column
piq
d-vector. The random variable St which represents the price of asset i at time t is defined by the canonical
mapping
fi
» 0
ω1 ¨ ¨ ¨ ωT0
—
ffi
piq
piq
St : ω “ – ... . . . ... fl ÞÑ ωt .
ω1d ¨ ¨ ¨ ωTd
We assumed that the price of an asset only takes positive values. If it takes negative values, then we can
extend the sample space to Ω :“ pRd`1 qT and all the results of this section holds immediately. If asset 0 is
a risk-free asset, then we can remove its contribution in the sample space and write Ω :“ pRd` qT .
1
Arbitrage and trading in discrete-time markets
In this section, we define the portfolio and trading strategy for discrete-time market models.
p0q
pdq
A self-financing portfolio, or simply portfolio, is represented by a vectors θt “ pθt , ..., θt q with
d
ÿ
piq
θt “ 1, for t “ 0, ..., T ´ 1.
i“0
piq
piq
Here θt is the portion of value of the portfolio invested in asset i at time t. Notice that θt can be positive,
negative or zero, representing long, short, or no-investment positions in the assets, respectively. Given initial
wealth W0 , the initial portfolio investment is shown in the first row in Table 4; for the ease of calculations,
we consider the case d “ 1, i.e. i “ 0 and 1.
We rigorously define the portfolio below.
p0q
pdq
Definition 1.1. A self-financing portfolio strategy is given by a sequence of vector functions θt “ pθt , ..., θt q
p0q
pdq
such that θ0 P Rd is a real vector and for t “ 1, ..., T ´ 1, θt “ pθt , ..., θt q : Ω Ñ Rd is a function which
maps
fi
» 0
S1 ¨ ¨ ¨ St0
ffi
—
(1.1)
St :“ – ... . . . ... fl
S1d ¨ ¨ ¨ Std
into a vector in Rd .
piq
In other words, the function θt depends only on the prices of assets from time t “ 0 until time t ` 1, not
the future prices. This is inline with the intuition that the portfolio strategy depends only on the information
gathered until now.
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Summary of the lecture
Time
# of units of asset (0)
p0q
∆t
t
p0q
θt
:“
t ` 1 after rebalancing
p1q
∆t
p0q
p0q
θt
t ` 1 before rebalancing
# of units of asset (1)
Wt
St
p0q
∆t`1 :“
:“
p1q
Wt
θt
p0q
St
p1q
θt
Value of the portfoio
Wt
p0q
Wt
p0q
p0q
θt`1 Wt`1
p1q
∆t`1 :“
p0q
St`1
p1q
p1q
` ∆ t St
p0q
p1q
p0q
p1q
p1q
Wt`1 “ ∆t St`1 ` ∆t St`1
p1q
St
p0q
Wt “ ∆t St
p1q
St
p1q
θt`1 Wt`1
p0q
p1q
Wt`1 “ ∆t`1 St`1 ` ∆t`1 St`1
p1q
St`1
Table 4: Rebalancing a portfolio strategy from time t to time t ` 1.
We define the arithmetic return, or simply return, of an asset St at time t ` 1 by
St`1 ´ St
.
St
RSt :“
Then, the first two rows of Table 4 suggests that the return of portfolio is equal to the weighted combinations
of return of the assets with the same is given by
p0q
RW
t “
p0q
Wt`1 ´ Wt
p0q St`1 ´ St
“ θt
p0q
Wt
S
p1q
p1q St`1
` θt
p1q
´ St
p0q
“ θt RSt
p1q
St
t
p0q
p1q
p1q
` θt RSt .
(1.2)
In particular, if asset S p0q is a risk-free asset with interest rate R14 and S p1q is a risky asset, we have
p0q
p1q
p1q
S
RW
.
t “ θt R ` θt Rt
On the other hand, one can write a portfolio in terms of the total value of investment is each assets, i.e.
piq
∆t :“
piq
θt Wt
piq
St
piq
. Notice that ∆t is also a function of St in (1.1). Then, (1.2) suggests that the change in the
wealth satisfies
p0q
Wt`1 ´ Wt
l jh n
p0q
p0q
p1q
p1q
p1q
“ ∆t pSt`1 ´ St q ` ∆t pSt`1 ´ St q .
jh
n
jh
n
l
l
change in the value of the portfolio
change due to risky asset (0)
change due to risky asset (1)
If we sum up the above telescopic summation, we obtain
Wt “ W0 `
t´1
ÿ
p0q
p0q
p0q
∆i pSi`1 ´ Si q `
i“0
t´1
ÿ
p1q
p1q
p1q
∆i pSi`1 ´ Si q.
i“0
In the right hand side above, either of the summation corresponds to the cumulative changes in the value of
the portfolio due to investment in one of assets.
p0q
p0q
p0q
p1q
If asset S p0q is a risk-free asset, then ∆i pSi`1 ´ Si q “ RpWi ´ ∆i Si q. In this case, we can simply
drop the superscript of the risky asset and write
Wt`1 ´ Wt
l jh n
“
change in the value of the portfolio
RpWt ´ ∆t St q
jh
n
l
change due to risk-free asset
Wt “ W0 ` R
t´1
ÿ
i“0
14
Interest rate is defined by the return of the risk-free asset.
28
` ∆t pSt`1 ´ St q
l
jh
n
change due to risky asset
pWi ´ ∆i Si q `
t´1
ÿ
i“0
∆i pSi`1 ´ Si q.
(1.3)
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1 DISCRETE-TIME MARKETS
In (1.3), the first summation is the cumulative changes in the value of the portfolio due to investment in
risk-free asset and the second summation is the cumulative investment in the portfolio due to the changes in
risky asset.
An important consequence of this formula is that a self-financing portfolio is only characterized by trading
strategy ∆ “ p∆0 , ..., ∆T ´1 q and the initial wealth W0 , since there is no in-flow and out-flow of cash to the
portfolio. The term
t´1
ÿ
p∆ ¨ Sqt :“
∆i pSi`1 ´ Si q
i“0
is called a discrete stochastic
integral15 .
Exercise 1.1. Let W̃t :“ p1 ` Rq´t Wt and S̃t :“ p1 ` Rq´t St be respectively discounted wealth process
and discounted asset price. Then, show that
W̃t “ W0 `
t´1
ÿ
∆i pS̃i`1 ´ S̃i q,
Ŵ0 “ W0 .
i“0
To understand the meaning of stochastic integral p∆ ¨ Sqt , we provide the following example.
Example 1.1. Recall from Example B.8 that a random walk W is the wealth of a player in a game of chance
in which he wins or loses $1 in each round based on the outcome of flipping a coin.
Consider a modification of the game such that the player can bet on larger amount of money in each round
of the game. For example, he can bet of $3 in the first round and $5 in the second round. If he loses the first
round and wins the second, then his total wealth after two rounds is $2. On the other hand, he wins the first
round and loses the second round he is short $2.
If we denote the bet of the player in round i by ∆i´1 , then the total wealth W ∆ from the betting strategy
∆ “ p∆0 , ∆1 , ...q in t rounds is given by
Wt∆ “
t´1
ÿ
∆i ξi`1
i“0
where tξi ui is the same sequence of i.i.d. random variables as in Example B.8; ξi takes values `1 and ´1
based on the outcome of flipping a coin in the round i. Since ξi “ Wi`1 ´ Wi we have
Wt∆ “
t´1
ÿ
∆i pWi`1 ´ Wi q “ p∆ ¨ Sqt .
i“0
In particular, if ∆i ” 1, W ∆ “ W is merely a random walk.
Example 1.2 (Saint Petersburg paradox and doubling strategy). In Example 1.1, let ∆0 “ 1 and for i ą 0
∆i “ 2i if the player lost all previous rounds i ´ 1, ..., 1. Otherwise, ∆i “ 0 which means the player exits
the game. For example, if the players outcome in the first five round is given by "loss, loss, loss, loss, win",
then his bet is given by "1, 2, 4, 8, 16, 0 , ...". More precisely, he will not bet after he gains his first win, i.e.
∆i “ 0. Then, the wealth of the player after five rounds is given by
W5 “ 1p´1q ` 2p´1q ` 4p´1q ` 8p´1q ` 16p1q “ 1.
15
According to Phillip Protter, this notation is invented by the prominent french probabilist Paul André Méyer to simplify the
task of typing with old-fashion typewriters.
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1 DISCRETE-TIME MARKETS
Summary of the lecture
However, his wealth before the fifth round is always negative.
W1 “ ´1,
W2 “ ´3, W3 “ ´7,
W4 “ ´15,
Exercise 1.2.
a) Show that in Example 1.2 if the player has the opportunity to borrow with no limitation
and continue the game until he wants, he will always end up with terminal wealth equal to $1.
b) Assume that the player has a credit line, denoted by C. He stops playing if either he reaches his credit
limit or he wins for the first time. Find the possible values for the terminal wealth of the player.
c) Find the expected value of the terminal wealth of the player, given the probability of winning is p P
p0, 1q.
Arbitrage strategy
In this section, we present the definition arbitrage opportunity and a FTAP for the multi-step market. In
order to define arbitrage, we first need to determine the set of the relevant events.
Before we define arbitrage, we introduce the implausibility set N Ď B as the set of events that are impossible to happen, whether as a part of our belief or suggested by the experience and data. The implausibility
set N is often subjective and depends on the choice of model(s) and should satisfy
a) H P N .
b) N all members of N are open sets in Ω “ pRd` qT .
c) If A P N then AC P N .
d) If A P N and A Ď B, the B P N .
The σ-field generated by N includes all trivial events, i.e. the events which must have probability zero or
one in our modeling.
Example 1.3 (implausibility set). In the binomial model in Section 3.1, Ω “ R` is the canonical space and
the implausibility set N is given by all open sets A which does not contain any of the points S0 u or S0 l.
Similarly in trinomial model in Section 3.2, Ω “ R` is the canonical space and the implausibility set N is
given by all events A which which does not contain any of the points S0 u, S0 m or S0 l.
The arbitrage in the context of implausibility set N is defined below.
Definition 1.2. A (weak) arbitrage opportunity is a portfolio ∆ such that
a) W0 “ 0,
b) WT ě 0, and
c) tWT ą 0u R N .
A strong arbitrage opportunity is a portfolio ∆ such that
a) W0 ă 0,
b) WT ě 0, and
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1 DISCRETE-TIME MARKETS
For the distinction between two notions of arbitrage see Remark 3.1. Here, we recall that strong arbitrage
is not subject to the choice of implausibility set N and is model independent.
In the above discussion, we did not use any probability concept. However, to extend FTAP to the discretetime multi-period models, we need the probabilistic notion of martingale which is introduced in Section
B.3. Risk-neutral probability can be defined in terms of martingale property for the discounted asset price;
Ŝt :“ p1 ` Rq´t St .
Definition 1.3. We call a probability P̂ a risk-neutral probability if the discounted asset price is a martingale
with respect to P.
ÊrŜt`1 | Ŝt , Ŝt´1 , ..., Ŝ1 s “ ÊrŜt`1 | Ŝt s “ Ŝt .
(1.4)
Here Ê is the expectation with respect to P̂.
For single-period binomial model, the martingale property with respect to risk neutral probability is expressed and verified in Section 3.1; see (3.5).
Exercise 1.3. Show that if the discounted price Ŝt “
St
p1`Rqt
is a martingale with respect to a probability P̂,
then stochastic integral p∆ ¨ Ŝqt and the discounted wealth process W̃t are martingales with respect to P̂.
The following result extends Theorem 3.1 into multi-period. In order to have a fundamental theorem of
asset pricing in a general form, we need to impose the following assumption.
Assumption 1.1. There exist a probability P such that A P N if and only if PpAq “ 0.
The probability P can be regarded as the physical probability in the market. Therefore, one can equivalently
define implausibility set by the set of events which has non-zero-or-one probability under P. However, as
seen in Theorem 1.1, the only relevant information about probability P in no-arbitrage condition is the
implausibility set; the actual value of probability of a relevant event does not matter as long as it is non-zeroor-one. In this case we call that the implausibility set is generated by P.
Definition 1.4. Two probabilities P and P̂ are called equivalent if they generate the same implausibility set.
We denote the equivalency by P ” P̂.
Theorem 1.1 (Fundamental theorem of asset pricing (FTAP)). Let Assumption 1.1 holds. Then, there is no
weak arbitrage opportunity in the discrete-time model if and only if there exist a probability measure P̂ such
that
a) P̂ ” P, and
b) the discounted asset price Ŝt is a P̂-martingale16 , a.k.a. risk-neutral probability.
In addition, if no arbitrage holds, the market is complete if and only if there is a unique risk-neutral
probability.
One way of the proof is easy. Assume that a risk-neutral probability P̂ exists and consider an arbitrage
strategy ∆ with the corresponding discounted wealth process Wt . Then,
Ŵt “ p∆ ¨ Ŝqt . p Recall
W0 “ 0q.
Since Ŝt is a P̂-martingale, then by Exercise 1.1, Ŵt is a martingale and we have
ÊrŴt s “ W0 “ 0.
16
martingale under P̂
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2 BINOMIAL MODEL
This is in contradiction with condition (c) in the definition of arbitrage 1.2. For a complete proof of this
result see [13, Pg. 7, Theorem 1.7]. A very general form of this theorem can be found in the seminal paper
of Delbaen and Schechermayer [9].
Remark 1.1. Assumption 1.1 can be relaxed by assuming that the implausibility set is generate by a convex
collection of probabilities P. More precisely A P N if and only if PpAq “ 0 for all P P P. Then, the
fundamental theorem of asset pricing should be modified: there is no weak arbitrage opportunity in the
discrete-time model if and only if
a) Q :“ tP̂ : Ŝt is a P̂-martingaleu is non-empty,
b) P and Q generate the same implausibility set.
For more on the relaxation of Assumption 1.1, see [6].
2
Binomial model
Let H1 , H2 , ... be a i.i.d. sequence of random variables with values u and l17 . Let the maturity be T “ n
and the time step is discrete and are 0, 1, ..., n. At time 0, the price of asset is S0 . At time t ě 1, the price of
the asset satisfies St “ St´1 Ht . Binomial model is shown in Figure 2.1.
S0 u T
S0 u 3
S0 uT ´1 l
S0 u 2 l
S0 uT ´2 l2
S0 u 2
S0 u
S0
..
.
S0 ul
S0 l
S0 ul2
S0 u2 lT ´2
S0 l 3
S0 ulT ´1
S0 l 2
S0 l T
Figure 2.1: Asset price in binomial model
We label the nodes of the binomial tree by the time and the state of the asset price. For example, at time t
when the asset price is equal to St,j`1 :“ S0 uj lt´j , the node is labeled pt, j ` 1q. The only node at time 0
is labeled 0 for simplicity. See Figure 2.2.
17
The probability of this values are irrelevant at this moment.
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Summary of the lecture
p3, 4q
p2, 3q
p1, 2q
p3, 3q
p2, 2q
p0q
p1, 1q
p3, 2q
p2, 1q
p3, 1q
Figure 2.2: Label of the nodes in three-period binomial model
2.1
No arbitrage
The no arbitrage condition for multi-period binomial model is the same as single period. There is no arbitrage in multi-period binomial model if and only if there is no arbitrage for the single period model with the
same parameters.
Proposition 2.1. There is no arbitrage in multi-period binomial model if and only if l ă 1 ` R ă u. In this
case, the multi-period binomial market is complete and the risk neutral probability is given by assigning the
following distribution to each Hi .
P̂pHi “ uq “
1`R´l
u´1´R
and P̂pHi “ lq “
.
u´l
u´l
Sketch of the proof. If there is an arbitrage portfolio ∆, then there is a time t and state i such that the wealth
process Wt associated with arbitrage portfolio is zero at state i for the last time. State i is a node of the
binomial tree at time t.
At time t, the arbitrage strategy is a single-period arbitrage for one-period binomial tree. There for the
condition l ă 1 ` R ă u must be violated. The inverse trivial. The remaining of the theorem follows from
the single period.
As shown in Figure 2.1, the random variable Sn takes values S0 un´k lk for k “ 0, ..., n. Under the risk
neutral probability,
ˆ ˙
n
n´k k
P̂pSn “ S0 u
l q“
pπ̂u qn´k pπ̂l qk .
k
` ˘
To see this, notice that in the binomial tree in Figure 2.1, there are nk paths from the node S0 to node
S0 un´k lk and the probability of each path is pπ̂u qn´k pπ̂l qk 18 . We say that at time t state i of the market
occurs if St “ S0 ut´i li for i “ 0, ..., t. For simplicity, we denote S0 ut´i li by St piq.
2.2
Basic properties of binomial tree
Binomial tree described above has some properties which are somehow features of many models in financial
mathematics.
Time homogeneity
Since the sequence of random variables tHi uqt “ 18 are i.i.d., then for t ą s, the conditional distribution
of St given Ss “ S is the same as conditional distribution of St´s given S0 “ S. To see this, notice that
18
This is an elementary combinatorics problem.
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2 BINOMIAL MODEL
given Ss “ S
St “ Ss Hs`1 ¨ ¨ ¨ Ht “ SHs`1 ¨ ¨ ¨ Ht
has the same distribution as St´s “ SH1 ¨ ¨ ¨ Ht´s .
Martingale property
Under no arbitrage conditions, the risk neutral probability P̂ exists and one can show
1
ÊrSt`1 | St , ..., S1 s “ St ,
1`R
(2.1)
where Ê is the expectation under risk neutral probability. For the detail of proof of (2.1), see also Example
B.13. Thus, under risk neutral probability, the discounted price of of the asset, Ŝt :“ p1 ` Rq´t St is a
martingale.
In other words, the martingale property asserts that the risk adjusted value of future return of the asset
is the same as a risk free bond! This is rather a surprising property, because it suggests that in pricing
derivatives such as options, the return of the asset does not matter at all. All that matters is the level of
riskiness measured by the risk neutral probability, which only depends on the value of u and l.
Martingale property is in the core of financial models for market risk and is also an indicator of lack of
arbitrage in the market.
Markovian property
The first property is that binomial model is Markovian, i.e. given all states of binomial tree at all times
t “ 0, 1, ..., j, in order to determine the probability of future scenarios of a binomial tree, for example the
possible states at time t “ j ` 1, the only relevant information from the past movements of the price is the
most recent price at time t “ j, i.e.
P̂pSt`1 “ S0 ut´k lk | St , ..., S1 q “ P̂pSt`1 “ S0 ut´k lk | St q.
This is a common property of all Markov chains. Markovian property is often useful is reducing the computational effort of the algorithms in a model. The Markovian property is the result of independence of the
sequence tHi u8
i“1 which implies that the conditional distribution of St`1 given St , ..., S1 is the same as the
conditional distribution of St`1 given St . See Figure2.3.
St`1 pi ` 1q :“ St piqu
St piq
St`1 piq :“ St piql
Figure 2.3: Conditioning of the binomial tree
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Summary of the lecture
Markovian property can equivalently be expressed in terms of expectation; i.e. for any function g : R Ñ R
ÊrgpSt`1 q | St , ..., S1 s “ ÊrgpSt`1 q | St s.
Exercise 2.1. How many paths are there is a binomial model from time t “ 0 to time t “ n? How many
nodes (states) are there?
Why Markovian property is important? The answer lies on the solution to the above exercise. In fact,
the models with Markovian property are computationally less costly . Therefore, one can do the task of
evaluation in a reasonable time.
2.3
Pricing and replicating contingent claims in binomial model
We start by illustrating the pricing method in a two-period binomial model.
Example 2.1. Consider a two-period binomial model with S0 “ 1, u “ 2, l “ 12 and R “ .5 (for simplicity).
We consider a European call option with strike K “ .8; the payoff is gpS2 q “ pS2 ´.8q` . Therefore, π̂u “ 32
p4 ´ Kq` “ 3.2
S2 p3q “ 4
V1,2
S1 p2q “ 2
S0 “ 1
S2 p2q “ 1
p1 ´ Kq` “ .2
V0
S1 p1q “ 1
2
V1,1
S2 p1q “ 1
4
´ Kq` “ 0
p1
4
Figure 2.4: European call option in two-period binomial model. Left: asset price. Right: Option price
and π̂l “ 13 . At node p1, 2q, where t “ 1 and state is 2, the value of the option is equal to the discounted
expectation with respect to risk neutral probability, i.e.
ˇ
1
1
ˇ
ÊrpS2 ´ Kq` | S1 “ S0 us “
pπ̂u pS1 u ´ Kq` ` π̂l pS1 l ´ Kq` q ˇ
(2.2)
V1,2 “
1`R
1`R
S1 “S0 u
ˆ
˙
1
2
1
4.4
“
p3.2q ` p.2q “
« 1.4666.
1.5 3
3
3
Notice that in (2.2), the right hand side is a function of S1 , in which we plugged S1 p2q “ 2 for S1 to obtain
the price. Similarly at node p1, 1q, where t “ 1 and state is 1, we have
ˇ
1
1
ˇ
ÊrpS2 ´ Kq` | S1 “ S0 ls “
pπ̂u pS1 u ´ Kq` ` π̂l pS1 l ´ Kq` q ˇ
1`R
1`R
S1 “S0 l
ˆ
˙
1
2
1
.8
“
p.2q ` p0q “
« 0.8888.
1.5 3
3
9
V1,1 “
To evaluate the option at node p0, 1q, where t “ 0, we take discounted risk neutral expectation of the values
of the option in different states of time t “ 1, i.e.
ˆ
˙
1
2 2 4.4
1 .8
V0 “
pπ̂u V1,2 ` π̂l V1,1 q “
p q ` p q « .6716.
1`R
3 3 3
3 9
To replicate, we need to solve the same system of equations as in 3.7 at each node of the binomial tree in a
35
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Summary of the lecture
backward manner. At node p1, 2q,
#
θ0 p1 ` Rq ` θ1 S1 p2qu
θ0 p1 ` Rq ` θ1 S1 p2ql
“ p4 ´ Kq`
“ p1 ´ Kq`
2 BINOMIAL MODEL
#
or
1.5θ0 ` 4θ1
1.5θ0 ` θ1
“ 3.2
.
“ .2
Thus, θ1 “ 1 and θ0 “ ´ 1.6
3 . In other words, to replicate the contingent claim at node p1, 2q we need to
keep one unit of risky asset at borrow 1.6
3 units of risk free bond. This leads to the price θ0 ` θ1 S1 p2q “
4.4
2 ´ 1.6
“
the
same
price
as
we
found
with risk neutral probability.
3
3
The same method should be used in other node, p1, 1q, to obtain the system of equation
#
θ0 p1 ` Rq ` θ1
“ .2
.
1
θ0 p1 ` Rq ` 4 θ1 “ 0
` ˘
Thus, θ1 “ .83 and θ0 “ ´ .49 . The price θ0 θ1 S1 p1q “ 12 .83 ´ .49 “ .89 is again the same as the risk neutral
price.
At node 0, the replicating portfolio needs to reach the target prices of the contingent claim at time t “ 1,
i.e.
#
#
θ0 p1 ` Rq ` 2θ1 “ 4.4
θ0 ` θ1 S0 u “ V1,2 “ 4.4
3
3 .
or
θ0 ` θ1 S0 l “ V1,1 “ .89
θ0 p1 ` Rq ` 12 θ1 “ .89
20
By solving the above system we obtain θ1 “ 24.8
27 units of risky asset and θ0 “ ´ 81 units of risk free bond in
the replicating portfolio. This is the structure of the replicating portfolio at the beginning of the replication.
If the price moves up, we have to restructure the portfolio to keep one unit of risky asset and ´ 1.6
3 units of
.8
risk free bond. If it moves down, we need to readjust the position to 3 units of risky asset and ´ .49 units of
risk free bond.
We now move on to the pricing method in a general European contingent claim with payoff gpST q, where
g : R` Ñ R is a function which assigns a value to the payoff based on the price ST of the asset at terminal
time T . Such European contingent claims are also called Markovian claims. By using Markovian property
of binomial model, we shall obtain the following.
i) At state i and time t, the price Vt,i of the European contingent claim and the number of units of assets
∆t,i in the replicating portfolio are both functions of the price of underlying asset St piq and time t
only. In other word, the only relevant information from past is the most recent one represented by
St piq.
ii) The price V0 of the European contingent claim gpST q at time 0 is given by
Vt,i “
1
ÊrgpST q | St piqs.
p1 ` RqT ´t
(2.3)
1
ÊrgpST qs.
p1 ` RqT
(2.4)
In particular,
V0 “
provided that there is no arbitrage.
We comprehensively study above claims in this Section.
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2 BINOMIAL MODEL
Markovian property of the option price and the hedging components
First notice that the price of the contingent claim at maturity time T is equal gpST piqq if the state i occurs.
Then, we follow a backward scheme to find the price of the contingent claim at the remaining nodes of the
tree. Assume that the price of contingent claim Vt`1,j is known at time t ` 1 for all states j “ 0, ..., t ` 1
and is a function V of pt ` 1, St`1 pjqq, i.e. V pt ` 1, St`1 pjqq. Conditional on the state i at time t, the state
of the market at time t ` 1 is either i ` 1 or i; see Figure 2.3. Therefore, we can use the result of the single
period binomial tree to conclude that under no arbitrage condition l ă 1 ` R ă u we have
Vt,i “
1
1
pVt`1,i`1 π̂u ` Vt`1,i π̂l q “
ÊrV pt ` 1, St`1 q | St “ St piqs.
1`R
1`R
(2.5)
Here, we are using the fact that no arbitrage implies the existence of risk neutral probability with conditional
u´1´R
probabilities π̂u “ 1`R´l
u´l and π̂l “ u´l given state i at time t.
The above expression also shows that at any time t and state j, the price Vt,j of the contingent claim is a
function V pt, Sq of St pjq. In addition, this function is given by taking discounted conditional expectation
from the price of contingent claim at time t ` 1, i.e.
V pt, Sq :“
1
ÊrV pt ` 1, St`1 q | St “ Ss.
1`R
(2.6)
As you can see from the above, the price of a European contingent claim at time t, only depends on the
price of the underlying asset St , and not on the price of the underlying asset before time t. This is not unique
to the binomial model. The reason for such simplification is the Markovian property of the binomial model.
Other Markovian models also have the same property.
Remark 2.1. Since binomial model is time-homogeneous, ÊrgpST q | St piq “ Ss in (2.3) is equal to
ÊrgpST ´t q | S0 “ Ss.
This suggest that the the price function V pt, Sq in (2.6) is a function of S and time-to-maturity τ :“ T ´ t.
Time-to-maturity is often use instead of time in financial literature of pricing contingent claims.
To replicate the payoff gpST q at time t ` 1, one needs to solve the following system of equations
#
θ0 p1 ` Rq ` θ1 St`1 pi ` 1q “ Vt`1,i`1
.
θ0 p1 ` Rq ` θ1 St`1 piq
“ Vt`1,i
The solution is given by
θ1 “
Vt`1,i`1 ´ Vt`1,i
V pt ` 1, St piquq ´ V pt ` 1, St piqlq
“
St`1 pi ` 1q ´ St`1 piq
St piqpu ´ lq
which is the number of units of risky asset in the replicating portfolio and
θ0 “
uVt`1,i ´ lVt`1,i`1
uV pt ` 1, St piqlq ´ lV pt ` 1, St piquq
“
pu ´ lqp1 ` Rq
pu ´ lqp1 ` Rq
which is the number of units of risk free bond in the replicating portfolio. In other words, the replicating portfolio is a self-financing portfolio given by (1.3) with w0 “ V0 and portfolio strategy given by
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2 BINOMIAL MODEL
Summary of the lecture
∆p0, S0 q, ..., ∆pT ´ 1, ST ´1 q
∆pt, Sq :“
V pt ` 1, Suq ´ V pt ` 1, Slq
.
Spu ´ lq
(2.7)
The number of units of risky asset in the replicating portfolio, given by (2.7), is called the Delta of the
contingent claim at time t. Basically, the Delta of a European Markovian contingent claim is a function of
price on underlying asset given by (2.7).
Remark 2.2. As you can see from (2.7), Delta of the contingent claim at time t measures the sensitivity of
the value of the contingent claim with respect to the changes in the underlying price, i.e. changes in the
price of option due to changes in the price of underlying.
By (1.3), the replicating portfolio for the binomial model takes the form
V pt, St q “ V p0, S0 q ` R
t´1
ÿ
pV pi, Si q ´ ∆pi, Si qSi q `
i“0
t´1
ÿ
∆pi, Si qpSi`1 ´ Si q,
i“0
where
ř V pi, Si q is the price of contingent claim at time i when the underlying price is Si . The term
R t´1
i“0 pV pi, Si q ´ ∆i pSi qSi q represents
řt´1 accumulated changes in the risk free bond part of replicating portfolio caused by interest rate, and i“0 ∆i pSi qpSi`1 ´ Si q represents the accumulated changes in the replicating portfolio caused by the changes in the risky asset price. The act of constructing replicating portfolio
for a contingent claim is ofter referred to as Delta hedging.
The above discussion is summarized in the following algorithm.
Backward pricing and replicating European options in binomial model
1:
2:
3:
4:
5:
6:
7:
At time T , the value of the option is gpST pjqq.
for each t “ T ´ 1, ..., 0 do
for each j “ 1, ..., t ` 1 do
1
The value of the option V pt, St pjqq “ 1`R
ÊrV pt ` 1, St`1 q | St “ St pjqs.
The replicating portfolio is made of ∆pt, St pjqq units of risky assets and V pt, St pjqq ´ St pjq∆pt, St pjqq
is risk free bond.
end for
end for
Remark 2.3. Give functions ∆ and V are calculated, one has to plug in time t and asset price St into the
function to find the price and adjust the replicating portfolio of the contingent claim. However, there is no
guarantee that the quoted prices in the market match the prices in the binomial (or any other) model. In
such case, interpolation techniques can be exploited to fine the price and adjust the replicating portfolio.
Exercise 2.2. In the binomial model, show that the Delta of a call option ∆call and the Delta of a put option
∆put with the same maturity and strike satisfy
∆call
´ ∆put
“ 1, for all t “ 0, ..., T ´ 1.
t
t
Is this result model-independent? Hint: put-call parity.
Exercise 2.3. Consider a two-period binomial model for a risky asset with each period equal to a year and
take S0 “ $1, u “ 1.03 and l “ 0.98.
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2 BINOMIAL MODEL
gpST q
K2 ´ K1
K1
K2
ST
Figure 2.5: Payoff of Exercise 2.3
a) If the interest rate for both periods is R “ .01, find the price of the option with the payoff shown in
Figure 2.6 with K1 “ 1.00 and K2 “ 1.05 at all nodes of the tree.
b) Find the replicating portfolio at each node of the tree.
Exercise 2.4. Consider a two period binomial model for a risky asset with each period equal to a year and
take S0 “ $1 u “ 1.05 and l “ 1.00. Each year’s interest rate comes from Exercise 1.3.
a) Is there any arbitrage? Why? Give one arbitrage portfolio.
Now consider a two period binomial model for a risky asset with each period equal to a year and take
S0 “ $1 u “ 1.05 and l “ 0.95.
b) Find the price of the option with the payoff shown in Figure 1.7 with K1 “ 1.00 and K2 “ 1.05 at all
nodes of the tree. Find the interest rates for each period from information in part (a).
c) Find the replicating portfolio and specifically ∆ at all nodes of the tree.
Remark 2.4. 2.6 suggest that the price of a Markovian claim in Binomial model does not depend on the
past movements of the price and it only depends on the current price S. This is not indeed true for nonMarkovian claims. For example a look-back option with payoff pmaxt“0,...,T St ´ Kq` or Asian option
¯
´ ř
T
1
S
´
K
are non-Markovian options with the price which depends to some extent in the past
t“0 t
T
`
history of the price movement rather than only the current price of the underlying. See for instance Exercise
2.5. Evaluation of these type of non-Markovian payoffs, a.k.a. path-dependent payoffs, cannot benefit fully
from the Markovian property of the model.
Exercise 2.5. Consider the setting of Exercise 2.3. How do you price a look-back option with payoff
pmaxt“0,1,2 St ´ 1q` ? Hint: The price is path dependent and each path has a payoff.
Martingale property of discounted option price and hedging strategy components
As shown in Section 2.2, under risk neutral probability, the discounted price of the underlying asset is a
martingale, i.e.
ÊrŜt`1 | Ŝt s “ Ŝt .
In this section we show that the martingale property under risk neutral probability also holds for the price of
a European contingent claim on the asset.
Vt,i
Notice that by (2.5), the discounted price of a contingent claim, V̂t,i “ p1`Rq
t , satisfies
V̂t,i “ ÊrV̂t`1 | St “ St piqs.
39
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Summary of the lecture
Or in other words, the function V̂ pt, Sq “
V pt,Sq
p1`Rqt
2 BINOMIAL MODEL
satisfies
V̂ pt, St q “ ÊrV̂ pt ` 1, St`1 q | St s.
Therefore, the discounted price of the contingent claim is a martingale under risk neutral probability.
As a result of martingale property and the tower property of conditional expectation in Proposition B.2, we
conclude that
T ˆ ˙
ÿ
1
n
V0 “ V̂0 “ ÊrV̂T s “
pπ̂u qn´k pπ̂l qk gpSTk q.
(2.8)
p1 ` RqT k“0 k
In other words, the price of the European Markovian contingent claim is equal to expectation of discounted
payoff under risk neutral probability, and therefore, (2.4) is established.
Martingale property of the price contingent claim does not show its full computational capacity in the
binomial model. However, in the continuous time models including Black-Scholes, this property is a very
useful tool to reduce the pricing and replication to solving a partial differential equation for the function
V pt, Sq which determines the price of the contingent claim. We will discuss this matter in Chapter III.
Example 2.2. Consider a four-period binomial model for a risky asset with each period equal to a year and
take S0 “ $1, u “ 1.06, l “ 0.98 and R “ .02. We shall find the price of the option with the payoff shown
in Figure 2.6.
gpST q
1
0
10
11
S
12 T
Figure 2.6: Payoff of Example 2.2
The value V0 of the option is the discounted expected value price of the payoff gpST q. Since the asset ST
takes values 9.2236816, 9.9766352, 10.7910544, 11.6719568, and 12.6247696, the only non-zero value,
gpST q “ 1, of the payoff
when ST “ 10.7910544. The risk-neutral probability of ST “
`2˘ is 2obtained
3
2
10.7910544 is simply 4 pπ̂u q pπ̂l q “ 8 . Thus,
V0 “
3
« 0.34644203476.
8p1.02q4
Example 2.3 (Call and put option). Consider a call option with strike K and maturity T . Let k0 ě 0 be
such that S0 uk0 ´1 lT ´k0 `1 ă K ď S0 uk0 lT ´k0 . Then by (2.8), we have
V0call “
T ˆ ˙
ÿ
1
n
pπ̂u qn´k pπ̂l qk pSTk ´ Kq.
p1 ` RqT k“k k
0
Similarly, one can use (2.8) to obtain the price of a put option. However, given that we already have the
40
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2 BINOMIAL MODEL
Summary of the lecture
VT,T `1 “ S0 uT ´ K
V1,2
Vt,t`1
..
.
Vt,t
VT,k0 `1 “ S0 uk0 lT ´k0 ´ K
..
.
V0
V1,1
Vt,2
0
Vt,1
..
.
0
Figure 2.7: Payoff of a call option in binomial tree; k0 is such that
S0 uk0 ´1 lT ´k0 `1 ă K ď S0 uk0 lT ´k0 .
price of a call option in the above, one can find the price of a put option by using Proposition 2.5.
K
´ S0
p1 ` RqT
T ˆ ˙
ÿ
1
n
K ´ ÊrST s
“
pπ̂u qn´k pπ̂l qk pSTk ´ Kq `
T
p1 ` Rq k“k k
p1 ` RqT
0
ˆ ˙
kÿ
0 ´1
n
“
pπ̂u qn´k pπ̂l qk pK ´ STk q.
k
k“0
V0put “ V0call `
In the above, we used martingale property of discounted asset price, (1.4), to write S0 “ ÊrS̃T s “
and we expanded the expectation as
ÊrST s
p1`RqT
T ˆ ˙
ÿ
n
K ´ ÊrST s “ ÊrK ´ ST s “
pπ̂u qn´k pπ̂l qk pK ´ STk q.
k
k“0
2.4
Dividend paying stock
Stock usually pays cash dividend to the share holders. Then, it is on the individual share holders to decide
whether to consume the cash divided or invest it back into the market. The dividend policy is decided by
the management of the company but it is influenced by the preference of the share holders. Although there
41
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Summary of the lecture
2 BINOMIAL MODEL
is not a simple formula how to pay the dividends, but the dividend is usually considered as a percentage of
the asset price, which is referred to as dividend yield. If the asset price at time t is St , then after paying
dt P r0, 1q portion of asset price as dividend, the asset price is reduced to p1 ´ dt qSt . Therefore, under the
dividend policy d1 , ..., dn , the asset price dynamics follows
t
ź
St “ S0 H1 ...Ht p1 ´ di q,
i“1
where Hi is a sequence of i.i.d. random variables with distribution
P̂pHi “ uq “
u´1´R
1`R´l
and P̂pHi “ lq “
.
u´l
u´l
If we define H̃i “ Hi p1 ´ di q, the dividend policy makes the binomial model look like
St “ S0 H̃1 ...H̃t .
See Figure 2.8. Then, H̃i is distributed as
P̂pH̃i “ up1 ´ di qq “
u´1´R
1`R´l
and P̂pH̃i “ lp1 ´ di qq “
.
u´l
u´l
Remark 2.5. It is important to notice that when no arbitrage is condition l ă 1 ` R ă u is not changing.
However, the dynamics of the dividend paying underlying asset is different.
A smart way of pricing contingent claims on dividend paying asset is to transform the payoff in the following way.śConsider the payoff of a European contingent claim given by gpST q. Recall than ST “
S0 H1 ...HT Ti“1 p1 ´ di q. Notice that without paying dividend, S̄t :“ S0 H1 ...Ht is an ordinary binomial
` ś
˘
model. Therefore, the payoff gpST q can be written as a new payoff g S̄T Ti“1 p1 ´ di q on the ordinary
binomial asset S̄T . Then, the pricing of a contingent claim with payoff gpST q at time t given St “ S is
given by
T
” `
ı
ź
˘
1
1
ÊrgpS
q
|
S
“
Ss
“
Ê
g
S̄
p1
´
d
q
|
S̄
“
S
t
i
t
T
T
p1 ` RqT ´t
p1 ` RqT ´t
j“t`1
ˆ
˙
T
T
´
¯
ÿ T ´t
ź
1
i T ´t´i
i T ´t´i
“
π̂
π̂
g
Su
l
p1
´
d
q
.
j
u
l
p1 ` RqT ´t i“t
i
j“t`1
V pt, Sq :“
Remark 2.6. Notice that there is a minor difference between
` śa European ˘contingent claim V pS̄T q on the
underlying asset S̄ and the modified contingent claim g S̄T Ti“1 p1 ´ di q defined above. The price of a
contingent claim V`pS̄T ś
q on non-dividend
˘ paying asset S̄T at time t is a function of T ´ t. But, The price of
T
contingent claim g S̄T i“1 p1 ´ di q is not.
Exercise 2.6. Consider a two-period binomial model for a risky asset with each period equal to a year and
take S0 “ $1, u “ 1.15 and l “ 0.95. The interest rate for both periods is R “ .05.
a) If the asset pays 10% divided yield in the first period and 20% in the second period, find the price of
a call option with strike K “ .8.
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3 CALIBRATION
Summary of the lecture
S0 un p1 ´ d1 q...p1 ´ dn q
S0 u3 p1 ´ d1 qp1 ´ d2 qp1 ´ d3 q
S0 un´1 lp1 ´ d1 q...p1 ´ dn q
S0 u2 p1 ´ d1 qp1 ´ d2 q
S0 up1 ´ d1 q
S0
S0 u2 lp1 ´ d1 qp1 ´ d2 qp1 ´ d3 q
S0 un´2 l2 p1 ´ d1 q...p1 ´ dn q
.
.
.
S0 ulp1 ´ d1 qp1 ´ d2 q
S0 lp1 ´ d1 q
S0 ul2 p1 ´ d1 qp1 ´ d2 qp1 ´ d3 q
S0 u2 ln´2 p1 ´ d1 q...p1 ´ dn q
S0 l2 p1 ´ d1 qp1 ´ d2 q
S0 l3 p1 ´ d1 qp1 ´ d2 qp1 ´ d3 q
S0 uln´1 p1 ´ d1 q...p1 ´ dn q
S0 ln p1 ´ d1 q...p1 ´ dn q
Figure 2.8: Dividend paying asset in binomial model
b) Consider a more complicated dividend strategy which pays 10% divided yield only if the price moves
up and no dividend if the price moves down at each period. Find the price of a call option with strike
K “ .8.
Remark 2.7. Assume that the divided strategy is random, i.e. at each node of the binomial
` śtree at time ˘t
and state i, the divided is a random variable dpt, iq. Is the modified contingent claim g S̄T Ti“1 p1 ´ di q
easy to price? The two-period binomial model in Exercise 2.6 part (b), is a special case of this case when
pricing is not a difficult job.
In general, even for a Markovian asset price model and a Markovian contingent claims, the pricing is
similar to pricing a path dependent option if the dividend strategy is path dependent. This feature makes
path dependent pricing a very important topic in financial mathematics.
3
Calibrating the parameters of the model by the market date: Binomial
model
Calibrating a model is the practice of setting the parameter of the model to make it consistent to the data.
In binomial model, the parameters are interest rate R, u and l. Calibrating the risk free interest rate R is a
separate job and usually uses the price quotes of the risk free (sovereign) bonds. Here, we assume that R is
already calibrated and is given by 1 ` R “ 1 ` rδ ` opδq in terms of short rate r, where δ is the duration of
one period in the binomial market.
Assume that the asset price quotes are collected at δ time lapse, i.e. S´mδ , ..., S0 be the past quotes of
the asset price from time ´mδ until now. To calibrate u and l, we need to define some statistic from the
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3 CALIBRATION
Summary of the lecture
financial date. Let Then for each t “ ´mδ, ..., ´δ, the arithmetic return at time t is defined by
Rarth
:“
t
St`δ ´ St
,
St
and the logarithmic return is defined by
Rlog
t
ˆ
:“ ln
St`δ
St
˙
.
arth are very close20 .
If the time step is small so that the price movements are also small19 , then Rlog
t and Rt
However, we will see in Section 3.1 that the small difference between the two returns will show up when
we derive continuous-time models as a limit of discrete-time models. For the moment, we focus on the
arithmetic return and drop the superscript "arth" for simplicity.
Notice that in binomial model the sequence of return tRt ut are making a sequence of i.i.d. random variables. So, to proceed with calibration, we need to assume the following stronger condition.
Assumption 3.1. The return statistic tRt ut is a sequence of i.i.d. random variables with mean and the
variance given by by µδ ` opδq and σ 2 δ ` opδq, respectively.
The dimension-less quantities σ and µ are respectively called the volatility and mean return rate of the
price.
Remark 3.1. The assumption that volatility is a constant is not very realistic. However, this assumption,
which is widely used in practice back in 70s and 80s, makes the problems more tractable. We will try later
to test some approaches which relax this assumption in different directions.
Exercise 3.1 (Project). Go to google finance, yahoo finance or any other database which provides free asset
price quote. Download a spreadsheet of the daily price of a highly liquid asset such as IBM, Apple, Alphabet,
etc. Assuming Assumption 3.1, find the volatility σ and the average return rate µ of the asset. Then, use
these quantities to find the daily, weekly, and yearly standard deviation and mean of the return.
The collected data is coming from the physical probability and not the risk neutral probability. Therefore,
volatility σ and mean return rate µ are parameters of the distribution of return Rt under physical probability, and while calibrating the data with the binomial model, we need to specify the frequencies (physical
probability) p and 1 ´ p of the states u and l of the price; see Remark 3.5. Therefore given that mu and σ
are estimated from the data, in the binomial model we need to find three parameters: u, l, p, where p is the
probability that St`1 “ St u. See Figure 3.1.
p
S0 u
Rt “
S0
1´p
$
’
U :“ u ´ 1
’
’
’
&
with probability p
’
’
’
’
%L :“ l ´ 1
with probability 1 ´ p
S0 l
Figure 3.1: Left: binomial model under the physical probability. Right: The arithmetic return Rt .
19
20
This means in a short time step in markets clock!
lnp1 ` xq « x for small x
44
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3 CALIBRATION
Summary of the lecture
We match the first and second momentum of the binomial model with the mean return rate and volatility:
#
pU ` p1 ´ pqL “ µδ ` opδq
pU 2 ` p1 ´ pqL2 ´ ppU ` p1 ´ pqLq2 “ σ 2 δ ` opδq
Or equivalently
#
pU ` p1 ´ pqL “ µδ ` opδq
pp1 ´ pqpU ´ Lq2 “ σ 2 δ ` opδq
(3.1)
In the above system of two equations, there are three unknowns U , L and p, which gives us one degree of
freedom. We are going to use this degree of freedom by assuming that the variance of return Rt under
risk neutral probability is also σ 2 δ, i.e.
π̂u π̂l pU ´ Lq2 “ σ 2 δ ` opδq,
R´L 21
where π̂u “ 1`R´l
u´l “ U ´L . To simplify further, we also drop the opδq term from the equations. Therefore,
we have a system of three equations and three unknown.
$
’
&pU ` p1 ´ pqL “ µδ
pp1 ´ pqpU ´ Lq2 “ σ 2 δ
’
%
pU ´ RqpR ´ Lq “ σ 2 δ
For the sake of simplicity, we set new variables
U ´R
,
α“ ?
δσ
R´L
β“ ?
,
δσ
and r “
R
,
δ
(3.2)
where r is the annual interest rate (APR) calculated at periods δ. Thus, we have
$
?
’
´ p1 ´ pqβ “ λ δ
&pα
a
pp1 ´ pqpα ` βq “ 1
’
%
αβ “ 1
Here λ :“
(3.3)
µ´r
σ .
Remark 3.2 (Risk premium). The quantity µ´r
σ is referred to as risk premium of the asset and measures the
excess mean return of the asset adjusted with its level of riskiness measured by the volatility.
The solution of (3.3) is given by
c
?
1´p
α“
` λ δ,
p
c
β“
?
p
´ λ δ, and
1´p
p“
1
1 ` x20
(3.4)
where x0 is the unique positive solution of equation (see Figure 3.2)
x´
?
1
“ λ δ.
x
(3.5)
21
The reason behind this choice, will be more clear later when we study continuous time models, specifically Black-Scholes
model.
45
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3 CALIBRATION
Summary of the lecture
x ´ x´1
?
λ δ
x
x0
Figure 3.2: Positive solution of equation x ´
1
x
?
“ λ δ.
Then, u, l are given by
u “ 1 ` δr `
?
δσα, and l “ 1 ` δr ´
?
δσβ.
(3.6)
Remark 3.3. The above calibration is robust in the sense that for any possible value of parameters σ ą 0,
µ and r, one can find proper u and l such that l ă 1 ` R ă u in a unique fashion. Notice that in (2.8),
pricing contingent claims is not affected by p and thus, p is a least important parameter in this context.
3.1
Time-varying return and volatility
Assumption 3.1, tRt ut is a sequence of i.i.d. random variables, is not realistic and must be relaxed. Several
empirical studies show that the volatility is not constant. This removes the "identical distribution" of the
return sequence. The independence condition also does not have an empirical basis. In this section, we keep
the independence assumption, but remove the identically distributed on tRt ut .
Assumption 3.2. The return statistic tRt ut is a sequence of independent random variables with mean
ErRt s “ µt δ and varpRt q “ σt2 δ.
Assumption 3.2 allows for the parameters µ and σ to vary over time. To see more details on the estimation
of time-varying parameters, see [14]. In addition, we can also allow for interest rate to depend of time
Rt “ rt δ. Therefore, the calibration in the previous section, Section 3, should be modified in the way that
α, β, and p are time varying, and satisfy
$
?
’
p
α
´
p1
´
p
qβ
“
λ
δ
t
t
t
t
t
&a
pt p1 ´ pt qpαt ` βt q “ 1
’
%
αt βt “ 1
Here λt :“
µt ´rt
σt .
(3.4) and (3.5) also become time-varying and we have
ut “ 1 ` δrt `
?
δσt αt , and lt “ 1 ` δrt ´
46
?
δσt βt .
(3.7)
(Spring) 2016
1 CONTINUOUS-TIME MARKETS
Summary of the lecture
Part III
Modeling financial assets in continuous-time
Louis Bachelier in 1900, [3], introduced the first asset price model and pricing method for the derivatives.
Although this model is now considered impractical, the educational implications of this model is still important. Bachelier modeled the discounted asset price by a Brownian motion. As seen in Section B.7, Brownian
motion is weak limit of normalized random walk. At the time of Bachelier, the Brownian motion was not
even rigorously defined. However, many properties of Brownian motion was well understood. Bachelier
contribution in the theory of probability and stochastic processes is to use heat equation in derivative pricing. His contribution was undermined for about 30 years until Nikolai Kolmogorov who partial differential
equations to describe a class of stochastic processes, called diffusion processes. Nikolai Kolmogorov is the
first mathematician who brought probability into rigor by establishing its mathematical foundation. Other
mathematicians built up on Kolmogorov’s work; Norbert Wiener was the first to define Brownian motion,
and Kyiosi (read Kiyoshi) Itô introduced a simple representation of diffusion processes in terms of Brownian
motion.
In finance, the Bachelier model has been out of record for more than 50 years. Paul Samuelson suggested
Geometric Brownian motion (GBM), which never takes non-positive values, to model the asset price. GBM
is also known as Black-Scholes model, names after Fischer Black and Myron Scholes. Black and Scholes
in [5] and Robert Merton in [20] independently developed a pricing method for derivatives based on GBM.
For a through review of Bachelier’s effort and contribution see [25]. For a brief history of asset price models
see [23]. For the biography of Bachelier see [26].
1
Trading and arbitrage in continuous-time markets
Recall form Section II.1 that in a discrete-time market where trading occurs at times t0 “ 0 ă t1 ă ... ă
tN “ T the value of the portfolio generated by the strategy ∆0 , ∆t1 , ..., ∆tN ´1 is given by
Wtn “ W0 `
n´1
ÿ
Ri pWti ´ ∆ti Sti q `
n´1
ÿ
∆ti pSti`1 ´ Sti q.
i“0
i“0
Here Ri is the interest rate for the period of time rti , ti`1 s which can be taken to be rpti`1 ´ ti q. For
i “ 0, ..., N ´ 1, the strategy ∆ti is a function of the history asset price, i.e. Su for u ď ti . Therefore,
Wtn “ W0 ` r
If we take ti`1 ´ ti :“ δ “
T
N
lim r
δÑ0
n´1
ÿ
n´1
ÿ
i“0
i“0
pWti ´ ∆ti Sti qpti`1 ´ ti q `
∆ti pSti`1 ´ Sti q.
and let δ Ñ 0, we obtain the Reimann integral
Nÿ
´1
żT
i“0
0
pWti ´ ∆ti Sti qpti`1 ´ ti q “ r
pWt ´ ∆t St qdt,
which is the accumulated net change in the portfolio due to investment on the risk-free asset. The limit of
the second term
n´1
ÿ
∆ti pSti`1 ´ Sti q
i“0
47
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1 CONTINUOUS-TIME MARKETS
Summary of the lecture
does not necessarily exist unless we enforce proper assumptions on the asset price S. For instance if the
asset price follows Brownian motion (Bachelier model) or GBM (Black-Scholes model), then the limit exists
and is interpreted as stochastic Itô integral. Given the stochastic integral above is well-defined, the wealth
generated by the trading strategy t∆utě0 follows
żt
Wt “ W0 ` r
żt
pWs ´ ∆s Ss qds `
0
∆s dSs .
(1.1)
0
şt
We choose the same notion p∆ ¨ Sqt :“ 0 ∆s dSs for the stochastic integral in continuous-time. Notice
that in the discrete-time setting, the strategy ∆ at time ti is the function of the past history of asset price
S0 , ..., Sti . When we pass to the limit, the trading strategy at time t, ∆t , depends only on the paths of asset
until time t, i.e. tSu : u ď tu. IN other words, a trading strategy does not use any informations from the
future.
Following the discussion in Section C.1, specifically (C.1), the stochastic integral is defined only in the
almost surely sense and can only be defined on the sample paths of the asset price model S. Therefore, a
probability space pΩ, Pq (or at least the set of sample paths of the asset price) is needed. Then, the notion of
arbitrage is define das follows:
Definition 1.1. A (weak) arbitrage opportunity is a portfolio ∆ such that
a) W0 “ 0,
b) WT ě 0, and
c) WT ą 0 of a set of sample paths with positive P probability.
A strong arbitrage opportunity is a portfolio ∆ such that
a) W0 ă 0,
b) WT ě 0, and
şt
Given the stochastic integral p∆ ¨ Sqt “ 0 ∆s dSs is defined in a probability space pΩ, Pq, the fundamental
theorem of asset pricing (FTAP) for continuous-time is as follows. Two probabilities P̂ and P are called
equivalent if any event with probability zero under one of them has probability zero under the other, i.e.
P̂pAq “ 0 iff PpAq “ 0.
Theorem 1.1 (Fundamental theorem of asset pricing (FTAP)). There is no weak arbitrage opportunity in the
continuous-time model if and only if there exist a probability P̂, a.k.a. risk-neutral probability, equivalent to
P such that S is a P̂-(local) martingale22 .
In addition, the market is complete if and only if there is only there is a unique risk-neutral probability.
Recall that a market model is called complete if any contingent claim is replicable, i.e. for any payoff XT
which is a function over all paths tSu : u ď T u of the asset price until time T , there exists a strategy
∆ :“ t∆t uTt“0 such that the wealth W generated by ∆ in (1.1) satisfies XT “ WT .
Theorem 1.2 (Fundamental theorem of asset pricing (FTAP)). Under the same setting as in Theorem 1.1,
the market is complete if and only if there is only there is a unique risk-neutral probability.
22
local martingale is roughly a martingale without condition (a) in Definition B.3.
48
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2
Summary of the lecture
2 BACHELIER MODEL
Continuous-time market of Bachelier
We start with recalling the properties of a Brownian motion from Section B.7. A Brownian motion is a
stochastic process characterized by the following properties.
1) B has continuous sample paths,
2) B0 “ 0,
3) when s ă t, the increment Bt ´ Bs is a normally distributed random variables with mean 0 and
variance t ´ s and is independent of Bu ; for all u ď s.
The Bachelier model is based on an assumption made by Bachelier himself in his PhD thesis“Théorie de
la spèculation” (Pg 33 [3]):
L’espèrance mathématique de l’acheteur de prime est nulle
(2.1)
which translates into “Mathematical expectation of the buyer of the asset is zero”. In the modern probabilistic language, what Bachelier meant is that the discounted asset price is a martingale under risk neutral
probability, which implies no arbitrage by the FTAP in [9]. As the simplest martingale in continuous-time,
Bachelier model simply takes the discounted asset price under risk-neutral probability a factor of a standard
Brownian motion, i.e.
Ŝt “ e´rt St “ S0 ` σBt , σ ą 0.
(2.2)
Therefore under risk neutral probability, Ŝt has Gaussian distribution with constant mean S0 and variance
σ 2 t, i.e. the pdf of Ŝt is given by
ˆ
˙
1
px ´ S0 q2
exp ´
fŜt pxq “ ?
for x P R and t ą 0.
(2.3)
2σ 2 t
σ 2πt
The price of the underlying asset in Bachelier model is given by St “ ert pS0 ` σBt q. Under risk-neutral
probability, St is also a Gaussian random variable with
ÊrSt s “ S0 ert , and
varpSt q “ σ 2 e2rt t.
Therefore, the risk-neutral expected value of the asset price increases in a similar fashion as a risk-free
asset. The variance of St increases exponentially fast too. It follows form applying Itô formula (C.4) on
f pt, xq “ ert pS0 ` σxq that St satisfies the SDE
dSt “ rSt dt ` σert dBt .
(2.4)
Inherited from Brownian motion, Bachelier model possesses the same properties as binomial model in
Section 2.2; i.e.
1) time-homogeneity,
conditional distribution of Ŝt given Ŝs “ S equals distribution of Ŝt´s
with S0 “ S
2) Markovian,
Êrf pŜt q | Ŝu , u ď ss “ Erf pŜt q | Ŝs s
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2 BACHELIER MODEL
3) martingale,
ÊrŜt | Ŝu , u ď ss “ Ŝs .
The conditioning on tŜu , u ď su (or Ŝs ) in the above conditional expectations can simply be replaced by
tSu , u ď su (or Ss ). Since having the knowledge on the asset price is equivalent of having the knowledge
on the discounted asset price. Therefore, we have
Êr¨ | Ŝu , u ď ss “ Êr¨ | Su , u ď ss
(or Êr¨ | Ŝs s “ Êr¨ | Ss s).
It is important for the reader to know that Bachelier model is not practically interesting in modeling financial markets. However, for educational purposes, it has all the basic components of the more practical model
of Black-Scholes. In addition, pricing a European option in Bachelier model is equivalent to solving a heat
equation, which on one hand, can also be used in Black-Scholes model, and on the other hand, is a special
case of parabolic partial differential equations which appear in more general models. Section ?? discusses
the drawbacks of Black-Scholes model.
2.1
Pricing and replicating contingent claims in Bachelier model
As a result of quote (2.1), Bachelier concluded that the price of a European contingent claim with payoff
gpST q is simply the discounted expectation of the payoff under risk neutral probability, i.e.
V0 “ e´rT ÊrgpST qs.
(2.5)
In addition, given the past history of asset price tSu : u ď tu, the price of the option at time t is given by
Vt “ e´rpT ´tq ÊrgpST q | Su : u ď ts.
(2.6)
For simplicity in the rest of this section, we denote the past history of asset price by Ft :“ tSu : u ď tu,
i.e.
Êr¨ | Su : u ď ts “ Êr¨ | Ft s.
(2.7)
Price of contingent claim at time zero
Since under risk neutral probability ST “ erT ŜT is a Gaussian random variable with mean erT S0 and
variance e2rT σ 2 T , one can explicitly calculate V0 in cases where the following integral can be given in a
closed-form,
ż8
`
˘ px´S0 q2
e´rT
V0 “ ?
g erT x e´ 2σ2 T dx.
(2.8)
σ 2πT ´8
We start off by providing a closed form for the Bachelier price of a call option.
Example 2.1 (Price of call and put in Bachelier model). Let gpST q “ pST ´ Kq` “ erT pŜT ´ K̂q` , where
K̂ “ e´rT K. Since ŜT „ N pS0 , σ 2 T q, the price V0call can be calculated in closed form.
V0call
ż8
px´S0 q2
` rT
˘
e´rT
“ ?
e x ´ K ` e´ 2σ2 T dx
σ 2πT ´8
ż8 ´
¯
px´S0 q2
1
“ ?
x ´ K̂ e´ 2σ2 T dx
`
σ 2πT ´8
ż8´
¯
px´S0 q2
1
“ ?
x ´ K̂ e´ 2σ2 T dx.
σ 2πT K̂
50
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2 BACHELIER MODEL
Summary of the lecture
By the change of variable y “
V0call
x´S
? 0,
σ T
we obtain
ż8
´ ?
¯ y2
1
“?
σ T y ` S0 ´ K̂ e´ 2 dy
?
2π pK̂´S0 q{pσ T q
? ż8
ż
2
y2
S0 ´ K̂ 8
σ T
´
´ y2
2 dy `
?
ye
e
dy
“ ?
2π pK̂´S0 q{pσ?T q
2π pK̂´S0 q{pσ?T q
The second integral above can be calculated in terms of the standard normal cdf Φpxq “ ?12π
i.e.
ż8
´ K̂ ´ S ¯
´ S ´ K̂ ¯
2
1
0
0
´ y2
?
?
?
e
dy
“
1
´
Φ
“
Φ
.
?
σ 2πT pK̂´S0 q{pσ T q
σ T
σ T
Notice that here we used Φpxq “ 1 ´ Φp´xq.
The first integral can be evaluated by the change of variable u “
1
?
2π
ż8
?
pK̂´S0 q{pσ T q
ye´
y2
2
1
dy “ ?
2π
“
ż8
şx
´8 e
´y 2 {2 dy,
y2
2 .
2
´ y2
? ye
|S0 ´K̂|{pσ T q
ˇ
1 ´ y2 ˇˇ8
2
´? e
ˇ
?
2π
|S0 ´K̂|{pσ T q
dy
´ S ´ K̂ ¯
pS0 ´K̂q2
1
0
?
“ ? e´ 2σ2 T “ Φ1
,
2π
σ T
x2
where Φ1 pxq “ ?12π e´ 2 is the pdf of standard normal.
To summarize, we have
? `
˘
V0call “ σ T Φ1 pdq ` dΦpdq ,
where d “
S0?
´K̂
.
σ T
(2.9)
By the put-call parity we have,
? `
˘
V0put “ K̂ ´ S0 ` V0call “ σ T Φ1 pdq ´ dΦp´dq .
Example 2.2. A digital option with is an option with payoff
#
1 ST ě K
gpST q “ 1tST ěKu
.
0 ST ă K
The Bachelier price of the digital option is given by
”
ı
´
¯
V0digit “ e´rT Ê 1tST ěKu “ e´rT P̂ ST ě K .
Notice that ST ě K is equivalent to
Bt
?
t
ě
e´rt K´S
? 0.
σ t
Since
Bt
?
t
„ N p0, 1q, we can write
´
`
˘¯
` ˘
V0digit “ e´rT 1 ´ Φ ´ d “ e´rT Φ d .
Exercise 2.1. Find a closed form solution for the Bachelier price of a European option with payoff in Figure
(2.1)
51
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2 BACHELIER MODEL
Summary of the lecture
gpSq
K
a
15
S
Figure 2.1: All slopes are 0 or 1.
Markovian property of the option price
Recall from (2.6) that the option price Vt at time t is a random variable given by
Vt “ e´rpT ´tq ÊrgpST q | Ft s “ e´rpT ´tq ÊrgpST q | Su : u ď ts.
Since Brownian motion B is a Markovian process, the only relevant information form the past is the most
recent asset price St . Therefore, Vt is given by V pt, St q, where function V pt, xq is given by
V pt, xq “ e´rpT ´tq ÊrgpST q | St “ xs.
(2.10)
The function V pt, xq is called pricing function which provides the price of the contingent claim in terms of
time t and underlying price St . The pricing function on discounted price Ŝt is simply obtained by a change
of variable x̂ “ e´rt x, i.e.
V̂ pt, x̂q :“ e´rpT ´tq ÊrgpST q | Ŝt “ x̂s “ V pt, ert x̂q.
Remark 2.1. For the simplicity of the notation, we use V to denote a function of time t and underlying
asset price St . When we write the pricing function as a function of time and discounted asset price Ŝt , we
denote it by V̂ . Similarly, if we use the time-to-maturity τ instead of time, we modify the notion for pricing
function V and V̂ to U and Û , respectively. Finally, if the variable representing underlying asset price is x,
the discounted underlying asset price is represented by x̂; indeed we have ert x̂ “ x.
Time homogeneity of the option price
By the time homogeneity of the Brownian motion, the pricing function is actually a function of time-tomaturity τ “ T ´ t and the discounted underlying price Ŝt “ x̂, i.e. since Ŝτ given Ŝ0 “ x̂ has the same
distribution as ST given Ŝt “ x̂, then
V̂ pt, x̂q “ e´rτ ÊrgpSτ q | Ŝ0 “ x̂s “: Û pτ, x̂q.
As a function of time-to-maturity τ “ T ´ t and the underlying price St “ x, the option price is given by
U pτ, xq :“ Û pτ, e´rt xq “ Û pτ, e´rT erτ xq “ V pt, xq.
Example 2.3. Given St is known, we can write ŜT “ Ŝt ` σpBT ´ Bt q, or equivalently e´rτ ST “
52
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2 BACHELIER MODEL
St ` ert σpBT ´ Bt q, with τ “ T ´ t. The payoff of the call option can be written as
pST ´ Kq` “ erτ pSt ` ert σpBT ´ Bt q ´ e´rτ Kq` .
Therefore, similar to Example 2.1,
U
call
ż8
` rτ
˘
x2
e´rτ
pτ, St q “ ?
e pSt ` ert σpBT ´ Bt qq ´ K ` e´ 2σ2 τ dx
σ 2πτ ´8
ż8
`
˘
x2
1
?
“
St ` ert x ´ e´rτ K ` e´ 2σ2 τ dx
σ 2πτ ´8
ż8
`
˘
x2
1
St ` ert x ´ e´rτ K e´ 2σ2 τ dx
“ ?
σ 2πτ ´e´rt pSt ´e´rτ Kq
˘
`
˘˘
? ` `
“ ert σ τ Φ1 dpτ, St q ` dpτ, St qΦ dpτ, St q
Closed form for Bachelier price of a call option with strike K and maturity T at time t as a function of
τ “ T ´ t and St is given by
˘
`
˘˘
? ` `
U call pτ, St q “ erpT ´τ q σ τ Φ1 dpτ, St q ` dpτ, St qΦ dpτ, St q ,
where
dpτ, xq :“ e´rpT ´τ q
x ´ e´rτ K
?
.
σ τ
As a function of discounted underlying asset price Ŝt and time-to-maturity τ we can rewrite
˘
`
˘¯
? ´ `
ˆ Ŝt q ` dpτ,
ˆ St qΦ dpτ,
ˆ Ŝt q ,
Û call pτ, Ŝt q “ erpT ´τ q σ τ Φ1 dpτ,
where
´rT
ˆ x̂q :“ x̂ ´ e? K .
dpτ,
σ τ
Exercise 2.2. By mimicking the method in Example 2.3, show that
`
˘
U call pτ, St q “ e´rτ Φ dpτ, St q
is a closed form for Bachelier price Û digit pτ, St q of a digital option with payoff 1tST ěKu .
Exercise 2.3. Find a closed form for Bachelier price Û put pτ, Ŝt q and U put pτ, St q of a put option with strike
K and maturity T .
Exercise 2.4. What is the Bachelier price of at-the-money put option (K “ S0 ) with T “ 10, σ “ .5,
R0 p10q “ .025 (yield), and S0 “ 1? What is the probability that the asset price takes a negative value at
T?
Exercise 2.5. What is the Bachelier price of the payoff in Figure 2.5 with T “ 1, σ “ .1, R0 p1q “ .2 (yield),
and S0 “ 2? What is the probability that the option ends up out of the money?
53
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2 BACHELIER MODEL
gpST q
1
ST
1
2
Figure 2.2: Payoff of Exercise 2.5.
Martingale property of the option price
By the tower property of conditional expectation, for t ą s we have
Êre´rt Vt | Fs s “ e´rt Êre´rpT ´tq ÊrgpST q | Ft s | Fs s
“ e´rT ÊrgpST q | Fs s “ e´rs Êre´rpT ´sq gpST q | Fs s “ e´rs Vs .
Therefore, Bachelier’s quote (2.1) implies that in the discounted price of the option is a martingale. Following the Markovian property of the Bachelier model we have
e´rt ÊrV̂ pt, Ŝt q | Fs s “ e´rs V̂ ps, Ŝs q.
If we assume that the pricing function V̂ pt, x̂q (or equivalently V pt, xq) is continuously differentiable on t
and twice continuously differentiable on x̂, then by Itô formula we obtain
˙
ˆ
´
¯
σ2
´rt
´rt
d e V̂ pt, Ŝt q “ e
Bt V̂ ` Bx̂x̂ V̂ ´ rV̂ pt, Ŝt qdt ` e´rt Bx̂ V̂ pt, Ŝt qdŜt
2
(2.11)
ˆ
˙
σ2
´rt
´rt
“e
Bt V̂ ` Bx̂x̂ V̂ ´ rV̂ pt, Ŝt qdt ` σe Bx̂ V̂ pt, Ŝt qdBt
2
¯
¯
´
´
Notice that in the above, we used Bt e´rt V̂ “ e´rt Bt V̂ ´ rV̂ and dŜt “ σdBt .
Thus, e´rt V̂ pt, Ŝt q is a martingale if and only if
Bt V̂ `
σ2
Bx̂x̂ V̂ ´ rV̂ “ 0.
2
A partial differential equation (a.k.a. PDE) need appropriate boundary conditions to make sense. The
boundary condition here is a terminal condition given by the payoff g of the contingent claim;
`
˘
V̂ pT, x̂q “ g erT x̂ .
Therefore, the problem of finding the pricing function V̂ pt, x̂q reduces to solving the boundary value problem
(BVP) below.
#
2
Bt V̂ ` σ2 Bx̂x̂ V̂ ´ rV̂ “ 0
`
˘
(2.12)
V̂ pT, x̂q “ g erT x̂
The BVP (2.12) is a backward heat equation , i.e. we video the evolution of heat over time and playback it
54
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2 BACHELIER MODEL
in reverse. If we do the change of variable τ “ T ´ t and Û pτ, x̂q “ V̂ pt, x̂q, then U satisfies the regular
heat equation
#
2
Bτ Û “ σ2 Bx̂x̂ Û ´ rÛ
`
˘
(2.13)
Û p0, x̂q “ g erT x̂
Therefore, the price of a contingent claim at any time can be obtained by solving the BVP (2.13).
Example 2.4. By bare hand calculations, we can show that the function
˘
`
˘¯
? ´ `
ˆ x̂q ` dpτ,
ˆ x̂qΦ dpτ,
ˆ x̂q ,
Û pτ, x̂q “ erpT ´τ q σ τ Φ1 dpτ,
where
´rT
ˆ x̂q :“ x̂ ´ e? K .
dpτ,
σ τ
satisfies
Bτ Û “
σ2
Bx̂x̂ Û ´ rÛ .
2
By using chain rule, the pricing function in terms of time and non-discounted price St , V pt, xq “ V̂ pt, e´rt xq,
satisfies
#
2rt 2
Bt V ` rxBx V ` e 2 σ Bxx V ´ rV “ 0
V pT, xq “ gpxq
Exercise 2.6. Show that the pricing function in terms of time-to-maturity τ and non-discounted price St ,
U pτ, xq “ V pt, xq, satisfies
#
2rT ´2rτ 2
Bτ U “ rxBx U ` e e 2 σ Bxx U ´ rU
U p0, xq “ gpxq
Exercise 2.7. Show that the function
˘
`
˘˘
? ` `
U pτ, xq “ erpT ´τ q σ τ Φ1 dpτ, xq ` dpτ, xqΦ dpτ, xq ,
where
dpτ, xq :“ erpT ´τ q
x ´ e´rτ K
?
.
σ τ
satisfied
Bt U “ rxBx U `
e2rT e´2rτ σ 2
Bxx U ´ rU.
2
Exercise 2.8. Show that the pricing function on non-discounted price St ,
V pt, xq “ V̂ pt, e´rt xq,
satisfies
#
Bt V ` rxBx V `
V pT, xq “ gpxq
e2rt σ 2
2 Bxx V
´ rV “ 0
Exercise 2.9. Show that the discounted pricing function upτ, x̂q “ e´rτ Û pτ, x̂q satisfies the standard form
55
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Summary of the lecture
of heat equation below
#
Bτ u “
σ2
2 Bxx u
(2.14)
´
¯
up0, xq “ g̃pxq :“ g erT x
Remark 2.2 (On regularity of the pricing function). To be able to apply Itô formula in (2.11), the pricing
function V pt, xq needs to be continuously differentiable on t and twice continuously differentiable on x.
While the payoff of the option may not be differentiable or value function n continuous, the V pt, xq is regular
for all t ă T and all x.
Replication in Bachelier model: Delta Hedging
By (1.1), the dynamics of a portfolio in Bachelier model is given by
żt
żt
Wt “ W0 ` r pWs ´ ∆s Ss qds ` ∆s dSs .
0
0
Similar to Exercise I.1.1, the discounted wealth from portfolio is given by Ŵt “ e´rt Wt satisfies
żt
Ŵt “ W0 ` ∆s dŜs ,
0
and is a martingale. On the other hand, by applying Itô formula to the discounted option price e´rt V̂ pt, Ŝt q,
we obtain (2.11)
ˆ
˙
żt
σ2
´rs
´rt
Bt V̂ ` Bx̂x̂ V̂ ´ rV̂ ps, Ŝs qds
e V̂ pt, Ŝt q “ V p0, S0 q ` e
2
0
żt
` e´rs Bx̂ V̂ ps, Ŝs qdŜs
0
żt
“ V p0, S0 q ` e´rs Bx̂ V̂ ps, Ŝs qdŜs .
0
The last inequality above is by martingale property of discounted option price.
A
replicating
portfolio is a portfolio such that the terminal wealth WT is equal to the payoff V pT,`ŜT q “
` rT
˘
˘
g e Ŝ . Since both Ŵt and e´rt V̂ pt, Ŝt q are martingales with ŴT “ e´rT V̂ pT, ŜT q “ e´rT g erT Ŝ ,
then we must have Ŵt “ e´rt V̂ pt, Ŝt q for all t P r0, T s. Therefore, V p0, S0 q “ W0 and
∆t “ e´rt Bx̂ V̂ pt, Ŝt q.
Since ∆t represents the number of units of the underlying in the portfolio, then it follows from (2.11) that
and ∆t is a function of t and Ŝt and is given by
∆pt, Ŝt q “ e´rt Bx̂ V̂ pt, Ŝt q “ Bx V pt, St q.
Notice that since V is a function of τ “ T ´ t, so is ∆ and
∆pτ, Ŝt q “ e´rt Bx V pt, Ŝt q “ e´rpT ´τ q Bx U pτ, Ŝt q.
To summarize, the issuer of the option must exactly keep ∆t “ e´rt Bx V pt, Ŝt q number of units of underlying asset at time t in the replicating portfolio. ∆t also accounts for the sensitivity of the option price with
respect to the change in the price of underlying.
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Example 2.5. The replicating portfolio for a call option in Bachelier model is obtained by taking partial
derivative Bx of the function
˘
`
˘˘
? ` `
V call pt, xq “ U call pτ, xq “ erpT ´τ q σ τ Φ1 dpτ, St q ` dpτ, St qΦ dpτ, St q ,
with
dpτ, xq :“ e´rpT ´τ q
Since Bx dpτ, xq “
´τ q
e´rpT
?
,
σ τ
x ´ e´rτ K
?
.
σ τ
we have
∆pτ, xq “ Bx U call pτ, xq
`
˘
`
˘
`
˘¯
? ´
“ e´rpT ´τ q σ τ Bx dpτ, xqΦ2 dpτ, xq ` Bx dpτ, xqΦ dpτ, xq ` dpτ, xqBx dpτ, xqΦ1 dpτ, xq
`
˘
`
˘
`
˘
“ Φ2 dpτ, xq ` Φ dpτ, xq ` dpτ, xqΦ1 dpτ, xq
`
˘
“ Φ dpτ, xq .
(2.15)
Here Φ1 pxq “
2
x
?1 e´ 2
2π
is the pdf of standard normal and we used Φ2 pxq “
2
´x ´ x2
?
e
2π
“ ´xΦ1 pxq.
Example 2.6. To find the Bachelier price of an option with payoff gpxq “ ex , we need to solve equation the
following BVP for the pricing function.
#
2
Bτ U “ σ2 Bxx U ´ rU
`
˘ .
U p0, xq “ exp erT x
¯
`
A possible solution for this problem is U pτ, xq “ eλτ exp erT x . If we check this solution by plugging it in
the equation, we obtain
˙
ˆ
2
2rT σ
` r U pτ, xq “ 0.
λ´e
2
2
Thus, for λ “ e2rT σ2 ´ r, U pτ, xq satisfies the equation and the initial
The
is
´ condition.
¯
´ Delta-hedging
¯
` rT ˘
σ2
´rT
rτ
pλ`rqτ
2rT
rT
obtain by ∆pτ, xq “ e
e Bx U pτ, xq “ e
exp e x “ exp e
2 τ exp e x
?
Exercise 2.10. Find a closed form solution for the Bachelier price of an option
with payoff gpxq “ 2 cosp 2xq´
?
3 sinp´xq. Hint: Search for the solution of the form U pτ, xq “ α1 eλ1 τ cosp 2xq ` α2 eλ2 τ sinp´xq.
Example 2.7. Let S0 “ $10, σ “ .03, r “ 0.03. The Bachelier Delta of the following portfolio of
vanilla options given in the table below is the linear combination of the Daltas, i.e. 3∆call pτ “ .5, K “
10q ´ 3∆put pτ “ 1, K “ 10q ´ ∆call pτ “ 2, K “ 8q.
position
long
short
short
units
3
3
1
type
call
put
call
strike
$8
$10
$8
maturity
.5
1
2
The maturities are given in year. Then, (2.15) for Delta of call option in Example 2.5 should be used to
evaluate ∆call pτ “ .5, K “ 10q, 3∆put pτ “ 1, K “ 10q and ∆call pτ “ 2, K “ 8q.
Exercise 2.11. Let S0 “ 10, σ “ .03, r “ 0.03. Consider a portfolio below.
57
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Summary of the lecture
position
long
long
units
3
4
type
call
put
strike
$10
$5
maturity
.25 yrs
.5 yrs
How many units x of underlying is required to eliminate any sensitivity of the portfolio with respect to the
change in price of underlying?
Example 2.8. Let S0 “ 10, r “ .01, σ “ .02 and T “ 1. Consider the payoff in Figure 2.3
gpST q
1
9
10
11
12
ST
Figure 2.3: Payoff of Exercise 2.8.
a) Find the delta of the payoff gpST q at t “ 0.
b) Find an appropriate portfolio of call and/or put option such that a portfolio made of a short position
in payoff gpST q and these call/put options has a constant ∆ over time.
(a) The payoff gpST q can be written as the following combination of call options
gpST q “ pST ´ 9q` ´ pST ´ 10q` ´ pST ´ 11q` ` pST ´ 12q` .
Therefore,
∆g pt “ 0, x “ 10q “∆call pτ “ 1, K “ 9q ´ ∆call pτ “ 1, K “ 10q ´ ∆call pτ “ 1, K “ 11q
` ∆call pτ “ 1, K “ 12q.
Then, (2.15) for Delta of call option in Example 2.5 should be used to evaluate ∆call pτ “ 1, Kq, for
K “ 9, 10, 11, and 12.
(b) By part (a), if we add put options all with maturity T “ 1; long in puts with strikes K “ 9 and K “ 12
and short in puts with strikes K “ 10 and K “ 11. Then, the total payoff of the portfolio will be
p9 ´ ST q` ´ p10 ´ ST q` ´ p11 ´ ST q` ` p12 ´ ST q` ´ gpST q “ 0.
Thus, the replicating portfolio for the total payoff is made of 2 units short in cash and zero position in
underlying over time.
Exercise 2.12. Let S0 “ 9, r “ .01, σ “ .05 and T “ 1. Consider the payoff gpST q shown in Figure ??.
a) Find the delta of the payoff gpST q at time t “ 0.
b) Find an appropriate portfolio of call and/or put option such that a portfolio made of a short position
in payoff gpST q and these call/put options has a constant ∆ over time.
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Summary of the lecture
gpST q
1
8
10
9
ST
gpST q
K2 ´ K1
ST
K1
K2
Figure 2.4: Payoff of Exercise 2.13.
Exercise 2.13. Consider the payoff gpST q shown in Figure 2.4.
Part a) Write this payoff as a linear combination of the payoffs of some call options with different strike and
same maturity T .
Part b) Take T “ 10, σ “ .05, R0 p10q “ .01 (yield), and S0 “ 1. In addition, assume that K1 “ .80, but K2
is unknown. However, the Bachelier Delta of the contingent claim at time 0 is equal to .3243. Find
K2 .
Part c) Find the Bachelier price of the contingent claim at time 0.
2.2
Numerical methods for option pricing in Bachelier model
The BVP for heat equation in (2.12), (2.13) and (2.14) generates closed form solutions in some specific cases,
such as a linear combination of call or put options, exponential payoff in Example 2.6, or sin-cos payoff in
Exercise 2.10. In general, closed-form solution can be obtained if the integral in (2.8) can precisely be
evaluated or equivalently the BVP for heat equation has a closed form solution. The class of payoffs with
closed-form solution is narrow, and therefore, one needs to study numerical method for solving heat equation
in Bachelier model. Although Bachelier model is very far from being practically valuable, the numerical
methods presented in this section are not only applicable to Black-Scholes model directly, but also provide
a solid framework for evaluations of more complicated models. Therefore, the reader should rather interpret
the title of this section as “Numerical methods for heat equation”.
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2 BACHELIER MODEL
Finite-difference scheme for heat equation
In this section, we introduce finite-difference method for the classical heat equation (2.14), i.e.
#
2
Bτ u “ σ2 Bxx u
´
¯
up0, xq “ g̃pxq :“ g erT x
For educational purposes, despite the availability of analytical formulas, we restrict the discussion to the
call and put options only. Other type of payoffs has to be treated by a similar but yet different analysis.
We denote the price of call (put) option with strike K as a function of time-to-maturity τ and the current
discounted price of underlying x by upτ, xq “ Cpτ,
` xq (upτ,˘xq “ P pτ, xq),` which is the
˘ solution of the heat
equation (2.13) with initial condition up0, xq “ erT x ´ K ` (up0, xq “ K ´ erT x ` ). One last piece of
setting, we set S0 “ 0 by considering the change of variable Xt “ Ŝt ´ S0 , i.e. the shifted price equal to
the difference between the discounted price Ŝt and initial price S0 .
As the actual domain of heat equation is infinite, in order to apply finite-difference scheme, first we need
to choose a finite computational domain, i.e. pτ, xq P r0, T s ˆ r´xmax , xmax s for a suitable choice of
xmax ą 0. Now since the computational domain is bounded, it induces more boundary conditions to the
problem at the boundaries x “ xmax and x “ ´xmax . Recall to solve a BVP analytically or numerically,
the boundary condition is necessary at all the boundaries. We should find appropriate boundary conditions
at both points xmax and ´xmax , which usually relies on the terminal payoff of the option. This type of
boundary conditions which are induced by the computational domain and does not exist in the original
problem are called artificial boundary conditions, or shortly ABC.
` rT
To learn
how
to
set
the
ABC,
let’s
study
the
case
for
a
call
option
with
payoff
gpxq
“
e x ´ pK `
˘
rT
rT
rT
e S0 q `, i.e. up0, xq “ pe x ´ pK ` e S0 qq` . The idea is simply as follows. If x is a very small
negative number, then up0, xq “ 0. If the current discounted price of the underlying is a sufficiently small
negative number, the chance that the price at maturity enters the in-the-money region rpK ` erT S0 q, 8q is
significantly small. For instance, since the shifted price Xt “ σBt is a Gaussian random variable, for A ą 0
the probability that Xt ě A (or equivalently Xt ď ´A) is given by
ż8
y2
A2
1
1
?
e´ 2τ σ2 dy „ e´ 2τ σ2 , as A Ñ 8,
2
σ 2τ π A
?
which is smaller than .006 for A ą 3σ T . In other words, far out-of-money options should almost have
zero price. On the other hand, when Xt “ x is sufficiently large, the chance that the discounted price of
underlying drops below K ` erT S0 at the maturity T (out-of-money), is significantly small and therefore
perT ŜT ´ Kq` « erT ŜT ´ K. Far in-the-money options should almost have almost the same price as the
price of payoff ST ´ K, i.e. one unit of asset minus K units of cash.
For a more rigorous discussion, we actually need the following estimation:
?
ż8
2
σ τ ´ A22
1
´ y 2
?
ye 2τ σ dy „ ? e 2τ σ , as A Ñ 8.
σ 2τ π A
2π
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2 BACHELIER MODEL
Summary of the lecture
If we set A :“ e´rT K ` S0 ` e´rT xmax , for xmax sufficiently large we have
ż8
py`xmax q2
e´rτ
upτ, ´xmax q “ ?
perT y ´ pK ` erT S0 qq` e´ 2τ σ2 dy
σ 2τ π ´8
ż8
py`xmax q2
e´rτ
?
“
perT y ´ pK ` erT S0 qqe´ 2τ σ2 dy
σ 2τ π e´rT K`S0
ż
y2
erpT ´τ q 8
?
“
py ´ Aqe´ 2τ σ2 dy
σ 2τ π A
?
ż
erpT ´τ q σ τ ´ A22
erpT ´τ q 8 ´ y2 2
?
ď ?
e 2τ σ , as A Ñ 8.
ye 2τ σ dy „
σ 2τ π A
2π
In other words, far out-of-money options should have zero price. This suggests to set upτ, ´xmax q « 0 for
xmax sufficiently large.
On the other hand, if we set B :“ e´rT K ` S0 ´ e´rT xmax
ż8
py´xmax q2
e´rτ
upτ, xmax q “ ?
perT y ´ pK ` erT S0 qq` e´ 2τ σ2 dy
σ 2τ π ´8
ż
y2
erpT ´τ q 8
py ´ Bqe´ 2τ σ2 dy.
“ ?
σ 2τ π B
Notice that since
1
?
σ 2τ π
and
ż8
ye
B
´
y2
2τ σ 2
1
dy “ ?
σ 2τ π
1
?
σ 2τ π
ż8
żB
ye
´
y2
2τ σ 2
´8
?
σ τ ´ B22
dy „ ? e 2τ σ ,
2π
y2
e´ 2τ σ2 dy „ 1
B
as B Ñ ´8, we have
upτ, xmax q „ ´erpT ´τ q B “ e´rτ pxmax ´ pK ` erT S0 qq.
Following this observation, we choose ABC for (2.16) for the call option is given by
upτ, xmax q “ e´rτ perT xmax ´ pK ` erT S0 qq and upτ, ´xmax q “ 0.
For put option, put-call parity Proposition 2.5 implies that the ABC is given by upτ, xmax q “ 0 and
upτ, ´xmax q “ e´rτ ppK ` erT S0 q ´ erT xmax q.
To summarize, we have to solve the following BVP to numerically price a call option.
$
2
’
Bτ upτ, xq
“ σ2 Bxx upτ, xq for x P p´xamx , xmax q, t ą 0
’
’
’
&up0, xq
“ perT x ´ pK ` erT S0 qq` for x P p´xamx , xmax q
.
(2.16)
’
upτ, xmax q
“ e´rτ perT xmax ´ pK ` erT S0 qq for t ą 0
’
’
’
%upτ, ´x q “ 0 for τ ą 0
max
Next step is to discretize the BVP (??) in time and space. For time discretization, we choose N as the
T
number of time intervals and introduce the time step h :“ N
and discrete times τi for i “ 0, ..., N ´
1, N . Then, we choose a computational domain by choosing , and discretize the computational domain
r´xmax , xmax s by k :“ xmax
M and obtain discrete points xj for j “ ´M, ..., M . The discretization leads to
61
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Summary of the lecture
a grid including points pti , xj q for i “ 0, ..., N and j “ ´M, ..., M , shown in Figure 2.5.
x
xmax
T
τ
k
´xmax
h
Figure 2.5: Finite difference grid for heat equation. In the explicit scheme the CFL condition should be
satisfied, i.e. kh2 ď σ1 . Artificial boundary conditions are necessary on both xmax and ´xmax .
Next, we need to introduce derivative approximation. There are two ways to do this: explicit and implicit.
In the both methods, the first derivative of a function upτi , xj q with respect to time τ at any discrete time τi ,
i “ 1, ..., N , and any discrete point xj , j “ ´M ` 1, ..., M ´ 1, is approximated by
Bt upτi , xj q «
upτi`1 , xj q ´ upτi , xj q
h
Then, the second derivative with respect to x can be approximated by
Bxx upτi , xj q «
upτi , xj`1 q ` upτi , xj´1 q ´ 2upτi , xj q
.
k2
Now, we have all the ingredients to present the explicit scheme for heat equation. The scheme is simply
obtained from the heat equation (3.24) by simply plugging the above approximations for derivatives, i.e.
upτi`1 , xj q ´ upτi , xj q
σ 2 upτi , xj`1 q ` upτi , xj´1 q ´ 2upτi , xj q
“
¨
.
h
2
k2
We can simplify the scheme by writing
ˆ
˙
hσ 2
hσ 2
upτi`1 , xj q “ 1 ´ 2 upτi , xj q ` 2 pupτi , xj`1 q ` upτi , xj´1 qq .
k
2k
62
(2.17)
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2 BACHELIER MODEL
Summary of the lecture
x
xj`1
xj
xj´1
τi
τi`1
t
Figure 2.6: Possible active points in the finite-difference scheme to evaluate upτi`1 , xj q, marked with a
square. The function u is unknown at dark nodes and known at light nodes. All six nodes are active for
implicit scheme with θ ‰ 1. For explicit scheme, θ “ 1, only filled-in nodes are active.
In order to use explicit finite-difference scheme in (2.17), we need to have the CFL23 condition
h
1
ď 2.
k2
σ
Otherwise, the scheme does not converges. The right hand side of CFL condition is always 12 times the
inverse of the coefficient of second derivative in the equation. For implicit schemes, this condition can be
relaxed.
Notice that in problem (2.16), at τ0 “ 0, the initial condition is known. Therefore, we set
up0, xj q :“ e´rT gperT xj q for j “ ´M, ..., M.
Then, if upτi , xj q is known for all j “ ´M, ..., M , the explicit scheme (2.17) suggests that upτj`1 , xj q can
be found for all j “ ´M ` 1, ..., M ´ 1. For j “ ´M and M , one can use ABC to assign values to
upτi , x´M q and upτi , xM q.
Implicit scheme is a little more difficult than explicit scheme to implement. But, it has its own advantages,
e.g. the CFL condition is not necessary. To present the implicit method we need to modify the approximation
of the second derivative as follows.
upτi , xj`1 q ` upτi , xj´1 q ´ 2upτi , xj q
k2
upτi`1 , xj`1 q ` upτi`1 , xj´1 q ´ 2upτi`1 , xj q
`θ
.
k2
Bxx upτi , xj q « p1 ´ θq
In the above, θ P r0, 1s is a parameter. If θ “ 0, then the scheme is the same as the explicit scheme. If θ “ 1,
we call it a pure implicit scheme. Then for θ ‰ 0, we can present the implicit scheme as follows.
23
Courant-Friedrichs-Lewy
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ˆ
2 BACHELIER MODEL
Summary of the lecture
hσ 2
1`θ 2
k
˙
hσ 2
upτi`1 , xj q ´ θ 2 pupτi`1 , xj`1 q ` upτi`1 , xj´1 qq “
2k
ˆ
˙
hσ 2
hσ 2
1 ´ p1 ´ θq 2 upτi , xj q ` p1 ´ θq 2 pupτi , xj`1 q ` upτi , xj´1 qq .
k
2k
(2.18)
If upτi , xj q is known for all j “ ´M, ..., M , then the right hand side above is known. Lets denote the right
hand side by
ˆ
˙
hσ 2
hσ 2
Rpτi , xj q :“ 1 ´ p1 ´ θq 2 upτi , xj q ` p1 ´ θq 2 pupτi , xj`1 q ` upτi , xj´1 qq .
k
2k
For j “ M ´ 1, upτi`1 , xj`1 q on the left hand side in known. Thus, we move this term to the other side
˙
ˆ
hσ 2
hσ 2
1 ` θ 2 upτi`1 , xM ´1 q ´ θ 2 upτi`1 , xM ´2 q “
k
2k
hσ 2
Rpτi , xM ´1 q ` θ 2 upτi`1 , xM q.
2k
Similarly for j “ ´M ` 1 we have
˙
ˆ
hσ 2
hσ 2
1 ` θ 2 upτi`1 , x´M `1 q ´ θ 2 upτi`1 , x´M `2 q “
k
2k
hσ 2
Rpτi , x´M `1 q ` θ 2 upτi`1 , x´M q.
2k
To find upτi`1 , xj q, one needs to solve the following tridiagonal equation for upτi`1 , xj q, j “ ´M `
1, ..., M ´ 1.
AUi`1 “ Ri ´ Yi
(2.19)
where A is a 2M ´ 1-by-2M ´ 1 matrix given by
2
1 ` θ hσ
k2
— ´θ hσ2
—
2k2
—
0
—
—
..
A :“ —
.
—
—
—
0
—
–
0
0
»
2
´θ hσ
2k2 2
1 ` θ hσ
k22
´θ hσ
2k2
0
2
´θ hσ
2k2 2
1 ` θ hσ
k2
¨¨¨
¨¨¨
¨¨¨
´θ hσ
2k2
0
0
2
0
0
2
´θ hσ
2k2
..
.
0
0
0
..
.
¨¨¨
¨¨¨
¨¨¨
2
1 ` θ hσ
k2
hσ 2
´θ 2k2
0
2
´θ hσ
2k2 2
1 ` θ hσ
k22
´θ hσ
2k2
0
2
´θ hσ
2k2 2
1 ` θ hσ
k2
Yi is a column 2M ´ 1-vector
Yi :“ θ
hσ 2
pupτi`1 , x´M q, 0, ¨ ¨ ¨ , 0, upτi`1 , xM qqT ,
2k 2
Ri is a column 2M ´ 1-vector
Ri :“ pRpτi , x´M `1 q, ¨ ¨ ¨ , Rpτi , xM ´1 qqT ,
64
fi
ffi
ffi
ffi
ffi
ffi
ffi,
ffi
ffi
ffi
ffi
fl
(Spring) 2016
Summary of the lecture
2 BACHELIER MODEL
and the unknown column 2M ´ 1-vector
Ui`1 :“ pupτi`1 , x´M `1 q, ¨ ¨ ¨ , upτi`1 , xM ´1 qqT .
Notice that at the end points upτi`1 , x´M q and upτi`1 , xM q are given by the ABC:
upτi`1 , xM q “ e´rτ perT xM ´ pK ` erT S0 qq and upt, x´M q “ 0.
The CFL condition for implicit scheme with θ P r0, 1q is given by
1
h
ď
.
2
k
p1 ´ θqσ 2
For the pure implicit scheme (θ “ 1), no condition is necessary for convergence.
Monte Carlo approximation for Bachelier model
The fundamental of the Monte Carlo method is to generate samples based on the underlying probability
distribution; in the Bachelier model Gaussian distribution with mean S0 and variance σ 2 T . The pricing
formula
U pτ, xq “ e´rτ ÊrgperT Ŝτ q | Ŝ0 “ xs.
Given S0 “ x, the random variable Ŝτ is N px, σ 2 τ q. If txj : j “ 1, ..., N u are i.i.d. samples of N p0, 1q,
then the expectation ÊrgperT Ŝτ q | Ŝ0 “ xs can be approximated by
N
?
1 ÿ
gperT px ` σ τ xi qq.
N j“1
Hence, the price of the option U pτ, t, xq can be approximated by
U approx pτ, xq “
N
?
e´rpT ´tq ÿ
gperT px ` σ τ xi qq.
N
j“1
(2.20)
The larger the number of samples N is, the more accurate the approximation U approx pτ, xq is obtained. Plain
Monte Carlo method is not as efficient as the finite-difference, at least where there is only one single risky
asset. However, some methods such as variance reduction can be used to increase its performance.
Quadrature approximation for Bachelier model
Quadrature methods are based on (non-random) approximations of the integral. In the Bachelier model the
price of the option is given by
ż8
`
˘ py´xq2
e´rτ
U pτ, xq “ ?
g erT y e´ 2σ2 τ dy.
σ 2πτ ´8
As an example of quadrature method, one can first approximate the improper integral above by the proper
integral
ż x`A
`
˘ py´xq2
e´rτ
?
g erT y e´ 2σ2 τ dy
σ 2πτ x´A
65
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Summary of the lecture
2 BACHELIER MODEL
and then use Riemann sums to approximate the price of option by
U
approx
˚
2
N ´1
e´rτ ÿ ` rT ˚ ˘ ´ pyj ´xq
2
2σ
τ
g e yj e
pτ, xq “ ?
pyj`1 ´ yj q,
σ 2πτ j“0
where y0 “ x ´ A ă y1 ă ¨ ¨ ¨ ă yN “ x ` A. If the discrete points yj for j “ 0, ..., N are carefully choses,
the quadrature method outperforms plain Monte Carlo method.
Exercise 2.14 (Project). Consider the initial price S0 , σ, r and payoff assigned to your group.
Group 1
1
2
3
4
5
T
1
10
1
2
1
S0
10
100
2
2
.5
r
.2
.1
.2
.5
.1
σ
1
5
.5
.5
.001
Step 1. Choose a computational domain around the initial price, rS0 ´ xmax , S0 ` xmax s.
Step 2. Set appropriate artificial boundary conditions (ABC) at the boundary points S0 ´xmax and S0 `xmax .
Step 3. Write a program which implement the implicit finite difference code. The time and space discretization
parameters ph, kq must be set to satisfy
h
1
ď
.
2
k
p1 ´ θqσ 2
Step 4. To make sure your code is correct, run the program for call (or put) option and compare it for the
closed form solution (2.9).
Step 5. Run the program for θ “ 0 (explicit), θ “
1
2
(semi-implicit) and θ “ 1 (implicit) and record the result.
Step 6. Simulate a discrete sample path of the price of the underlying asset with the same time discretization
parameter h. By the following algorithm:
Simulating a sample paths of underlying in Bachelier model
Discretize time by t0 “ 0, ti “ ih, T “ hN .
2: for each j “ 1, ..., N do
?
3: Generate a random number wj from standard normal distribution N p0, 1q to represent pBtj ´ Bj q{ h
?
4: ŝj “ ŝj´1 ` σ hwj
5: sj “ ertj ŝj
6: end for
Output: vector ps0 “ S0 , st1 , ..., stN ´1 , sT q is a discretely generated sample paths of Bachelier model.
1:
Step 7. Recall that the hedging is given by the derivative Bx V̄ pt, Ŝt q “ ert Bx V pt, ert St q. Evaluate the hedging strategy discretely at each node of the discretely generated sample paths ps0 “ S0 , st1 , ..., stN ´1 q
in Step 7, i.e.
pBx V p0, ert s0 q, Bx V pt1 , ert1 st1 q, ..., ertN ´1 Bx V ptN ´1 , ertN ´1 stN ´1 qq.
Notice that some interpolation may be needed in this step.
66
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Please submit the following outputs:
Output 1. The Program
Output 2. Comparing with the closed form solution for call at p0, S0 q.
Output 3. Price and hedging strategy at the points of discrete grid.
gpST q
gpST q
Group 1
5
10
ST
15
Group 2
-5
80
40
100
140
ST
-40
gpST q
1
gpST q
1
Group 3
Group 4
1
2
3
ST
1
2
3
ST
gpST q
.25
.75
Group 5
.25
.5
ST
-.25
Figure 2.7: Differenet payoff for the project
2.3
Discussion: drawbacks of Bachelier model
One of the drawbacks of the Bachelier model is the possibility of negative realization for asset price. However, this is not the main concern, as in many other application, Gaussian random variables are used to
model positive quantities such as human weight. Negative asset price can be problematic if the ratio σS?0T is
not sufficiently large. For instance if the ratio σS?0T “ 1, then the chance of negative price at maturity T is
significant, i.e. higher that .3.
The main concern about Bachelier model is the mismatch with the empirical studies on the asset return.
The return of an asset in Bachelier model is given by
Rarth
:“
t
erpt`δq Ŝt`δ ´ ert Ŝt
ert Ŝt
To be continued!
67
“
erδ Bt`δ ´ Bt
.
Bt
(Spring) 2016
3
3 BLACK-SCHOLES MODEL
Summary of the lecture
Continuous-time market of Black-Scholes
Black-Scholes model can be obtained by asymptotic methods from the binomial model. To do this, we first
present some asymptotic properties of binomial model.
3.1
Black-Scholes model: limit of binomial under risk neutral probability
Let T ą 0 be a real number and N be a positive integer. We divide T units of time into N time intervals,
T 24
each of size δ :“ N
. Then, consider a binomial model with N periods given by the times t0 “ 0 ă t1 “
δ, ..., tk “ kδ, ...tN “ T . Recall from binomial model that
Spk`1qδ “ Skδ Hk`1 , for k “ 0, ..., N ´ 1
where, in accordance to the Assumption 3.1, the random variables tHk uN
k“1 is a sequence of i.i.d. random
variables with the following distribution
#
u with probability π̂
Hk “
l
with probability 1 ´ π̂
Asymptotics of parameters u, l, and p
The goal of this section is to show the following approximations:
u “ 1 ` δr `
l “ 1 ` δr ´
?
?
δσα “ eδpr´
δσβ “ e
?
σ2
q`
2
δσα
?
2
δpr´ σ2 q´ δσβ
` opδq,
(3.1)
` opδq.
?
?
To obtain this approximation, one should notice that from (3.5), we have x0 “ 1 ` λ 2 δ ` op δq. Thus, it
follows from (3.4) that
?
?
?
?
1 λ δ
1´p
2
“ x0 “ 1 ` λ δ ` op δq and p “ ´
` op δq.
p
2
4
Then,
?
1´p
α “
` 2λ δ
p
2
c
?
?
?
1´p
` op δq “ x20 “ 1 ` 3λ δ ` op δq.
p
On the other hand, one can easily see that25
e
24
?
2
δpr´ σ2 q` δσα
?
2
?
pδpr ´ σ2 q ` δσαq2
σ2
“ 1 ` δpr ´ q ` δσα `
` opδq
2
2
?
σ2
δσ 2 α2
“ 1 ` δpr ´ q ` δσα `
` opδq
2
2
?
δpσ 2 ´ 1qα2
` opδq
“ 1 ` δr ` δσα `
2
?
“ 1 ` δr ` δσα ` opδq.
Each time unit is divided into 1{δ small time intervals.
2
e “ 1 ` x ` x2 ` opx2 q
25 x
68
(3.2)
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3 BLACK-SCHOLES MODEL
Summary of the lecture
?
In the above we used (3.2), i.e. σ 2?´ 1 “ Op
δq.
?
2
Similarly, we have β “ 1 ´ 3λ δ ` op δq and
eδpr´
?
σ2
q´
2
δσβ
“ 1 ` δr ´
?
δσβ ` opδq.
Arithmetic return versus log return
This asymptotics yields to the relation between the arithmetic return and log return in binomial model. While
the arithmetic return is given by
#
?
δr ` δσα
arth
?
Rt “
δr ´ δσβ
the log return is given by
Rlog
t
#
δpr ´
“
δpr ´
σ2
2 q
σ2
2 q
?
` δσα ` opδq
?
´ δσβ ` opδq
.
The probabilities of the values in both returns are given by pπ̂, 1´π̂q for risk neutral probability and pp, 1´pq
for physical probability. In particular, if Ê and E are respectively expectation with respect to risk neutral
probability and physical probability, then we have
ÊrRarth
s “ rδ, and ÊrpRarth
q2 s “ σ 2 δ ` opδq
t
t
ErRarth
s “ µδ, and ErpRarth
q2 s “ σ 2 δ ` opδq
t
t
1 2
log 2
2
ÊrRlog
t s “ pr ´ σ qδ, and ÊrpRt q s “ σ δ ` opδq
2
1 2
log 2
2
ErRlog
t s “ pµ ´ σ qδ, and ErpRt q s “ σ δ ` opδq.
2
(3.3)
Weak convergence of binomial model to geometric Brownian motion
From the asymptotics in (3.3), the log return of calibrated binomial model is given by
#
?
2
δpr ´ σ2 q ` δσα with probability π̂
log
?
Rt “ lnpHk q “
2
δpr ´ σ2 q ´ δσβ with probability 1 ´ π̂
Indeed,
lnpSpk`1qδ q “ lnpSkδ q ` lnpHk`1 q,
or
lnpSt q “ lnpS0 q `
N
ÿ
lnpHk q.
k“1
Let
tZk uN
k“1
is a sequence of i.i.d. random variables with the following distribution
#
α
with probability π̂
Zk “
´β with probability 1 ´ π̂
Then, we have
N
?
1 ÿ
σ2
?
lnpST q “ lnpS0 q ` pr ´ qT ` σ T ¨
Zk .
2
N k“1
69
(3.4)
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3 BLACK-SCHOLES MODEL
Summary of the lecture
Next, we want to show that the normalized summation
?1
N
řN
k“1 Zk
converges weakly to a random variable
(distribution) as the number of time intervals N approaches to infinity26 . To show this, from Theorem B.3
from the appendix, we only need to find
lim χ ?1
N Ñ8
řN
k“1
N
Zk pθq.
Notice that here the characteristic function is under risk neutral probability, i.e. χX pθq “ ÊreiθX s.
Since tZt uN
k“1 is a sequence of i.i.d. random variables, we have
χ ?1
řN
N
k“1
Zk pθq
“
N
ź
ˆ
χ Z1
k“1
θ
?
N
˙
.
On the other hand,
χZ1 pθq “ ÊreiθZ1 s “ 1 ` iθÊrZ1 s ´
θ2 ÊrZ12 s
` opθ2 q.
2
Notice that by (3.2), we can write
π̂ “
R´L
β
“
.
U ´L
α`β
(3.5)
Therefore, straightforward calculations show that ÊrZ1 s “ 0, and ÊrZ12 s “ 1. By using Taylor expansion
of the characteristic function, we obtain
ˆ
˙
θ
θ2
1
χZ1 ?
“1´
` op q,
2N
N
N
and
ˆ
χ ?1
řN
N
k“1
1´
Zk pθq “
1
θ2
` op q
2N
N
˙N
,
By sending n Ñ 8, we obtain
ˆ
lim χ ?1
N Ñ8
řN
N
k“1
Zk pθq “ lim
N Ñ8
θ2
1
1´
` op q
2N
N
˙N
θ2
θ2
“ e´ 2 .
Since e´ 2 is the characteristic function of standard normal random variable, we conclude that
converges weakly to N p0, 1q.
?1
N
řN
k“1 Zk
Exercise 3.1. In the above calculations, explain why we cannot apply central limit theorem (Theorem B.1)
directly.
(3.4) suggests that we define a continuous-time model for the price St of the asset at time T as the weak
limit of the binomial model by
lnpST q “ lnpS0 q ` pr ´
σ2
qT ` σN p0, T q
2
or equivalently
˙
σ2
ST “ S0 exp pr ´ qT ` σN p0, T q .
2
ˆ
26
Or equivalently δ Ñ 0.
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3 BLACK-SCHOLES MODEL
Summary of the lecture
Calibrating binomial model: revised
In the above, we see that the choice of parameters u, l, and R leads to we made a perfect choice of ÊrZ1 s “ 0
and ÊrZ12 s “ 1. However, in (3.4), the only criteria for the convergence of binomial model to Black-Scholes
model is that the random variables Zk , k “ 1, ..., N must satisfy ÊrZ1 s “ opδq and ÊrZ12 s “ 1 ` op1q, i.e.
N
1 ÿ
then ?
Zk convergens to N p0, 1q weakly.
N k“1
(3.6)
To avoid the calculation of α and β, one can choose different parameters for the binomial tree. Notice
that the binomial model has three parameters u, l and R while the Black-Scholes parameters are only two.
This degree of freedom provides us with some modifications of binomial three which still converges to the
Black-Scholes formula. This also simplifies the calibration process in Section 3 significantly simpler. Here
are some choices:
If ÊrZ1 s “ opδq, and ÊrZ12 s “ 1 ` op1q,
a) Symmetric probability:
u “ eδpr´
σ2
q`
2
?
δσ
, l “ eδpr´
Then;
σ2
q´
2
?
δσ
, and R “ rδ,
1
π̂u “ π̂l “ .
2
Notice that (3.4) should be modified by setting thetZk uN
k“1 distribution of i.i.d. sequence
#
1
with probability 12
Z1 “
´1 with probability 12
and ÊrZ1 s “ 0 and ÊrZ12 s “ 1.
b) Subjective return:
u “ eδν`
?
δσ
, l “ eδν´
?
δσ
, and R “ rδ,
Then;
1
π̂u “
2
˜
? r ´ ν ´ 1 σ2
2
1` δ
σ
¸
1
and π̂l “
2
˜
? r ´ ν ´ 1 σ2
2
1´ δ
σ
¸
.
In this case, (3.4) should be modified by setting thetZk uN
k“1 distribution of i.i.d. sequence
$?
& δ ν´r` 21 σ2 ` 1 with probability π̂
u
σ
Z1 “ ? ν´r`
1 2
σ
% δ
2
´ 1 with probability π̂
l
σ
and ÊrZ1 s “ 0 and ÊrZ12 s “ 1 `
´
ν´r` 21 σ 2
σ
¯2
δ.
Exercise 3.2. Show
ÊrZ1 s “ opδq, and ÊrZ12 s “ 1 ` op1q.
in the following cases.
a) symmetric probability
71
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3 BLACK-SCHOLES MODEL
Summary of the lecture
b) subjective return
Exercise 3.3. Consider risk neutral trinomial model with N periods presented by
Spk`1qδ “ Skδ Hk`1 , for k “ 0, ..., N ´ 1
where δ :“
T
N
and tHk uN
k“1 is a sequence of i.i.d. random variables with distribution
Hk “
$
?
2
δpr´ σ2 q` 3δσ
’
e
’
&
1
6
with probability π̂ “
2
δpr´ σ2 q
e
’
’
% δpr´ σ2 q´?3δσ
2
e
with probability 1 ´ 2π̂ “
2
3
1
6
with probability π̂ “
and π̂ ă 12 . Show that as δ Ñ 0, this trinomial model converges to the black-Scholes model in the weak
sense.
?
2
Hint: Find Zk such that lnpHk q “ pr ´ σ2 qδ ` σ δZk . Then, show (3.6)
3.2
Pricing contingent claims in Black-Scholes model
Recall from the last section that the limit of binomial model under risk neutral probability yields the geometric Brownian motion
˙
ˆ
σ2
(3.7)
St “ S0 exp pr ´ qt ` σBt .
2
We can use this random variable St to price an option in this continuous setting. We start with a call option
with maturity T and strike price K.
Inspired by Arrow-Debreu theory, the price of this call option is the discounted expected value of pST ´
Kq` under risk neutral probability. To calculated this price, we only need to know the distribution of St
under risk neutral probability which is given by (3.7). Since St is a function of a standard normal random
variable, we obtain
˙
˙
ˆ 2˙
ˆ
ż8 ˆ
?
x
1
σ2
exp ´
dx.
ÊrpST ´ Kq` s “ ?
S0 exp pr ´ qT ` σ T x ´ K
2
2
2π ´8
`
´
Notice that when x ď x˚ :“ σ?1 T lnpK{S0 q ´ pr ´
pST ´ Kq` “ St ´ K. Therefore,
ÊrpST ´ Kq` s “ S0 e
“ S0 e
2
pr´ σ2 qT
ż8
?
rT
ż8
e
?
“ S0 erT
2π
2π
2
´ x2
ż8
?
x˚ `σ T
x2
´ 2
e?
2π
e
?
, the the integrand is zero and otherwise
2
? 2
´ px´σ2 T q
?
¯
T x´ x2
x˚
x˚
ş8
eσ
σ2
2 qT
2π
ż8
dx ´ K
x˚
ż8
dx ´ K
x˚
ż8
dx ´ K
x˚
x2
e´ 2
? dx
2π
x2
e´ 2
? dx
2π
x2
e´ 2
? dx
2π
Notice that x˚
dx is the probability that a standard normal random variable is greater than x˚ and
2
?
ş8
´x
? e? 2 dx is the probability that a standard normal random variable is greater than x˚ `σ T . Simple
˚
x `σ T
2π
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3 BLACK-SCHOLES MODEL
Summary of the lecture
calculation shows that
?
1
x `σ T “ ?
σ T
˚
ˆ
σ2
lnpK{S0 q ´ pr ` qT
2
˙
.
In other words, the price of a European call option is given by
CpT, K, S0 , 0q :“ e´rT ÊrpSt ´ Kq`s “ S0 Φ pd1 q ´ e´rT KΦ pd2 q ,
ˆ
ˆ
˙
˙
(3.8)
1
σ2
1
σ2
d1 “ ?
lnpS0 {Kq ` pr ` qT and d2 “ ?
lnpS0 {Kq ` pr ´ qT .
2
2
σ T
σ T
Here Φpxq “
y2
´ 2
e?
´8
2π
şx
dy is the standard normal distribution function.27
Exercise 3.4. Use put-call parity to show Black-Scholes formula for the price of a put option with maturity
T and strike K is given by
P pT, K, S0 , 0q :“ e´rT KΦ p´d2 q ´ S0 Φ p´d1 q .
For a general contingent claim, the price of derivative with payoff gpSt q in Black-Scholes model is given
by
„ ˆ
˙ȷ
?
2
´rT
´rT
pr´ σ2 qT `σ T N p0,1q
V0 pS0 q :“ e
ÊrgpST qs “ e
Ê g e
˙
ż8 ˆ
?
σ2
x2
1
pr´
qT
`σ
T
x
2
g e
e´ 2 dx.
“?
2π ´8
As a consequence of Assumption 3.1, the random variables if we repeat the calculations in Section 3.1, we
obtain
˙
ˆ
˙
ˆ
σ2
σ2
and ST “ St exp pr ´ qpT ´ tq ` σN p0, T ´ tq .
St “ S0 exp pr ´ qt ` σN p0, tq
2
2
Now we would like to explain the relation between the two normal random variables N p0, T ´ tq and
N p0, tq in the above. Thus, in Black-Scholes model, SSTt is independent of St . This, in particular, implies
that Black-Scholes model is Markovian28 and the price of a call option with strike K and maturity T given
the price of underlying asset at time t is equal to St is a function of St and t but not Su for u ă t. As a result
the price of call option at time t given St “ S is given by
CpT, K, S, tq :“ e´rpT ´tq ÊrpST ´ Kq` | St “ Ss “ SΦ pd1 q ´ e´rpT ´tq KΦ pd2 q ,
ˆ
˙
σ2
1
lnpS{Kq ` pr ` qpT ´ tq and
d1 “ ?
2
σ T ´t
˙
ˆ
1
σ2
d2 “ ?
lnpS{Kq ` pr ´ qpT ´ tq .
2
σ T ´t
In general, Markovian property implies that for a general contingent claim, the price of derivative with
payoff gpSt q, the Black-Scholes price at time t is a function V pt, St q :“ e´rpT ´tq ÊrgpST q | St s. We
will study some more properties of this function V : r0, T s ˆ R` Ñ R in the future. In Black-Scholes
market, contingent claims that have payoff is gpST q, a function of underlying price at time T , St , are called
27
28
In the above calculations, we use 1 ´ Φpxq “ Φp´xq.
Future movements of the price are independent of the past.
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3 BLACK-SCHOLES MODEL
Markovian claims and the price of a Markovian claim is given by
„ ˆ
˙ȷ
?
2
´rpT ´tq
´rpT ´tq
pr´ σ2 qpT ´tq`σ T ´tN p0,1q
V pt, Sq :“ e
ÊrgpST q | St “ Ss “ e
Ê g e
ˆ
˙
ż
` pr´ σ2 qpT ´tq`σ?T ´tx ˘ ´ x2
e´rpT ´tq 8
2
g S e
e 2 dx.
“ ?
2π
´8
(3.9)
Remark 3.1. Price of a Markovian claim in Black-Scholes model does not depend on the past movements of
the price and it only depends on the current price St . This is not true for non-Marovian
´ ş claims, ¯indeed. For
T
example a look-back option with payoff pmax0ďtďT St ´ Kq` or Asian option T1 0 St ´ K
are non`
Markovian options with the price which depends to some extent in the past history of the price movement
rather than only the current price of the underlying.
As seen in (3.9), the Black-Scholes price of a Markovian European option is always a function of T ´ t
rather than t and T separately. Therefore, we can introduce the new variable τ :“ T ´ t which is called
time-to-maturity. Then, one can write the value of the Markovian European option as a function of τ and S
by
ˆ
˙
ż
` pr´ σ2 qτ `σ?τ x ˘ ´ x2
e´rτ 8
´rτ
2
g S e
e 2 dx.
V pτ, Sq :“ e ÊrgpSτ q | S0 “ Ss “ ?
2π ´8
For call option the Black-Scholes formula in terms of τ is given by
1
d1 “ ?
σ τ
3.3
ˆ
Cpτ, K, Sq “ SΦ pd1 q ´ e´rτ KΦ pd2 q ,
˙
˙
ˆ
1
σ2
σ2
and d2 “ ?
lnpS{Kq ` pr ` qτ
lnpS{Kq ` pr ´ qτ .
2
σ τ
2
(3.10)
Delta Hedging
As seen in the binomial model, to hedge the risk of issuing an option, one has to construct a replicating
portfolio. The replicating portfolio contains ∆t pSt q units of risky asset at time t if the price of the asset is
equal to St where by (2.7)
Vt`δ pSuq ´ Vt`δ pSlq
.
∆t pSq :“
Spu ´ lq
Using (3.1), we have
∆t pSq “
Vt`δ pS ` Spδr `
?
δσαqq ´ Vt`δ pS ` Sp1 ` δr ´
?
S δσpα ` βq
?
δσβqq
Heuristically, as δ Ñ 0, we obtain the Delta of the Black-Scholes model as
∆BS
t pSq “ BS V pt, Sq,
(3.11)
where V pt, Sq is the Black-Scholes price of a general contingent claim with any given payoff.
We can also find another heuristic derivation of this result. In the binomial model, we can write
ÊrgpST q | St`δ “ Sus “ ÊrgpST ´δ uq | St “ Ss and ÊrgpST q | St`δ “ Sls “ ÊrgpST ´δ lq | St “ Ss.
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This is because binomial model is time homogeneous. Therefore,
∆t pSq “
Vt`δ pSuq ´ Vt`δ pSlq
Spu ´ lq
“ e´rpT ´tq
ÊrgpST q | St`δ “ Sus ´ ÊrgpST q | St`δ “ Sls
` Opδq
Spu ´ lq
“ e´rpT ´tq
ÊrgpST ´δ uq | St “ Ss ´ ÊrgpST ´δ lq | St “ Ss
` Opδq
Spu ´ lq
“ e´rpT ´tq
ÊrST ´δ V 1 pST ´δ q | St “ Sspu ´ lq
` Opδq.
Spu ´ lq
In the above we used (3.1) to obtain pu ´ lq2 “ Opδq and
V pxq ´ V pyq “ V 1 pxqpy ´ xq ` Oppx ´ yq2 q.
Hence,
e´rpT ´tq
ÊrST ´δ V 1 pST ´δ q | St “ Ss ` Opδq
S
Now by weak convergence of underlying asset price in binomial model to asset price in Black-Scholes
model, as δ Ñ 0, we obtain
∆t pSq “
∆BS
t pSq “
e´rpT ´tq
ÊrST V 1 pST q | St “ Ss.
S
Notice that given St “ S we have
˙
σ2
ST “ S exp pr ´ qpT ´ tq ` σN p0, T ´ tq .
2
ˆ
Therefore,
BS
´
˙
ˆ
¯
ST 1
σ2
V pST q.
gpST q “ exp pr ´ qpT ´ tq ` σN p0, T ´ tq V 1 pST q “
2
S
This implies that
”S
ı
T 1
´rpT ´tq
∆BS
pSq
“
e
Ê
V
pS
q
|
S
“
S
“ e´rpT ´tq ÊrBS pgpST qq | St “ Ss
t
T
t
S
´
¯
“ BS e´rpT ´tq ÊrgpST q | St “ Ss “ BS V pt, Sq.
As for a call option with maturity T and strike K, by taking derivative with respect to S from (3.10), we
obtain
∆BS
t pSq “ Φpd1 q.
Here we used
SBS d1 Φ1 pd1 q ´ e´rpT ´tq KBS d2 Φ1 pd2 q “ 0.
Exercise 3.5. Let S0 “ $10, σ “ .03, r “ 0.03. Find the Black-Scholes Delta of the following portfolio.
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position
long
short
short
units
3
3
1
type
call
put
call
strike
$10
$10
$10
maturity
60 days
90 days
120 days
The maturities are given in business days.
Exercise 3.6. Let S0 “ 10, σ “ .03, r “ 0.03. Consider a portfolio below.
position
long
long
?
units
3
4
x
type
call
put
underlying
strike
$10
$5
NA
maturity
60 days
90 days
NA
How many units x of underlying is required to eliminate any sensitivity of the portfolio with respect to the
change in price of underlying?
3.4
Completeness of Black-Scholes model
As a matter of fact, Black-Scholes is a complete market, i.e. every contingent claim is replicable. For the
moment, it is not our concern to show this fact. Instead, we accept this fact and would rather emphasize on
how to replicate a contingent claim. In order to replicate a Markovian contingent claim with payoff gpST q
in Black-Scholes model. We start by recalling from Section 2.3, that the replicating portfolio in binomial
model is determined by ∆bi
ti pSti q given by (2.7) and the replicating portfolio is written as
V bi pt, St q “ V bi p0, S0 q ` R
t´1
ÿ
pV bi pi, Si q ´ ∆bi
i pSi qSi q `
i“0
t´1
ÿ
∆bi
i pSi qpSi`1 ´ Si q,
(3.12)
i“0
By taking limit from (3.12), we obtain
żT
żT
∆t pSt qdSt .
pV pt, St q ´ ∆t pSt qSt qdt `
V pT, St q “ V p0, S0 q ` R
(3.13)
0
0
In the above V pt, Sq is the Black-Scholes price of the contingent claim and ∆t pSt q satisfies (3.11). The
first integral in (3.13) is a simple Riemann integral. The second integral is a more complicated stochastic
integral, i.e. the integrator dSt is stochastic. We will make sense of the stochastic integration in the future
by defining Itô integral. But
the moment, you can use the heuristics of Riemann-Stieltjes integral for
ř for
´1
stochastic integral, limit of N
i“0 ∆ti pSti qpSti`1 ´ Sti q.
3.5
Error of discrete hedging in Black-Scholes model and Greeks
Equation (3.13) suggests to adjust Delta continuously in time to replicate the contingent claim. On one hand,
this is a useful formula because in reality trading can happen with enormous speed which makes continuoustime a fine approximation. However, in practice, the time is still in discrete and hedging is only a time lapse.
Therefore, it is important to have some estimation on the error of discrete-time hedging in Black-Scholes
framework.
Let us consider that the Black-Scholes model is running continuous in time, but we only adjust our position
T
on the approximately replicating portfolio at times ti :“ δi where δ “ N
and i “ 0, 1, ..., N . By setting
aside the accumulated error until time ti , we can assume that our hedge has been perfect until time ti , for
some i, i.e.
V pti , Sti q “ ∆BS
ti pSti qSti ` Yti ,
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where V pt, Sq is the Black-Scholes value of the contingent claim, and Yt is the position in cash. At time
ti`1 “ ti ` δ, the value of the portfolio is
rδ
∆BS
ti pSti qSti `δ ` e Yti .
Since by (3.11), ∆BS
ti pSti q “ BS V pti , Sti q, the error is given by
”
ı
Errti pδq :“Ê V pti ` δ, Sti `δ q ´ BS V pti , Sti qSti `δ ´ erδ Yti | Sti
”
ı
“Ê V pti ` δ, Sti `δ q ´ BS V pti , Sti qSti `δ ´ erδ pV pti , Sti q ´ BS V pti , Sti qSti q | Sti
”
“Ê V pti ` δ, Sti `δ q ´ V pti , Sti q ´ BS V pti , Sti qpSti `δ ´ Sti q
ı
` perδ ´ 1qpV pti , Sti q ´ BS V pti , Sti qSti q | Sti
”
“Ê V pti ` δ, Sti `δ q ´ V pti , Sti q ´ BS V pti , Sti qpSti `δ ´ Sti q
ı
´ δrpV pti , Sti q ´ BS V pti , Sti qSti qq | Sti ` Opδ 2 q
By Taylor formula, the price of the option is
V pti ` δ, Sti `δ q “ V pti , Sti q ` Bt V pti , Sti qδ ` BS V pti , Sti qpSti `δ ´ Sti q
1
1
` BSS V pti , Sti qpSti `δ ´ Sti q2 ` Btt V pti , Sti qδ 2
2
2
` BSt V pti , Sti qpSti `δ ´ Sti qδ ` ....
Notice that since
?
Sti `δ ´ Sti “ σSti δN p0, 1q ` Opδq, with N p0, 1q independent of Sti .
(3.14)
3
the conditional expectation ÊrpSti `δ ´ Sti q2 | Sti s “ σ 2 St2i δs ` Opδ 2 q. Thus the conditional expectation
we can write for the error term29
Errti pδq :“ δpBt V pti , Sti q `
σ 2 St2i
3
BSS V pti , Sti q ` rBS V pti , Sti qSti q ´ rV pti , Sti qq ` Opδ 2 q.
2
(3.15)
3
where the term Opδ 2 q depends on the higher derivative BtS V , σ, Sti , and r.
Before finishing the error estimation, let us briefly explain some of the important terms which show up in
(3.15). The second derivative BSS V of the option, which is called Gamma and denoted by Γpt, Sq, determines
the convexity of the option value on the price of underlying. The time derivative Bt V , which is called time
decay factor or Theta and is denoted by Θpt, Sq, determines how the price of option evolves over time. As
a function of time-to-maturity τ “ T ´ t, by the abuse of notation, we define V pτ, Sq :“ V pT ´ τ, Sq and
therefore, we have
∆pτ, Sq “ BS V pτ, Sq
Γpτ, Sq “ BSS V pτ, Sq
Θpτ, Sq “ ´Bτ V pτ, Sq
Here we heuristically assumed N p0, 1q „ 1. A more rigorous treatment of this error is in by calculating the L2 error by
1
calculating ÊrpErrti pδqq2 s 2 .
29
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For example in the case of call option with strike K and maturity T , by taking derivatives BSS and Bτ in
(3.10), we have
Γpτ, Sq “
1
Sσ
? Φ1 pd1 q and Θpτ, Sq “ ´ ? Φ1 pd1 q ´ rKe´rτ Φpd2 q.
Sσ τ
2 τ
(3.16)
The ∆, Γ, and Θ of a call option in Black-Scholes model is shown in Figure 3.1.
Delta
Gamma
Theta
1
0.5
0
−0.5
6
8
10
S
12
14
Figure 3.1: Greeks of call option with τ “ 1, σ “ .1, r “ .05, and K “ 10. As you see the significant
sensitivity is near the ATM.
To continue with the error estimation, we need the following proposition.
Proposition 3.1. For a European Markovian contingent claim, the Black-Scholes price satisfies
Θpτ, Sq “ ´
σ2S 2
Γpτ, Sq ´ rS∆pτ, Sq ` rV pτ, Sq.
2
As a result of this proposition, the term f order δ in (3.15) vanishes and thus one step error is of order
3
3
Opδ 2 q. Provided that the term Opδ 2 q remains bounded, we have
«
ff
Nÿ
´1
?
Errpδq :“ Ê
Errti pδq “ Op δq,
i“0
which converges to 0 as fast as
?
δ when δ Ñ 0.
Discrete hedging without money market account
One reason to completely disregard the money market account is because the risk free interest rate r is not
exactly constant. The money market is also under several risks which is a different topic. To understand this
better, let us first consider the issuer of an option which is long in ∆BS pti , Sti q units of underlying. Then
the change in the portfolio from time ti to time ti ` δ is
ÊrV pti ` δ, Sti `δ q ´ V pti , Sti q ´ ∆BS pti , Sti qpSti `δ ´ Sti qs
“ pBt V pti , Sti q `
78
σ 2 St2i
BSS V pti , Sti qqδ ` opδq,
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3 BLACK-SCHOLES MODEL
where we used (3.14) in the right hand side. This error is related to loss/profit of not perfectly hedging and
is called slippage error; see Figure 3.2.
V pt, Sq
V pti ` δ,Sti `δ q
V pti ,Sti q
S
St i
Sti `δ
Figure 3.2: Slippage error shown in burgundy.
2.5
V (t, S)
2
1.5
1
0.5
0
K
S
Figure 3.3: Time decay for the price of call option in Black-Scholes model. . As τ Ñ 0, the color goes
darker.
The loss/profit from slippage can be calculated by using the same technology as in previous section; the
slippage error during the time interval rti , ti ` δs is given by
pΘpti , Sti q `
σ 2 St2i
Γpti , Sti qqδ ` opδq
2
As illustrated in Figure 3.4, when, for instance, the time decay factor is negative and Gamma is positive, for
small changes in the price of asset, we loose and for larger changes we gain. As seen in (3.16), it is a typical
situation to have negative Θ and positive Γ for call options (or put options or any European Markovian
option with convex payoff function). See Figure 3.3.
Exercise 3.7. Show that if the payoff function gpST q is a convex function on ST , then the Markovian European contingent claim with payoff gpST q has non-negative Γ, i.e. V pτ, Sq is convex on S for all τ .
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Profit/Loss
∆S
Figure 3.4: Loss/profit of discrete hedging in Black-Scholes model. The red graph shows the loss/profit
without time decay factor. The blue includes time decay factor too.
Let S̃t “ e´rt St and Ṽ pt, S̃t q “ e´rt V pt, St q “ e´rt V pt, ert S̃t q be respectively the discounted underlying
price and discounted option price. Then, we can show that
Bt Ṽ pt, S̃q “ ´
σ 2 S̃ 2
B Ṽ pt, S̃q.
2 S̃ S̃
Exercise 3.8. Use Proposition 3.1, to show the above equality.
This suggests that if the interest rate is nearly zero, then the lack of money market in replication does not
impose any error. Otherwise, when interest rate is large, the slippage error is significant and is equal to
pΘpti , Sti q `
σ 2 St2i
Γpti , Sti qqδ ` opδq “ rpV pti , Sti q ´ St2i ∆pti , Sti qqδ ` opδq,
2
which can accumulate to a large number.
Other derivatives
Two other Greeks are Rho, denoted by ρ, and Vega, denoted by V, and respectively measure the sensitivity
with respect to interest rate r and volatility σ, i.e.
´
¯
´
¯
ρpτ, Sq :“ Br e´rτ ÊrgpSτ q | S0 “ ss
and Vpτ, Sq :“ Bσ e´rτ ÊrgpSτ q | S0 “ ss .
For call option, these derivatives are given by
?
ρpτ, Sq “ e´rτ Kτ Φpd2 q and Vpτ, Sq “ S τ Φ1 pd1 q.
Figures (3.1) and 3.5 show the Greeks ∆, Γ, Θ, ρ and V for a call option as a function of S.
Exercise 3.9. The third derivative of Black-Scholes price with respect to S is called speed. Find a closed
form solution for speed.
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10
Rho
Vega
8
6
4
2
0
6
8
10
S
12
14
Figure 3.5: ρ and V of call option with τ “ 1, σ “ .1, r “ .05, and K “ 10.
Example 3.1. The payoff in Figure 3.6 can be written as pST ´ K1 q` ´ pST ´ K2 q` ´ pST ´ K3 q` `
pST ´ K4 q` . Therefore, the closed form solution for the Black-Scholes price of the option is given by
V pτ, Sq “ Cpτ, K1 , Sq ´ Cpτ, K2 , Sq ´ Cpτ, K3 , Sq ` Cpτ, K4 , Sq.
All the Greeks of the option are also linear combination of these call option Greeks. For instance,
gpST q
K1
K2
K3
K4
ST
Figure 3.6: Payoff of Example 3.1.
∆pt, Sq “ Φpd1 pτ, K1 , Sqq ´ Φpd1 pτ, K2 , Sqq ´ Φpd1 pτ, K3 , Sqq ` Φpd1 pτ, K4 , Sqq.
Exercise 3.10. Write the payoffs in Figure 3.7 as linear combination of call options and derive a closed
form formula for the Black-Scholes price, the Delta, and the Gamma of them. All the Greeks of the option
are also linear combination of these call option Greeks. For instance,
∆pt, Sq “ Φpd1 pτ, K1 , Sqq ´ Φpd1 pτ, K2 , Sqq ´ Φpd1 pτ, K3 , Sqq ` Φpd1 pτ, K4 , Sqq.
Exercise 3.11 (Bull and bear call spreads). Write the following payoffs as linear combination of call options
with different strikes and possibly some cash and give the closed form formula for them.
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gpST q
gpST q
K
K1
ST
K2
ST
Figure 3.7: Left: payoff of straddle. Right: payoff of strangle.
gpST q
K1
K2
ST
Figure 3.8: Red: Bull spread call. Blue: Bear spread call
3.6
Time-varying Black-Scholes model
Recall from Section 3.1 that the binomial model can be calibrated to the time dependent parameters. Let
T
δ“N
and consider the discrete time instances ti “ iδ. Time-varying binomial can be given by
lnpStk`1 q “ lnpSkδ q ` lnpHk`1 q,
where tHk uN
k“1 is a sequence of independent random variables with the following distribution under risk
neutral probability
#
?
1 ` δrtk ` δσtk αk with probability π̂k
?
Hk “
,
1 ` δrtk ´ δσtk βk with probability 1 ´ π̂k
where αk , βk are given by (3.4) and (3.5) for time dependent λt . Therefore, equation (3.4) takes the following time-dependent form:
k´1
ÿ
N
? ÿ
σt2i
lnpStk q “ lnpS0 q `
prti ´
qδ ` δ
σtk Zk .
2
i“0
k“1
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Analogous to (3.5), we have πk “
βk
αk `βk
and
ÊrZk s “ 0 and ÊrZk2 s “ 1.
Therefore,
χ?δσ
tk
iθσtk Zk
s “ 1 ` iθσtk δ ÊrZ1 s ´
Zk pθq “ Êre
θ2 σt2k δ ÊrZ12 s
` opδq.
2
This implies that
χ?δ řN
k“1
σtk Zk pθq
“
N ˆ
ź
k“1
˙ ź
N
2 řN
1 2 2
´ 1 θ2 σ2 δ
σ2 δ
´θ
2
1 ´ δθ σtk ` opδ q “
e 2 tk ` opδq “ e 2 i“1 tk ` opδq.
2
k“1
As δ Ñ 0,
χ?δ řN
k“1 σtk
which is the characteristic function of N p0,
şT
0
ˆż T
ST “ S0 exp
0
´
Zk pθq Ñ e
θ2
2
şT
0
σt2 dt
,
σt2 dtq. Thus in the limit, we have
σ2
prt ´ t qdt `
2
żT
˙
σt2 dtN p0, 1q
.
0
As a matter of fact, for any t we have
ˆ żt
˙˙
2
0, σu du
St “ S0 exp
0
˙˙
ˆż T
ˆ żT
2
σu
2
pru ´
σu du
.
ST “ St exp
qdu ` N 0,
2
t
t
´ ş
¯
´ ş
¯
T
t
and the random variables N 0, t σu2 du and N 0, 0 σu2 du are independent.
Usually the interest rate rt and volatility σt are not given and we have to estimate them from the data. In
the next section, we discuss some estimation methods of these two parameters.
Using the variable r and σ, we can rewrite Black-Scholes formula for a call option by
ˆż t
σ2
pru ´ u qdu ` N
2
0
şT
ÊrpST ´ Kq` | St “ Ss “ SΦ pd1 q ´ e´
˙ ˙
ˆ
żTˆ
σu2
1
du and
lnpS{Kq `
ru `
d1 “ b ş
T 2
2
t
σ
du
t u
ˆ
˙ ˙
żTˆ
1
σu2
d2 “ b ş
lnpS{Kq `
ru ´
du .
T 2
2
t
σ
du
t u
CpT, K, S, tq :“ e´
ru du
t
şT
t
ru du
KΦ pd2 q ,
For a general European payoff gpST q we have the Black-Scholes price given by
şT
ÊrgpSτ q | S0 “ Ss
ˆ
˙
b
ż
` ´ şT pru ´ σu2 qdu`x şT σ2 udu ˘ ´ x2
e´ t ru du 8
u
t
t
2
?
g S e
e 2 dx.
“
2π
´8
V pt, Sq :“ e´
t
ru du
şT
Remark 3.2. We shall see that in actual calculations, we only need to take one of the interest rate rt or
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3 BLACK-SCHOLES MODEL
volatility σt time-varying!
3.7
Black-Scholes with yield curve and forward interest rate
Recall from Section 1.4 that the yield curve Rt pT q and forward curve Ft pT q of zero-coupon bond are defined
by
Bt pT q “ e´pT ´tqRt pT q “ e´
şT
t
Ft puqdu
or or Rt pT q :“ ´
1
d ln Bt pT q
ln Bt pT q Ft pT q :“ ´
.
T ´t
dT
Since setting a model for forward rate is equivalent of setting a model for short rate rt , in Black-Scholes
formula with varying interest rate, one can equivalently use the forward rate or yield curve. Assume that σ
is constant. Then, the Black-Scholes pricing formula becomes
CpT, K, S, tq :“ e´
d1 “ a
d2 “ a
şT
t
şT
ÊrpST ´ Kq` | St “ Ss “ SΦ pd1 q ´ e´ t Ft puqdu KΦ pd2 q ,
ˆ
˙
żT
σ2
lnpS{Kq `
Ft puqdu ` pT ´ tq and
2
´ tq
t
ˆ
˙
żT
σ2
lnpS{Kq `
Ft puqdu ´ pT ´ tq .
2
´ tq
t
Ft puqdu
1
σ 2 pT
1
σ 2 pT
and
CpT, K, S, tq :“ e´Rt pT qpT ´tq ÊrpST ´ Kq` | St “ Ss “ SΦ pd1 q ´ e´Rt pT qpT ´tq KΦ pd2 q ,
˙
ˆ
1
σ2
d1 “ a
lnpS{Kq ` Rt pT qpT ´ tq ` pT ´ tq and
2
σ 2 pT ´ tq
˙
ˆ
1
σ2
a
d2 “
lnpS{Kq ` Rt pT qpT ´ tq ´ pT ´ tq .
2
σ 2 pT ´ tq
3.8
(3.18)
Black-Scholes model and Brownian motion
In Section B.7, we show that symmetric random walk converges
ř to the Brownian motion Bt . The same
principle shows that the linear interpolation of the summation N
k“1 Zk in logarithm of binomial model in
(3.4), also converges to the Brownian motion. Therefore, one can write the Black-Scholes model (3.7) by
using Brownian motion Bt ; i.e.
ˆ
˙
σ2
St “ S0 exp pr ´ qt ` σBt .
(3.19)
2
The above process is called a geometric Brownian motion (GBM for short)30 .
Markovian property of Black-Scholes model
Since for Brownian motion the increment Bs`t ´ Bt is independent of Bt ,
˙
ˆ
˙
ˆ
σ2
σ2
and St`s “ St exp pr ´ qs ` σpBt`s ´ Bt q
St “ S0 exp pr ´ qt ` σBt
2
2
30
Paul Samuelson first came with the idea of using GBM to model the risky asset price in 50’. The main motivation is that GBM
never generates negative prices. Long before Samuelson on 1900, Louis Bachelier modeled the price of a risky asset by a Brownian
motion.
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3 BLACK-SCHOLES MODEL
are independent. Verbally, this means that the future movements of the price are independent of the past
movements. In other words, given the history of the movements of asset price until time t, i.e. Su for all
u P r0, ts, the distribution of St`s only depends on St and the part of history during u P r0, tq is irrelevant.
As a result, for any function g : R` Ñ R, we have
„ ˆ
ˆ
˙˙ȷ
σ2
.
ÊrgpSt`s q | Su : @u P r0, tss “ ÊrgpSt`s q | St s “ Ê g St exp pr ´ qs ` σpBt`s ´ Bt q
2
The pricing formula (3.9) is precisely derived form Markovian property of Black-Scholes model.
As a result of Markovian property of GBM, one can write
ˆ
ˆ
˙
˙
σ2
St`dt ´ St “ St exp pr ´ qdt ` σpBt`dt ´ Bt q ´ 1
2
´
¯
σ2
1
“ St pr ´ qdt ` σpBt`dt ´ Bt q ` σ 2 pBt`dt ´ Bt q2 ` opdtq.
2
2
Therefore, the short term return of asset in Black-Scholes model is given by
St`dt ´ St
σ2
1
“ pr ´ qdt ` σpBt`dt ´ Bt q ` σ 2 pBt`dt ´ Bt q2 ` opdtq.
St
2
2
This is, in particular, consistent with Assumption 3.1 and the definition of mean return and volatility, i.e.
ȷ
„
σ2
1
St`dt ´ St
“ pr ´ qdt ` σ ÊrBt`dt ´ Bt s ` σ 2 ÊrpBt`dt ´ Bt q2 s ` opdtq “ rdt ` opdtq.
Ê
St
2
2
var pSt`dt ´ St St q “ σ 2 dt ` opdtq.
Inspired from above formal calculation, We often write the Black-Scholes formula as
dSt
“ rdt ` σdBt .
St
(3.20)
The above equation which describes the dynamics of the price movements in Black-Scholes model is the
differential form of writing (3.19).
Martingale property of Black-Scholes model
Similar to the binomial model, in the Black-Scholes model, the discounted asset price and the discounted
price of a European contingent claim are martingale. The discounted asset price is Black-Scholes model is
given by
˙
ˆ 2
σ
´rt
S̃t “ e St “ S0 exp ´ t ` σBt .
2
the conditional expectation of S̃t`s given S̃t is then given by
ˆ 2
„
ˆ 2
˙ˇ ȷ
˙ „
ˆ 2
˙ˇ ȷ
ˇ
ˇ
σ
σ
σ
S0 Ê exp ´ pt ` sq ` σBt`s ˇˇS̃t “ S0 exp ´ t ` σBt Ê exp ´ s ` σpBt`s ´ Bt q ˇˇS̃t
2
2
2
„
ˆ 2
˙ˇ ȷ
ˇ
σ
St Ê exp ´ s ` σpBt`s ´ Bt q ˇˇS̃t .
2
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By the independence of the increments of Brownian motion, we have
„
ˆ 2
˙ˇ ȷ
„
ˆ 2
˙ȷ
ˇ
σ
σ
ˇ
Ê exp ´ s ` σpBt`s ´ Bt q ˇS̃t “ Ê exp ´ s ` σpBt`s ´ Bt q ,
2
2
and therefore,
„
ˆ 2
˙ȷ
ˇ ı
”
σ
ˇ
Ê S̃t`s ˇS̃t “ S̃t Ê exp ´ s ` σpBt`s ´ Bt q .
2
On the other hand since Bt`s ´ Bt „ N p0, sq, we have
Ê rexp pσpBt`s ´ Bt qqs “ exp
ˆ
˙
σ2
s ,
2
ˇ ı
”
ˇ
and therefore, Ê S̃t`s ˇS̃t “ S̃t .
Recall from formula (3.9) that the price of a contingent claim with payoff gpST q is given by
V pt, St q “ e´rpT ´tq ÊrgpST q | St s.
If we define the discounted price by Ṽ pt, St q “ e´rt V pt, St q, then we can write the above as
Ṽ pt, St q “ e´rT ÊrgpST q | St s.
By the tower property of conditional expectation, we have
ˇ ı
ˇ ı
”
”
ˇ
ˇ
Ê Ṽ pt ` s, St`s qˇSt “ Ê e´rT ÊrÊrgpST q | St`s sˇSt “ e´rT Ê rgpST q|St s “ Ṽ pt, St q.
Therefore, the price of the contingent claim is a martingale under risk neutral probability.
3.9
Physical versus risk neutral in Black-Scholes model
Recall that the price of an asset in the Black-Scholes model is represented by (3.20) or simply
˙
ˆ
σ2
St “ S0 exp pr ´ qt ` σBt .
2
This model is obtained by taking limit from the binomial model in the risk neutral probability framework.
Therefore, the above representation is the risk neutral price of the asset in Black-Scholes model. Indeed, one
can see that the discounted price of the asset is a martingale, i.e.
e´rpt`sq ÊrSt`s | St s “ e´rt St .
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To show this one needs to use the fact that pBt`s ´ Bt and Bt are independent. Therefore, Bt`s ´ Bt and
St are independent and we have
„
ˆ
˙ˇ ȷ
ˇ
σ2
ÊrSt`s | St s “ Ê St exp pr ´ qs ` σpBt`s ´ Bt q ˇˇSt
2
„
ˆ
˙ˇ ȷ
ˇ
σ2
“ St Ê exp pr ´ qs ` σpBt`s ´ Bt q ˇˇSt
2
„
ˆ
˙ȷ
σ2
“ St Ê exp pr ´ qs ` σpBt`s ´ Bt q
2
˙ȷ
„
ˆ 2
σ
“ e´rs St .
“ St e´rs Ê exp ´ s ` σpBt`s ´ Bt q
2
σ2
The last equality in the above is because Bt`s ´ Bt „ N p0, sq and ÊreσN p0,sq s “ e´ 2 s .
Now recall the setting of Section 3.1, where we divide T units of time into N times intervals, each of size
T
δ :“ N
. Then, consider a binomial model with N periods given by the times t0 “ 0 ă t1 “ δ, ..., tk “
kδ, ...tN “ T . Then the binomial model under physical probability is given by
Spk`1qδ “ Skδ Hk`1 , for k “ 0, ..., N ´ 1.
However, the distribution of the sequence of i.i.d. random variables tHk uN
k“1 under physical probability is
now
#
u with probability p
Hk “
l
with probability 1 ´ p
Then, the dynamics of the asset price under physical probability is given by
lnpST q “ lnpS0 q `
N
ÿ
lnpHk q.
k“1
lnpHk q is the log return Rlog
kδ . Recall from (3.3) that the first two moments of log return are given by
1
ErlnpHk qs “ pµ ´ σ 2 qδ ` Opδ 2 q, ErlnpHk q2 s “ σ 2 δ ` opδq
2
In other words, if we define Zk1 :“
2 {2q
lnpHk q´pµ´σ
?
,
σ δ
one can write
˙
N
?
1 2
1 ÿ 1
lnpST q “ lnpS0 q ` µ ´ σ T ` σ T ¨ ?
Zk ,
2
N k“1
ˆ
where tZk1 u is a sequence of i.i.d. random variables satisfying
ErZk1 s “ opδq, and ErpZk1 q2 s “ 1 ` op1q.
ř
1
If follows from (3.6) that ?1N N
k“1 Zk converges weakly to N p0, 1q, and under physical probability, the
Black-Scholes model is described by
ˆ
˙
σ2
St “ S0 exp pµ ´ qt ` σBt .
2
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In other words, by switching from risk neutral probability to physical probability, we adjust the mean return
of the asset from r to the risk free interest rate µ.
Remark 3.3. For the purpose of derivative pricing, the physical probability is irrelevant. This is because
by the fundamental theorem of asset pricing, Theorem 3.1, the price of any derivative is determined by the
discounted expectation of payoff under risk neutral probability. However, for portfolio management the
physical probability is important, because it carries the long-term growth rate of the asset µ. For example,
an investor with x initial wealth wants to decide how to split his investment between risk free bond with
interest rate r and a risky asset given by Black-Scholes equation
ˆ
˙
σ2
St “ S0 exp pµ ´ qt ` σBt .
2
His objective is to maximize his expected wealth at time T subject to a constraint of the risk of portfolio
measured by the variance of the wealth at time T , i.e.
maxtErXTθ s ´ λvarpXTθ qu.
(3.21)
θ
where XTθ is the wealth of the investor at time T if he chooses to invest x0 ´ θ on risk free asset and θ on
risky asset. The wealth XTθ satisfies
XTθ “ erT px0 ´ θq ` θepµ´
σ2
qT `σBT
2
.
Therefore,
ErXTθ s “ erT px0 ´ θq ` θeµ and varpXTθ q “ θ2 e2µT peσ
2T
´ 1q.
Therefore, the portfolio maximization problem (3.21) is given by
)
!
2
max erT px0 ´ θq ` θeµT ´ λθ2 e2µT peσ T ´ 1q .
θ
and the solution is given by
θ˚ “
eµT ´ erT
.
2λe2µT peσ2 T ´ 1q
Volatility estimation
Notice that the log return of the Black-Scholes model satisfies
ln
´S
t`δ
¯
St
“ pµ ´
σ2
qδ ` σpBt`δ ´ Bt q
2
We consider the time discretization t0 “ 0 and ti`1 “ ti ` δ and tN “ t. Therefore, if follows form (B.10)
(quadratic variation of Brownian motion) that
Nÿ
´1
i“0
ln
´S
ti`1
St i
¯2
“ σ2
Nÿ
´1
pBti`1 ´ Bti q2 ` opδq Ñ σ 2 t
i“0
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One can use (3.22) to make an estimation of volatility. For any ti “ iδ evaluate
Yti :“
i´1
ÿ
ln
j“0
´S
tj`1
St j
¯2
.
If δ is small enough, then Yti should approximately be equal to ti σ 2 . This suggests that if we fit a line to the
data tpti , Yti q : i “ 0, 1, ...u. Then, the slope of line is σ 2 . The schematic picture of this fitting is shown in
figure 3.9.
Quadratic variation of log return
data
fitted line: Slope=σ 2
0
0.5
1
Time
Figure 3.9: Quadratic variation estimation of volatility
Exercise 3.12 (Project). With the same set of data from Exercise 3.1, calculate the volatility by using the
1
quadratic variation formula (3.22). Take δ “ 250
year. Then, plot the estimated quadratic variation Yti
versus time ti . Some examples of the the generated plots is given in Figure 3.10. Then fit a line to the data
by assuming Yt “ σ 2 t ` noise. To fit a line to the data points, you can use least square method. The slope
of the line gives you the volatility σ.
·10−2
Quadratic variation of log return
0.2
3
2
0.1
1
0
0
0.5
1
0
0.5
1
Time
Time
Figure 3.10: Quadratic variation estimation of volatility: Left: Tesla motors. Right: S&P500
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3.10 Black-Scholes equation
Recall from Proposition 3.1 that the price V pτ, Sq of the Markovian European contingent claim with payoff
gpST q at time t “ T ´ τ (τ is time-to-maturity) when the asset price is S satisfies
´Bτ V pτ, Sq “ ´
σ2 S 2
BSS V pτ, Sq ´ rSBS pτ, Sq ` rV pτ, Sq.
2
The above equation is a partial differential equation called Black-Scholes equation. One way to find the
pricing formula for an option is to solve the Black-Scholes PDE. As all PDEs a boundary condition and
a initial condition31 is required to solve a PDE analytically or numerically. The initial condition for the
Black-Scholes equation is the payoff of the derivative, i.e.
V p0, Sq “ gpSq.
Notice that here when τ “ 0, we have t “ T .
Notice that the above PDE holds in region τ P r0, T s and S ą 0. Therefore, we need a boundary condition
at S “ 0. This boundary condition is a little trick to make, because in Black-Scholes model, the price of the
asset St never hits zero, i.e. if price of an asset is initially positive, then the price stays positive at all times.
If an asset is price less, the price stays zero at all time. Therefore, if the price of the asset is initially 0, then
ST “ 0 and the price of the option with payoff gpST q is given by
V pτ, 0q “ e´rτ ÊrgpST q | ST ´τ “ 0s “ e´rτ gp0q.
Therefore, the boundary condition for at S “ 0 is given by
V pτ, 0q “ e´rτ gp0q.
To summarize, the Black-Scholes PDE for pricing Markovian European contingent claim with payoff
gpST q is given by the following boundary value problem
$
σ2 S 2
’
&Bτ V pτ, Sq “ 2 BSS V pτ, Sq ` rSBS V pτ, Sq ´ rV pτ, Sq
.
(3.23)
V pτ, 0q
“ e´rτ gp0q
’
%
V p0, Sq
“ gpSq
Exercise 3.13. Consider the option with payoff gpST q “ ?1S . Find the Black-Scholes price of this payoff at
T
time t “ 0 by solving PDE (3.23). Hint: Try to plug in V pτ, Sq “ e´rτ S a into the Black-Scholes equation,
for some constant a. Then, find the constant a. The boundary condition V pτ, 0q “ e´rτ gp0q is unnecessary
as gp0q “ 8.
Remark 3.4. The boundary condition V pτ, 0q “ e´rτ gp0q is some how redundant in equation (3.23).
Because in the Black-Scholes model, the price of underlying asset never hits zero. But, this condition is
important for solving Black-Scholes equation numerically.
31
For time-dependent PDEs.
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Heat equation and Black-Scholes model
Recall from Exercise 3.8 that the change of variable S̃ :“ e´rpT ´τ q S and Ṽ pτ, S̃q :“ e´rpT ´τ q V pτ, Sq “
e´rpT ´τ q V pτ, erpT ´τ q S̃q modifies the Black-Scholes equation to
Bτ Ṽ pτ, S̃q “
σ 2 S̃ 2
B Ṽ pτ, S̃q.
2 S̃ S̃
If we make a change of variable x :“ lnpS̃q and U pτ, xq :“ Ṽ pτ, ex q, then we obtain the heat equation
Bτ U pτ, xq “
σ2
Bxx U pτ, xq.
2
Unlike (3.23), the heat equation holds on the whole space, i.e.
#
2
Bτ U pτ, xq “ σ2 Bxx U pτ, xq
.
U p0, xq
“ e´rT V perT ex q
(3.24)
Notice that at time t “ 0 (or τ “ T ), the price of the the Markovian European contingent claim with payoff
gpST q is equal to V pT, S0 q “ Ṽ pT, S0 q “ U pT, lnpS0 qq.
Transforming the Black-Scholes equation into heat equation helps us to use numerical techniques from
heat equation in Section 2.2 for Black-Scholes equation.
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Solving Black-Scholes equation via finite-difference scheme for Black-Scholes equation
One can directly discretize Black-Scholes equation (3.23) to apply finite difference method described in
the previous section. See Figure 3.11 In this case, the computation domain has to be r0, Smax s for some
Smax ą 0. The boundary condition at 0 is already assigned at (3.23) by
V pτ, 0q “ e´rτ gp0q,
and the ABC at Smax is given by the growth of the payoff for large values of S. For the example of call
option, the ABC is
V pt, Smax q “ Smax for Smax sufficiently large.
The rest of the approximation goes the same as the previous section. However, one should be cautious
about applying the explicit schemes which need CFL condition. Recall that The right hand side of CFL
condition is always 21 times the inverse of the coefficient of second derivative in the equation. Therefore,
CFL condition translates to
1
h
ď 2 2.
k2
σ S
If Smax is very large, for a fixed space discretization k, one needs to set h very small. The downside of this
method is that the time of the algorithm increases significantly.
S
Smax
k
T
h
t
Figure 3.11: Finite difference grid for Black-Scholes equation. In the explicit scheme the CFL condition
1
should be satisfied, i.e. kh2 ď σS
. This requires a choice of very small h. Artificial boundary conditions are
necessary on both Smax and 0.
Binomial tree scheme for Black-Scholes
Recall from Section 3.1 that the Black-Scholes model is the limit of binomial model under risk neutral
probability. Therefore, if necessary, one can use the binomial model to approximate the Black-Scholes price
of the contingent claim. For implementation, one needs to choose a sufficiently large number of periods N ,
T
or equivalently a small δ “ N
. Suggested by (3.1), for a given interest rate r and volatility σ, we need to
choose l and l and the one period interest rate R as follows.
u “ eδpr´
σ2
q`
2
?
δσα
, l “ eδpr´
σ2
q´
2
?
δσβ
, and R “ rδ,
where α and β are given by (3.4). However, one can avoid the calculation of α and β by making some
different choices. Notice that the binomial model has three parameters u, l and R while the Black-Scholes
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parameters are only two. This degree of freedom provides us with some modifications of binomial three
which still converges to the Black-Scholes formula. This also simplifies the calibration process in Section 3
significantly simpler. Here are some choices:
a) Symmetric probability:
u “ eδpr´
σ2
q`
2
?
δσ
Then;
, l “ eδpr´
σ2
q´
2
?
δσ
, and R “ rδ,
1
π̂u “ π̂l “ .
2
b) Subjective return:
u “ eδν`
?
δσ
, l “ eδν´
?
δσ
, and R “ rδ,
Then;
1
π̂u “
2
˜
? r ´ ν ´ 1 σ2
2
1` δ
σ
¸
1
and π̂l “
2
˜
? r ´ ν ´ 1 σ2
2
1´ δ
σ
¸
.
To see why this choices work, recall (3.4) from Section 3.1. The only criteria for the convergence of binomial
model to Black-Scholes model is that when we write the log of asset price in binomial model as in (3.4),
lnpST q “ lnpS0 q ` pr ´
N
?
σ2
1 ÿ
qT ` σ T ¨ ?
Zk ,
2
N k“1
the random variables Zk , k “ 1, ..., N must satisfy
ÊrZ1 s “ opδq, and ÊrZ12 s “ 1 ` op1q.
In the Section 3.1, we made a perfect choice of ÊrZ1 s “ 0 and ÊrZ12 s “ 1.
Monte Carlo scheme for Black-Scholes
Recall from Section 3.2 that the price of a Markovian European contingent claim can be written as an
expectation or/and a single integral.
ˆ
˙
ż
? ˘
`
σ2
x2
e´rτ 8
g S epr´ 2 qτ `σ τ x e´ 2 dx.
V pτ, Sq :“ e´rτ ÊrgpSτ q | S0 “ Ss “ ?
2π ´8
One way to estimate V pT, S0 q is to generate a sample xp1q , ..., xpM q of N p0, 1q and approximate the above
expectation by
ˆ
˙
M
` pr´ σ2 qT `σ?T xpjq ˘
e´rT ÿ
2
g S0 e
.
M j“1
Another method of approximation is to directly target the integral by choosing a large numbers 0 ă xmax
and approximate the integral by a finite Riemann sum. More precisely, let ∆x :“ xmax
L and xpjq “ j∆x.
The the approximation goes as follows.
e´rT
L´1
ÿ
ˆ
˙ x2
?
`
˘
pjq
σ2
g S0 epr´ 2 qT `σ T xpjq e´ 2 ∆x.
j“´L
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There are more complicated and efficient Monte Carlo schemes and integration methods which we do not
discuss here.
Exercise 3.14 (Project). Consider a payoff bull spread call in Figure 3.8 with T “ 1, K1 “ 10 and K2 “ 20.
Assume that the parameters of the underlying asset are given by S0 “ 15, σ “ .02, and the interest rate
is r “ .01 (1%). We know the actual value of this option from Black-Scholes formula. Now, compare the
following estimation schemes for the price of bull spread call. Record the time of the algorithms for each
scheme to obtain the four digit accuracy.
a) Implicit finite-difference with parameter θ “ .5.
b) Implicit finite-difference with parameter θ “ 1.
c) Explicit finite difference (equivalently implicit with θ “ 0).
d) Symmetric binomial model.
e) Monte Carlo scheme.
f) Approximation by a Riemann sum.
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3.11 Stock price with dividend in Black-Scholes model
Here we consider two types of dividend strategies. If the dividend is paid continuously, then there is a
constant outflow of cash from the price of the asset. If the rate of dividend payment is Dt , the Black-Scholes
model in (3.20) has to be modified to
dSt “ rSt dt ` σdBt ´ Dt dt.
Choosing Dt :“ qSt for q ě 0, we obtain
dSt “ pr ´ qqSt dt ` σSt dBt .
Remark 3.5. For a continuous dividend rate q the dividend yield in time period rt, T s is given by e´qpT ´tq ˆ
100 percent.
Especially, this choice guarantees that the dividend is always less than the asset price and paying dividend
does not diminish the value of the asset. In the exponential form we have
˙
ˆ
σ2
St “ S0 exp ppr ´ qqt ` σBt q “ e´qt S0 exp pr ´ qt ` σBt .
2
In this case, the Black-Scholes price of a European contingent claim with payoff gpST q is given by
” ´
¯
ı
V pt, Sq “ e´rpT ´tq ÊrgpST q | St “ Ss “ e´rpT ´tq Ê g e´qpT ´tq S̄T | S̄t “ S ,
where S̄t satisfies the Black-Scholes equation without dividend, i.e.
ˆ
˙
σ2
S̄t “ S0 exp pr ´ qt ` σBt .
2
If the dividend strategy is time-varying qt , then one can write the above pricing formula as
” ´ şT
¯
ı
V pt, Sq “ e´rpT ´tq Ê g e´ t qs ds S̄T | S̄t “ S ,
For the continuous rate of dividend, the relation between Greeks of the option in Proposition 3.1 is modified
as in the following proposition.
Proposition 3.2. Let St follow Black-Scholes price with continuous dividend rate q. For a European Markovian contingent claim, the Black-Scholes price satisfies
Θpτ, Sq “ ´
σ2 S 2
Γpτ, Sq ´ pr ´ qqS∆pτ, Sq ` rV pτ, Sq.
2
Second type of dividend strategy is a discrete one. The discrete dividend is paid in times 0 ď t0 ă t1 ă
... ă tN ď T in amounts D0 , ..., DN . Then, between the time of dividend payments, the asset price follows
the Black-Scholes model, i.e.
ˆ
˙
σ2
St “ Stn´1 exp pr ´ qpt ´ tn´1 q ` σpBt ´ Btn´1 q , t P rtn´1 , qtn s.
2
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Just a moment before the payment of dividend at time tn ´32 the price of the asset is
ˆ
˙
σ2
Stn ´ :“ Stn´1 exp pr ´ qptn ´ tn´1 q ` σpBtn ´ Btn´1 q .
2
But after paying Dn for dividend, this price reduces to
ˆ
˙
σ2
Stn “ Stn ´ ´ Dn “ Stn´1 exp pr ´ qptn ´ tn´1 q ` σpBtn ´ Btn´1 q ´ Dn .
2
As mentioned in Section 2.4, the dividend is usually given as a percentage of the current asset price, i.e.
Dn “ dn Stn ´ , with dn P r0, 1q.
and we have
˙
σ2
“ p1 ´ dn qStn´1 exp pr ´ qptn ´ tn´1 q ` σpBtn ´ Btn´1 q .
2
ˆ
Stn “ p1 ´ dn qStn ´
Remark 3.6. For a discrete dividend, the dividend yield in time period rtn´1 , tn s is dn ˆ 100 percent.
Therefore,
N
ź
σ2
ST “
p1 ´ dn qS0 exp pr ´ qT ` σBT
2
n“1
ˆ
˙
.
Remark 3.7. If at time T there is a dividend payment, the payoff of a contingent claim gpST q takes into
account the price ST after the payment of dividend. In short, the time is always ex-dividend .
Proposition 3.3. Let St follow Black-Scholes price with discrete dividend policy given by d0 , ..., dN P r0, 1q
at times 0 ď t0 ă t1 ă ... ă tN ď T . For a European Markovian contingent claim, the Black-Scholes
price satisfies
Θpτ, Sq “ ´
σ2S 2
Γpτ, Sq ´ rS∆pτ, Sq ` rV pτ, Sq for τ P pT ´ tn , T ´ tn´1 q.
2
Then, the Black-Scholes price of a contingent claim with payoff gpST q is given by
” ´ ź
¯ˇ
ı
ˇ
V pt, Sq “ e´rpT ´tq ÊrgpST q | St “ Ss “ e´rpT ´tq Ê g
p1 ´ dn qS̄T ˇS̄t “ S ,
tătn ďT
where S̄t satisfies the simple no-dividend Black-Scholes model.
Exercise 3.15. Consider a portfolio of one straddle option with K “ 10 and one strangle option with
K1 “ 8 and K2 “ 14, both maturing at T “ 1. See Figure 3.7. Assume that the underlying asset has
parameters S0 “ 2, σ “ .2, and it y pays 4% quarterly dividend. The interest rate is r “ .01 (1%). Find
the price of this portfolio and it ∆ and Γ at time t “ 0.
Remark 3.8. The dividend strategies are sometimes not known upfront and therefore should be modeled by
random variables. If the dividend policy is a random policy which depends on the path of the stock price,
then, the pricing formula will be more complicated even in Black-Scholes model.
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gpST q
K2 ´K1
2
K1
K1 `K2
2
K2
ST
Figure 3.12: Butterfly spread payoff
Exercise 3.16 (Butterfly spread). Consider the payoff gpST q shown in Figure 3.12.
Consider Black-Scholes model for the price of a risky asset with T “ 1, r “ .04 and σ “ .02 and dividends
are paid quarterly with dividend yield %10. Take S0 “ 10, K1 “ 9, and K2 “ 11. Find the Black-Scholes
price, ∆, Γ, ρ, and V of this option at time t “ 0. Find Θ at time t “ 0 without taking derivatives with
respect to S.
Black-Scholes equation with dividend
In the case of dividend, the Black-Scholes equation in (3.23) is given by
$
σ2 S 2
’
&Bτ V pτ, Sq “ 2 BSS V pτ, Sq ` pr ´ qqSBS pτ, SqV pτ, Sq ´ rV pτ, Sq
.
V pτ, 0q
“ e´rτ gp0q
’
%
V p0, Sq
“ gpSq
This also allows us to solve the derivative pricing problem with more complicated dividend strategies. Let’s
assume that the dividend payment rate at time t is a function qpt, St q. Then, the Black-Scholes model with
dividend is given by the SDE
dSt “ St pr ´ qpt, St qqdt ` σSt dBt .
and Black-Scholes equation in (3.23) is given by
$
σ2 S 2
’
&Bτ V pτ, Sq “ 2 BSS V pτ, Sq ` pr ´ qpt, SqqSBS V pτ, Sq ´ rV pτ, Sq
.
V pτ, 0q
“ e´rτ gp0q
’
%
V p0, Sq
“ gpSq
After the change of variables described in Section 3.10, we obtain
#
2
Bτ U pτ, xq “ σ2 Bxx U pτ, xq ´ qpT ´ τ, erpT ´τ q ex qBx U pτ, xq,
.
U p0, xq
“ e´rT V perT ex q
The above equation is a heat equation with a drift term given by qpT ´ τ, erpT ´τ q ex qBx U pτ, xq.
Exercise 3.17 (Project). Consider a European call option with T “ 1 and K “ 2. Assume that the
parameters of the underlying asset are given by S0 “ 2, σ “ .2, and the interest rate is r “ .01 (1%). In
32
The moment before time t is denoted by t´.
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3 BLACK-SCHOLES MODEL
addition assume that the underlying asset pays dividend at continuous rate qpt, St q “ .05e´.01t St , i.e. 5%
of the discounted asset price at each time.
a) Write the Black-Scholes equation for this problem and convert it into heat equation with a drift term.
b) Solve this problem numerically by using a finite difference scheme.
Example 3.2. Consider a European option with payoff
gpST q “ eαST .
Assume that the the interest rate is r ą 0 and the volatility of the underlying asset is σ ą 0, and at time
0, it has value S0 , and pays dividend at continuous rate equal to qpt, St q “ qSt , where q ą 0. Then, the
Black-Scholes equation is given by
#
2 2
Bτ V pτ, Sq “ σ 2S BSS V pτ, Sq ` pr ´ qSqSBS V pτ, Sq ´ rV pτ, Sq
.
V p0, Sq
“ eαS .
The boundary condition for Black-Scholes equation at S “ 0 is given by
V pτ, 0q “ 1.
Function V pτ, Sq “ e´rτ eαS satisfies the Black-Scholes equation and the boundary conditions. Therefore,
the price of the contingent claim with payoff eαST at time-to-maturity τ (at time T ´τ ) is given by V pτ, Sq “
e´rτ eαS if the asset price takes value S at time time.
Exercise 3.18. Consider a European option with payoff
gpST q “ ST´5 e10ST .
Assume that the the interest rate is r “ .1 and the underlying asset satisfies S0 “ 2, σ “ .2, and pays
dividend at continuous rate equal to qpt, St q “ .2St .
a) Write the Black-Scholes equation for this problem.
b) Solve this problem analytically by the method of separation of variables. Plug into the equation a
solution candidate of the form eατ S ´5 e10S and determine α.
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4 VOLATILITY SMILE
Term structure of the volatility and volatility smile
For simplicity we assume that the interest rate is estimated to be constant. Then, given constant volatility σ,
the call option price is given by Black-Scholes formula.
Cpτ, K, S, σ, rq :“ SΦ pd1 q ´ e´rτ KΦ pd2 q ,
˙
ˆ
˙
1
σ2
1
σ2
d1 “ ?
lnpS{Kq ` pr ` qτ
and d2 “ ?
lnpS{Kq ` pr ´ qτ .
σ τ
2
σ τ
2
?
Since Bσ Cpτ, K, S, σ, rq “: V “ S τ Φ1 pd1 q is positive, then Cpτ, K, S, σ, rq is strictly increasing on σ. In
addition, we have
ˆ
lim Cpτ, K, S, σ, rq “ pS ´ e´rτ Kq` and
σÓ0
lim Cpτ, K, S, σ, rq “ S.
σÒ8
The following exercise shows that the price quotes of call option should be in the range rpS ´ e´rτ Kq` , Ss,
independent of the model.
Recall from Section 2 that the price of a call option satisfies
pS ´ Bt pT qKq` ď CpT, K, S, tq ď S.
independent of the model. Therefore, for any price quote C ˚ pτ, K, Sq of call option with strike K and
time-to-maturity τ , there exists a unique σ imp pτ, K, Sq such that
Cpτ, K, S, σ imp , rq “ C ˚ pτ, K, Sq.
σ imp pτ, K, Sq is called implied volatility. See Figure 4.1
C(τ, K, σ)
S
C ∗ (τ ,K)
(S − e−rτ K)+
σ
σ imp (τ ,K)
Figure 4.1: Implied volatility: the price of call option in Black-Scholes model is monotone on volatility σ.
Knowing S, σ imp is a function of τ and K. If Black-Scholes model is true, σ imp must be a constant
function, however this is not compatible with the data. For example, Figure 4.2 shows the implied volatility
for Tesla motor stock price which obviously is not constant. For closer maturities, the volatility is larger
than later maturities. In addition, for nearly ATM options, K « $175, the volatility is larger. In Figure 4.3,
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Summary of the lecture
Price
Bid
Ask
Volume
Open Interest
Strike
1.79
1.77
1.80
4636
6272
106.00
Table 5: A row of the call option data quote for FB. Source Google Finance. The large volume of trade and
open interest indicates that the price quote is probably more recent and more reliable.
Implied volatiltiy
we isolated some cross-sections of the volatility surface for time-to-maturity equal to 2 weeks and 4 months
and strike price equal to $185.
.8
.6
.4
.2
220
210
200
190
4 Mo
K
180
1.5 Mo
1 Mo
170
few days
τ
σ imp
Figure 4.2: Implied volatility surface at Feb 24, 2016 for Tesla stock NASDAQ:TSLA.
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
190
200
210
220
0.2
170
180
190
K
200
K
210
220
0
0.1
0.2
0.3
τ
Figure 4.3: Implied volatility at Feb 24, 2016 for Tesla stock NASDAQ:TSLA. Left τ “ 4 Months, middle
τ “ 2 weeks, and right K “ $185.
Exercise 4.1 (Project). Go to google finance, yahoo finance or any other database which provides free
option price data for one instant of time. In google finance it is often referred to as option chain. Pick a
stock which has a rich option chain with many quotes for different maturities and strike, such as IBM, Apple,
Alphabet, FB, etc. Then, single out rows of option price quotes which exhibit most recent option price which
show large values in traded volume and open interest. See Table 5.
Choose interest rate r from the data given on Table 6. Try to use options with maturities as close as possible
to the maturity of the yield curve for more accurate estimation.
Then, use Newton-Raphson method or bisection method to approximate the implied volatility from the call
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Summary of the lecture
date
02/24/16
1 Mo
0.28
3 Mo
0.33
6 Mo
0.46
1 Yr
0.55
2 Yr
0.75
Table 6: Annual yields of the government bonds. Source https://www.treasury.gov.
option data. Since you have yield curve data, use (3.18) as the Black-Scholes pricing formula.Create a chart
similar to the one below to insert the implied volatility inside.
First strike K1
Second strike K2
...
First maturity T1
Second maturity T2
..
.
Notice that, due to lack of data, you may have some cells unfilled at the end.
Unlike Black-Scholes market, the real market does not exhibit constant volatility behavior. Therefore, the
implied volatility is a non-constant function of strike K and time-to-maturity τ . The implied volatility as a
function of K is referred to as volatility smile33 . As you can see, the data is by no means generating a nice
implied volatility surface and a volatility smile. In practice, they fit some parametric surface to the implied
volatility. Unlike the data, the parametric surface is more consistent with the theoretical call option price
and implied volatility. In Section 4.1, we relax the assumption of constant volatility and introduce models
which can match with the volatility surface.
Some of the empirical facts about volatility smile can be observed from Figures 4.3 and 4.2. As a function
of K, the smile is more visible for shorter maturities and flattens out for longer maturities and it is decreasing
as time to maturity increases. As K grows away from the ATM value (here approximately $175), the smile
decreases flat.
Notice that modeling a future volatility surface “greatly simplifies the task of producing the future
option prices: since a smiley volatility is the wrong number to put in the wrong formula to get the
right price, we do not need a complex pricing model, and, by virtue of the very definition of smile, the
Black formula can be directly employed to calculate call prices”.34
Remark 4.1. In Section 3.9, we used quadratic variation to estimate the volatility in Black-Scholes model.
This estimation is based on the Assumption 3.1 of i.i.d. returns. Although volatility smile challenges this
problem, it is still possible to get a linear pattern in quadratic variation of the log return. For example, for
Tesla motors, despite the volatility smile in Figure 4.2, the quadratic variation shows a linear pattern, as
shown in Figure 3.10.
33
34
It sometimes looks like a smile :).
Quoted from [24, Section 17.2].
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4 LOCAL VOLATILITY
Summary of the lecture
Local volatility models
Recall from Section 4 that assumptions such as (3.1) fails to be valid and therefore, the Black-Scholes model
is not a good match for the data. See Figures 4.2 and 4.3. One way of capturing the volatility smile is to
introduce local volatility by assuming that the volatility is a function of the asset price, i.e. σpSt q. This leads
to a modification of the Black-Scholes model which is given by
dSt
“ rSt dt ` σpSt qdBt .
St
(4.1)
Constant elasticity of variance model
Constant elasticity of variance (CEV) represents a two-parameter class of local volatility models. In CEV,
the local volatility is chosen to be
σpSq “ σ0 S β σ0 ą 0 and β ą ´1.
If β “ 0, then the CEV model reduces to Black-Scholes model. If β ą 0, the volatility increases with
the asset price, which is referred to as leverage effect. Leverage effect is often observed in equity market.
Otherwise, ´1 ă β ă 0 represents inverse leverage effect. Inverse leverage effect usually happens in
commodity markets.
Calibration of local volatility models
Calibration of local volatility models is not an easy job. For example, assume that we know the volatility
is given by function σpSt q. Then, the quadratic variation of the asset price is given by a formula similar to
(3.22), i.e. for time discretization t0 “ 0 and ti`1 “ ti ` δ and tN “ t, we have
Nÿ
´1
i“0
ln
´S
ti`1
St i
¯2
“
Nÿ
´1
σ 2 pSti qpBti`1 ´ Bti q2 ` opδq Ñ
żt
σ 2 pSu qdu.
(4.2)
0
i“0
Unlike (3.22), where the limit is σ 2 t, the right hand side above is random, because Su is random. One may
think of trying to regress the summation of log returns on the discretized path S0 , St1 , ..., StN ´1 in order to
see some pattern. However, this is a high dimensional regression which is not a good idea. A better idea
could be to plot
¯
´S
ti`1 2
ln
“ σ 2 pSti qpBti`1 ´ Bti q2 ` op1q.
St i
against St to see if there is a pattern. The pattern can suggest a shape for the function σ 2 pSq. Notice that the
pBti`1 ´ Bti q2 term in the above right hand side is δ times a χ2 random variable. This makes it impossible
to apply simple statistical estimations. On the other hand, the a simple experiment shows that it sometimes
fails to give a reliable pattern. For instance, Figure 3.10 shows a clear pattern of approximately linear growth
in the quadratic variation of log return against time. If we try to do the same thing against the stock price St ,
we get Figure 4.4.
A better way to calibrate local volatility model is to go back to where it is coming from, i.e. the volatility
surface. A famous work of Bruno Dupire in [10] (see also [16])gives the local volatility which matches with
the volatility surface.
Theorem 4.1 (Dupire’s theorem). Suppose that at time t the underlying asset price is S. In addition, the
volatility surface is given, i.e. all call prices are given by CpK, T, S, tq.Then, there is at most one local
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Summary of the lecture
volatility model which matches with the given call prices. In this case, the volatility is given by
ˇ
BT C ` rKBK C ˇˇ
2
,
σ pt, sq “ 2
K 2 BKK C ˇpT,Kq“pt,sq
Quadratic variation of log return
provided that the derivatives BT C, BK C, and BKK C exists.
3
0.4
0.3
2
0.2
1
0.1
0
0
140 160 180 200 220 240 260 280
1,800 1,850 1,900 1,950 2,000 2,050 2,100 2,150
S
Figure 4.4: Plot of quadratic log return ln
S
´S
ti`1
¯2
Sti
against St . Left: Tesla motors. Right: S&P500.
As seen from Exercise 2.4, CpT, K, S, tq is convex with respect to K. Therefore, for the denominator we
have K 2 BKK CKK ě 0. The following exercise shows that independent of model, the numerator BT C `
rKBK C ě 0. Here we assume that the risk free interest rate is constant r.
Exercise 4.2. Show that
CpT ` δ, BT pT ` δqK, S, tq ě CpT, K, S, tq.
Therefore, the task of calibration can be simplified if we choose to fit a differentiable vol surface to the
implied volatility data which satisfies the following conditions
a) K 2 BKK CKK ą 0.
b) BT C ` rKBK C ě 0.
In addition, if the volatility surface is stationary over time, i.e. CpT, K, S, tq depends on τ “ T ´ t, then
the local volatility does not depend on t and we have
ˇ
Bτ C ` rKBK C ˇˇ
2
σ pSq “ 2
.
K 2 BKK C ˇpτ,Kq“p0,sq
Exercise 4.3. Consider a volatility surface given by CpT, K, S, tq “ e´αpT ´tq K β , where α and β are given
constants. What choices of α and β satisfy existence conditions of the local volatility (a) and (b)? Find the
local volatility for this model. Do you know this model?
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Summary of the lecture
5 AMERICAN OPTIONS
American options
American options are boosted version of European options that give their owner the right but not obligation
to exercise any date before or at the maturity. Therefore, at the time of exercise t1 P r0, T s, an American call
option has value equal to pSt1 ´ Kq` . The exercise time is determined by comparing the continuation value
to the exercise value pSt1 ´ Kq` . We will discuss this in details in the future. We use CAm pT, K, S, tq and
PAm pT, K, S, tq to denote the price of american call and american put at time t when the underlying asset
price is S with maturity T and strike K, respectively.
Remark 5.1. Consider an American option with payoff gpt, Sq, i.e. if the holder chooses to exercise the
option at time t when the underlying asset price is S , then the owner received gpt, Sq. At time T , we have
to different settings; the holder has right to exercise or she has the obligation. If she has the right, she
will never exercise when gpT, Sq ď 0. It is also true for all intermediate times t P r0, T q. Therefore, one
can simply assume that the payoff is g` pt, Sq :“ maxt0, gpt, Squ. For a non-negative payoff the right or
obligation to exercise does not make any difference; because the holder never chooses to exercises when the
payoff is negative and the obligation to exercise at time T is never against her will. Therefore, without loss
of generality, we can assume that the exercise at T is obliged.
5.1
Pricing American option in binomial model
The key to the pricing of American options is to compare to values at each node of the tree, i.e. continuation
value and exercise value. We will properly define these values in this section and use them to price American
option. We first present the pricing method in the naive case of one-period binomial model.
Example 5.1. Consider a one-period binomial model with S0 “ 1, u “ 2.1, l “ .6 and R “ .1 (for
simplicity). We consider an American put option with strike K; the payoff is pK ´ Sq` . Similar to the
European put, at the terminal time T “ 1, the value of the option is known. However, at time t “ 0 we are
facing a different situation; we can choose to exercise and get the exercise value of E :“ pK ´ 1q` , or we
can continue. If we continue, we will have a payoff of pK ´ 2.1q` or pK ´ .6q` depending of the future
events. The value of continuation is obtain via taking risk neutral expectation:
ˆ
˙
1
1
2
1
ÊrpK ´ ST q` s “
pK ´ 2.1q` ` pK ´ .6q` .
C :“
1`R
1.1 3
3
pK ´ 2.1q`
maxtC, Eu
pK ´ .6q`
Figure 5.1: American put option in one-period binomial model
Next, we should compare continuation value C and exercise value E. If C ą E, we should not exercise
at time 0 and should wait until time 1. Otherwise when C ď E, it is optimal to exercise. For example
when K “ 3, we have C “ 1.7273 ă E “ 2 and thus we exercise. However, when K “ 1, we have
C “ .2424 ą E “ 0 and thus we do not exercise.
As seen in Figure 5.2, only American put options with strike K in p.6, 21
13 q generate a larger continuation
value and therefore should not be exercises at time 0.
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Summary of the lecture
0.6
21
13
K
2.1
`1
˘
1
2
Figure 5.2: One-period binomial model: continuation value C “ 1.1
3 pK ´ 2.1q` ` 3 pK ´ .6q` (gray)
and exercise value E “ pK ´ 1q` (red) of an American put option as a function of strike price K.
To illustrate more, we consider a two period binomial model in the following example.
Example 5.2. Consider a two-period binomial model with the same parameters as in Exercise 5.1, i.e.
u “ 2.1, l “ .6 and R “ .1. We consider an American put option with strike K; the payoff is pK ´ Sq` .
The terminal time T “ 2, the value of the option is known. At time t “ 1, there are two nodes and at time
t “ 0 there is one node, at each of which we have to find exercise value and continuation value. The exercise
values are given by the payoff as seen in Figure 5.3
pK ´ 4.41q`
maxtC1,2 , pK ´ 2.1q` u
pK ´ 1.26q`
maxtC0 , Eu
maxtC1,1 , pK ´ .6q` u
pK ´ .36q`
Figure 5.3: American put option in two-period binomial model
Here C0 , C1,1 , and C1,2 are the continuation values at nodes t “ 0, t “ 1 price-down, and t “ 1 price-up,
respectively. Next, we should compare continuation value and exercise value at each node in a backward
manner. At time t “ 1, the continuation value is
ˆ
˙
1
1
2
1
C1,2 “
ÊrpK ´ ST q` | S1 “ 2s “
pK ´ 4.41q` ` pK ´ 1.26q`
1`R
1.1 3
3
ˆ
˙
1
1
1
2
1
C1,1 “
ÊrpK ´ ST q` | S1 “ s “
pK ´ 1.26q` ` pK ´ .36q`
1`R
2
1.1 3
3
Therefore, the price of the option at these nodes are
"
ˆ
1
1
V1,2 “ max
pK ´ 4.41q` `
1.1 3
ˆ
"
1
1
pK ´ 1.26q` `
V1,1 “ max
1.1 3
˙
*
2
pK ´ 1.26q` , pK ´ 2.1q`
3
˙
*
2
pK ´ .36q` , pK ´ .6q`
3
As you see from Figure 5.4, if we choose the strike of the option K in p1.26, 3.3q, then at node p1, 2q the
continuation value is larger, otherwise we exercise at this node. If we choose K in p.36, 12.6
13 q, at node p1, 1q
the the continuation value is larger.In this example, values of K that imposes the continuation in node p1, 1q
is disjoint from those which imposed continuation at node p1, 2q. Therefore, we exercise the American option
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Summary of the lecture
1.26 2.1
3.3
Node p1, 2q
.36 .6
4.41 K
12.6
13
1.26
K
Node p1, 1q
Figure 5.4: Two-period binomial model: at each nodes at time t “ 1, continuation value (gray) is compared
to exercise value (red) as a function of strike price K.
on at least one of these nodes.
Now let fix our put option by choosing K “ 2. Then, we continue at node p1, 2q and exercise at node p1, 1q.
Therefore, at time t “ 1, the option takes values
V1,2 “ C1,2 « 0.4485
and
V1,1 “ 1.4.
At time t “ 0, we need to see if it is optimal to exercise or if it is optimal to continue. The situation at node
p0, 0q is explained in Figure 5.5. The exercise value is E “ 1 but the continuation value is positive, i.e.
C “ .9844. Therefore, it is optimal to exercise.
0.4485
maxtC, Eu “ E “ 1
1.4000
Figure 5.5: American put option at time t “ 0 in two-period binomial model of Example 5.2.
Remark 5.2. As a general rule, when it is optimal to exercise, i.e. the continuation value is less than or
equal to exercise value, there is no point in continuing the option. This is because the continuation value
remains equal to exercise value since then after. This can be observed from Example 5.2 by taking for
example K “ 1. The schematic pattern of exercise and continuation nodes is presented in Figure 5.6.
Exercise 5.1. In Example 5.2, take the following values of K.
a) K “ 1.
b) K “ 3.
Similar to the European option, we start from the terminal time; at time T the American option must be
exercised, as explained in Remark 5.1. Therefore, the price of the contingent claim at maturity time T is
equal gpT, ST pjqq if the state j occurs. Therefore at time T , the American option costs gpT, ST q if the
j
Then, assume that the price of contingent claim Vt`1
is known at time t ` 1 for all states j “ 0, ..., t ` 1
and is a function V of time t ` 1 and asset price St`1 pjq, i.e. V pt ` 1, St`1 pjqq. Consider state i at time
t. In order to know if we should exercise or continue holding the option till next time step, we need to see
which one of continuation value or exercise value is larger. The exercise value at state i and time t is simply
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5 AMERICAN OPTIONS
(4.41,0)
(4.41,0)
(2.1,0)
(1,.2424)
(2.1,.4485)
(1.26,0)
(1,1)
(.6,.4)
K“1
(1.26,.74)
(.6,1.4)
(.36,.46)
K“2
(.36,1.64)
Figure 5.6: The pattern of continuation versus exercise in a two-period binomial model in Example 5.2.
The red nodes are the exercise nodes and blue nodes are continuation nodes. In the pair pa, bq, a is the asset
price and b is the American option price.
given by the payoff gpt, St piqq. If we choose not to exercise, the state of the market at time t ` 1 is either
i ` 1 or i. Therefore, the continuation value is given by
1
1
pVt`1,i`1 π̂u ` Vt`1,i π̂l q “
ÊrV pt ` 1, St`1 q | St “ St piqs.
1`R
1`R
(5.1)
The key observation in pricing American option is that when the continuation value is larger than the
exercise value, we should not exercise. In other words, the value of the option at time t and state i is given
by the maximum of the exercise value and continuation value, i.e.
"
*
1
V pt, St piqq “ max gpt, St piqq,
ÊrV pt ` 1, St`1 q | St “ St piqs .
1`R
The above discussion suggests the following algorithm in pricing American options.
Pricing American options in binomial model
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
At time T , the value of the option is gpT, ST pjqq.
for each t “ T ´ 1, ..., 0 do
for each j “ 1, ..., t ` 1 do
Exercise value = gpt, St pjqq.
˘
` i`1
1
1
i π̂ .
ÊrV pt ` 1, St`1 q | St “ St piqs “ 1`R
Vt`1 π̂u ` Vt`1
Continuation value = 1`R
l
!
)
1
The value of the option V pt, St piqq “ max gpt, St piqq, 1`R
ÊrV pt ` 1, St`1 q | St “ St piqs .
If V pt, St piqq “ gpt, St piqq, we exercise the option and stop.
If V pt, St piqq ą gpt, St piqq, we continue and the replicating portfolio is given by ∆t pSt pjqq units of
risky assets and V pt, St piqq ´ ∆t pSt pjqqSt pjq is cash.
end for
end for
Remark 5.3 (Path dependent American option). If the payoff of the American option depends on the path,
one can adjust the algorithm by considering the path dependent continuation value and path dependent
exercise value. Exercise 5.2 provides an example of such kind.
Example 5.3 (American call option on a non-dividend asset does not exist!). In this case the payoff is given
by gpt, Sq “ pS ´ Kq` . Then, the price of the American call option is the same as the price of European
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5 AMERICAN OPTIONS
call option! In fact, this is true in any model where the pricing of European claims is carried by the risk
neutral probability. See Proposition 5.2.
Before explaining the above exercise further, we want to introduce the exercise policies.
Definition 5.1 (Exercise policy). Exercise policy (or stopping time) is a random time τ such that at time t
the exercise decision τ “ t only depends on the paths of the underlying asset before or at time t.
For example, the above algorithm to price American options is suggesting optimal exercise policy for
American option is the first time that the continuation value is less than or equal to the exercise value, i.e.
*
"
1
˚
ÊrV pt ` 1, St`1 q | St “ St piqs .
T :“ inf t ď T : gpt, St piqq ě
1`R
Notice that T ˚ is a random time and the event T ˚ “ t depends on the paths of asset price at time t. Thus, T ˚
is a stopping time or exercise policy. The following proposition describes that the price of American option
is maximizer among all exercise policies.
Proposition 5.1. Consider an American option with payoff gpt, Sq. Then, the binomial price of American
option at time t is given by
ˇ
ˇ
„
ȷ
„
ȷ
ˇ
ˇ
1
1
ˇ
ˇ
˚
V pt, Sq “ sup Ê
gpτ, Sτ qˇSt “ S “ Ê
gpτ, ST qˇSt “ S ,
p1 ` Rqτ ´t
p1 ` RqT ˚ ´t
τ
where the supremum is over all stopping times τ .
One can equivalently describe optimal stopping time T ˚ by the first time that the value of the American
option is equal to the payoff, i.e.
T ˚ :“ inf tt ď T : gpt, St piqq “ V pt, St qu .
More precisely, we start with a continuation value strictly greater than the payoff. Over the course of time,
if the value of the option hit down the exercise value, we exercise and receive the payoff.
Proposition 5.2. Consider an American option with a convex payoff gpSq such that gp0q “ 0 on an asset
which pays no dividend. Assume that the price of a European contingent claim is given by
V Eu pt, Sq “
1
ÊrgpST q | St “ Ss,
Bt pT q
St
where Ê is expectation under risk neutral probability, and the discounted asset price S̃t “ p1`Rq
t is a
Am
martingale under risk neutral probability. Then, the price of American option V pt, Sq with payoff gpSq
is the same as V Eu , i.e.
V Eu pt, Sq “ V Am pt, Sq.
Remark 5.4. Proposition 5.2 is only true when the underlying asset does not pay dividend. See Exercise
5.3.
Exercise 5.2. Consider a two-period binomial model for a risky asset with each period equal to a year and
take S0 “ $1, u “ 1.5 and l “ 0.6. The interest rate for both periods is R “ .1.
a) Price an American put option with strike K “ .8.
b) Price an American call option with strike K “ .8.
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5 AMERICAN OPTIONS
c) Price an American option with a path dependent payoff which pays the running maximum35 of the
path.
Remark 5.5. A naive example of an American option is the case where the payoff is $1. In this case, if the
interest rate is positive, it is optimal to exercise the option right away. However, if interest rate is 0, the time
of exercise can be any time. This naive example has an important implication. If interest rate is zero, one
can remove condition gp0q “ 0 from Proposition 5.2, simply by replacing payoff g by g̃pSq “ gpSq ´ gp0q.
Since cash value of gp0q does not change value over time, the value of the American option with payoff gpSq
is gp0q plus the value of American option with payoff g̃pSq.
For negative interest rate, the exercise date will be postponed compared to the positive interest rate.
For example, if the interest rate is zero, the price of American put is equal to the price of European put.
Hedging American option in binomial model
Hedging American option in binomial model follows the same way as European option. The only difference
is that the hedging may not continue until maturity because of the early exercise. Given that we know the
price V pt ` 1, St`1 piqq of the American option at time t ` 1 at all states i “ 1, ..., t ` 2, to hedge at time
t and state j, we need to keep ∆t pSt pjqq units of risky asset in the replicating portfolio and V pt, St piqq ´
∆t pSt pjqqSt pjq in cash, where ∆t pSq is given by (2.7), i.e.
∆pt, Sq :“
V pt ` 1, Suq ´ V pt ` 1, Slq
for t ă T ˚ .
Spu ´ lq
Notice that hedging an American contingent claim is only matters before the time of the exercise. At the
time of the exercise or thereafter, there is no need to hedge. However, if the holder of the American claim
decides not to exercise at time T ˚ , the issuer can continue hedging with no hassle. For example, in Example
5.2 with K “ 2, it is optimal for the holder to exercise the option at the beginning, where the price of asset
is $1. Therefore, the issuer need $1 to replicate. However, if the holder continues, the replication problem
in the next period leads to solving the following system of equation.
#
2.1a ` 1.1b “ .4485
.6a ` 1.1b
“ 1.4000
which yields a “ ´1.903 and b “ 2.3107. Therefore, the issuer needs $a ` b “ .4077 to replicate the
option which is less than $1 if the holder exercises the option.
Exercise 5.3. Consider a two-period binomial model for a risky asset with each period equal to a year and
take S0 “ $1, u “ 1.2 and l “ 0.8. The interest rate for both periods is R “ .05.
a) If the asset pays 10% divided yield in the first period and 5% in the second period, find the price of
an American and European call options with strike K “ .8.
b) Construct the replicating portfolio for both American and European call option.
35
The running maximum at time t is the maximum of the price until or at time t.
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5 AMERICAN OPTIONS
Figure 5.7: A the exterior membrane of a tube (purple shape) is forcefully shaped into a dumbbell (red
shape). Upon release of the forces, the surface of the shape starts moving; each point moves at a speed
proportional to the curvature of the surface. Eventually, it returns back to the original shape. The picture is
adopted from [12].
5.2
Pricing American option in Black-Scholes model
In continuous time, including Black-Scholes model, the definition of exercise policy (stopping time) for
American options is not easy to define. The definition needs to use filtration and σ-algebra from measure
theory36 . Here we avoid technical discussion of stopping times and only present the solution for American
option in Black-Scholes model.
First notice that Proposition 5.2 implies that the American call option is Black-Scholes model has the same
price as European call if the underlying does not pay any dividend. Therefore, our focus here is on the
derivatives such as American put or American call on a dividend paying asset.
The key to solve the American option problem in Black-Scholes model is to set up a free boundary problem.
This type of problems are widely studied in physics. For example, in order to understand how an ice cube
is melting over time, we need to solve a free boundary problem. Or if we push an elastic object to a certain
shape, after releasing, the shape starts changing in a certain way which can be realized by solving a free
boundary problem. See Figure 5.7.
Another simple example occurs if we add obstacle underneath a hanging elastic string. An elastic string
fixed at two level points takes shape as a piece of parabola.
For the sake of simplicity, we only consider simple case where there is only one free boundary. This for
example occurs when we have an American put or an American call on a dividend paying asset. Below, we
list the important facts that you need to know about the free boundary.
a) The domain for the problem tpt, Sq : t P r0, T s, and S P r0, 8qu is divided into to parts separated
by a curve C :“ tpt, S ˚ ptqq : t P r0, T su. The curve C is called free boundary.
36
For more information see for example [21, 28] or for a more advanced text see [18, Chapter 1].
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5 AMERICAN OPTIONS
Figure 5.8: The blue curve shows a free string hanging at to points. If we position an obstacle underneath
the string such that the string is touched, then it changes the shape to the red curve. The equation satisfied
by the free string is △u “ c where the constant c depends on the physical properties of the string. This
equation is only satisfied by the obstacle-touching string in the region where it is not touching the obstacle.
b) On one side of the boundary it is not optimal to exercise the option. This side is called continuation
region, e.g. if the asset price Ss ą S ˚ psq for all s ď t, it has never been optimal to exercise the option
before or at time t.
c) The other side of the boundary is called exercise region, e.g. if the asset price St ď S ˚ ptq, then it is
optimal to exercise the option at time t. Therefore, the optimal stopping time is the first time that the
pair pt, St q hits the exercise region. More precisely, the asset price St hits the free boundary at time t
at point S ˚ ptq.
T ˚ :“ inf tt ď T : St “ S ˚ ptqu .
It is important to notice that the exercise boundary is an unknown in pricing American options in continuous
time. The other unknown is the price of the American option. We next explain that finding the price of the
American option also gives us the the free boundary. The relation between these two is lied in the following
representation of the stopping policy. For an American option with payoff gpt, St q, we have the optimal
stopping T ˚ given by
T ˚ “ inf tt ď T : V pt, St q “ gpt, St qu .
See [22] for more details. Therefore, finding the price of American option is the first priority here, which
will be explained in the sequel.
Finding the edges of exercise boundary
So far, we learned that the domain of the problem is split into continuation boundary and exercise boundary;
e.g. for American put at each time t, p0, S ˚ ptqq is the interior of exercise region and pS ˚ ptq, 8q is the interior
of the continuation region. determining continuation region is the matter of guess-and-check; we must look
at the payoff of the American option to guess the topology of the continuation region. One general rule is
that it is not optimal to exercise in the out of money region or where the payoff takes its minimum
value. We consider the following examples to clarify this rule:
a) American put. An American put option with strike K is out-of-money if St ą K, and since it is
not optimal to exercise in the out-of-money region, the exercise boundary should be inside the in-themoney region, i.e. S ˚ ptq ď K. See Figure 5.9 on the left. At the maturity (t “ T ), S ˚ pT q “ K.
b) American call. An American call option with strike K on a dividend paying asset is out-of-money
if St ă K, and since it is not optimal to exercise in the out-of-money region, the exercise boundary
should be inside the in-the-money region, i.e. S ˚ ptq ě K. See Figure 5.9 on the left. At the maturity
(t “ T ), S ˚ pT q “ K.
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c) Straddle. Several other options including strangle, bull and bear spread, etc can also be argued in the
similar fashion. However, we only explain it for straddle. First notice that one can shift the payoff of a
straddle option by cash amount of K2 so that the new payoff is positive. The key to analyze the straddle
is that it is the least desirable to exercise the option at and around the minimum point. Therefore, we
can guess that there are two free boundaries, S1˚ ptq and S2˚ ptq , located symmetrically on two sides
of the minimum point of the payoff K. At the maturity (t “ T ), S1˚ pT q “ S ˚ 2pT q “ K; the free
boundaries collapse to K.
K
K
St
St
S ˚ ptq
S ˚ ptq
Figure 5.9: Left: Free boundary of American put at time t. Right: Free boundary of American call (on
dividend paying asset) at time t. The continuation region in marked with |—|.
gpSt q
gpSt q
K
2
K
2
K
˚
S1
ptq
˚
S1
ptq
St
K
˚
S2
ptq
St
˚
S2
ptq
Figure 5.10: Two free boundaries of straddle.The continuation region in marked with |—|. The free boundaries can occur both on the positive regions of the payoff and on the negative region of the payoff. If the
time-to-maturity is long, we guess that the free boundaries are wides apart and as time-to-maturity decreases,
they get closer.
The position of the free boundary with respect time in the three examples is sketched in Figure 5.11.
Remark 5.6. The price of an American option with payoff equal to g1 pSq ` g2 pSq is not equal to, but only
smaller than, the summation of prices of an American option with payoff g1 pSq and an American option
with payoff g2 pSq Similar phenomenon is observed in Exercise 2.4.
Smooth fit
The main tool in finding the free boundary and to evaluate American options in the principle of smooth fit.
Let’s denote the price of the American option at time t when the asset price is equal to St “ S by V pt, Sq.
We provide the methodology for American put option. For other cases, the method can be adopted after
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2.8
2
Continuation region
Exercise region
2.5
2.6
1.8
Exercise region
2.4
1.6
1.4
2
Continuation region
2.2
Exercise region
1.5
Continuation region
Exercise region
1.2
2
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
1
0
0.2
0.4
0.6
0.8
1
Figure 5.11: Sketch of the position of American option exercise boundary in time (horizontal axis). Left:
American put. Middle: American call (dividend). Right: American Straddle.
necessary modifications. Before presenting this principle, we shall explain that in the continuation region,
the price function V pt, Sq of the American put option satisfies Black-Scholes equation, i.e.
$
σ2 S 2
˚
’
&Bt V pt, Sq ` 2 BSS V pt, Sq ` rSBS V pt, Sq ´ rV pt, Sq “ 0 for S ą S ptq
(5.2)
V pt, Sq “ gpt, Sq
for S ď S ˚ ptq .
’
%
V pT, Sq “ gpT, Sq;
Therefore, S ˚ ptq serves as a boundary for the above equation. However, S ˚ ptq itself is an unknown. With
the Black-Scholes equation, we have two unknowns but only one equation. Principal of smooth fit provides
a second equation.
Proposition 5.3 (Principal of smooth fit). Assume that the payoff gpt, Sq of the American option is twice
continuously differentiable with respect to S and continuous in t. Then, at the free boundary S ˚ ptq, we have
V pt, S ˚ ptqq “ gpt, S ˚ ptqq,
and
BS V pt, S ˚ ptqq “ BS gpt, S ˚ ptqq,
for all t ă T .
For example for American put we have gpt, Sq “ pK ´ Sq` . Therefore, principle of smooth fit implies
that V pt, S ˚ ptqq “ pK ´ S ˚ ptqq` and BS V pt, S ˚ ptqq “ ´1.
To see how smooth fit can be used in pricing American options, we provide some exactly solvable example
in the following. These examples are called perpetual American options with maturity T “ 8. As a result,
the price of the American option does not depend on time and the term Bt V in PDE (5.2) vanishes. Therefore,
the pricing function V pSq satisfies the ODE
σ2S 2 2
V pSq ` rSV 1 pSq ´ rV pSq “ 0,
2
in the continuation region.
Example 5.4 (Perpetual American put option). A perpetual American option is an option with maturity
T “ 8. In practice, there is no perpetual option. However, if the maturity is long (10 years), then one can
approximate the price with the price of a perpetual option.
The key observation is that since the Black-Scholes model is time-homogeneous, the free boundary of the
perpetual American put option does not depend on time, i.e. S ˚ ptq “ S ˚ for some unknown constant
S ˚ ă K. On the other hand, since the time horizon is infinite, the price V pt, Sq of the American put does
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Summary of the lecture
not depend on t, i.e. Bt V pt, Sq “ 0. Thus, we have
σ2S 2 1
V pSq ` rSV 1 pSq ´ rV pSq “ 0.
2
The general solution of the above equation is given by
2r
V pSq “ c1 S ` c2 S ´ σ2 .
One can argue that c1 must be equal to 0. Since as S Ñ 8, the option goes deep out-of-money and becomes
worthless. To find c2 , we use principle of smooth fit.
2r
c2 pS ˚ q´ σ2 “ K ´ S ˚
c2 “
Thus, S ˚ “
rK
2
r` σ2
σ 2 ˚ 2r2 `1
.
pS q σ
2r
and
$
’
&K ´ S
´
¯ 2r2 `1 2r
V pSq “
2
’
% σ2r rKσ2 σ
S ´ σ2
r`
Sě
rK
2
r` σ2
Să
rK
2
r` σ2
2
.
Example 5.5 (Perpetual American call option on continuous dividend paying asset). Consider a continuous
constant dividend rate q ą 0. The free boundary in this case is given by a constant S ˚ with S ˚ ą K and
the price of American option satisfies
σ2S 2 2
V pSq ` pr ´ qqSV pSq ´ rV pSq “ 0.
2
The general solution of the above equation is given by
V pSq “ c1 S γ1 ` c2 S γ2 ,
where γ1 and γ2 are roots of
σ2 2 ´
σ2 ¯
γ ` r´q´
γ ´ r “ 0.
2
2
Notice that γ1 and γ2 have opposite sign and the positive one is strictly larger than 1. With out loss in
generality, we assume that γ1 ă 0 ă 1 ă γ2 .
One can argue that c1 must be equal to 0. Since as S Ñ 0, the option goes deep out-of-money and becomes
worthless. To find c2 , we use principle of smooth fit.
c2 pS ˚ qγ2 “ S ˚ ´ K
1
.
c2 “
˚
γ2 pS qγ2 ´1
Thus, S ˚ “
γ2 K
γ2 ´1
and
V pSq “
#
S´K
Sď
1
˚ 1´γ2 S γ2
γ2 pS q
Są
γ2 K
γ2 ´1
γ2 K
γ2 ´1
.
Exercise 5.4. Formulate and solve the free boundary problem for the perpetual American options with
following payoffs.
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5 AMERICAN OPTIONS
a) pS ´ Kq` ` a where a ą 0.
b) pK ´ Sq` ` a where a ą 0.
c) Straddle
d) Strangle
e) Bull call spread
f) Bear call spread
American option with finite maturity
Unlike perpetual American option, when T ă 8, there is no closed-form solution for the free boundary
problem (5.2). Therefore, numerical methods should be used to approximate the solution.
The simplest among numerical methods is the binomial approximation. One needs to choose large number
of periods N and apply the algorithm of “Pricing American options in binomial model”. The parameters of
the binomial model u, l , and R can be chosen according to symmetric probability, subjective return, or any
other binomial which converges to the specific Black-Scholes model.
The finite difference scheme can be employed to solve free boundary problem numerically. Similar to
Section (3.10), one can apply the change of variables U pτ, xq “ e´rτ V pT ´ τ, erτ `x q, to derive a heat
equation with free boundary for U , i.e.
$
σ2
˚
’
&Bτ U pτ, xq “ 2 Bxx U pτ, xq ` rBx V pτ, xq ´ rU pτ, xq for x ą x pτ q
U pτ, xq “ gpT ´ τ, ex q
for x ď x˚ pτ q ,
’
%
U p0, xq “ gpT, ex q
where x˚ pτ q “ lnpS ˚ pτ qq.
For the simplicity, we only consider American put option, where we have U pτ, xq ě gpT ´τ, ex q. Whether
we want to apply explicit or implicit scheme to the above problem, because of the free boundary, we need
to add an intermediate step between step i and step i ` 1 of the scheme. Suppose that the approximate
solution Û pτi , xj q at τi is known for all j. Therefore, finite difference scheme (2.17) or (2.18) provides an
approximate solution, denoted by Û pτi` 1 , xj q for the heat equation without including the free boundary37 .
2
Then, to find an approximate solution Û pτi`1 , xj q for the free boundary problem at τi`1 , one only needs to
set
Û pτi`1 , xj q :“ maxtÛ pτi` 1 , xj q, gpT ´ τ, exj qu.
2
To summarize we have the following:
#
Û pτi` 1 , ¨q
2
:“ AÛ pτi , ¨q
Û pτi`1 , xj q :“ maxtÛ pτi` 1 , xj q, gpT ´ τ, exj qu @j
.
2
Here A can be implicit, explicit or mixed scheme.
The above method is in the category of splitting method, where there is one or more intermediate steps in
the numerical schemes to go from step i to step i ` 1.
37
The subscript in τi` 1 indicate that finite difference evaluation in each step in an intermediate step.
2
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5 AMERICAN OPTIONS
The splitting method described in this section can also be applied directly to Black-Scholes equation with
free boundary
$
σ2 S 2
˚
’
&Bτ V pτ, Sq “ 2 BSS V pτ, Sq ` rSBS V pτ, Sq ´ rV pτ, Sq for S ą S pτ q
V pτ, Sq “ gpT ´ τ, Sq
for S ď S ˚ pτ q .
’
%
V p0, Sq “ gpT, Sq;
The CFL condition is not different in the case of free boundary problems.
The Monte Carlo methods for American options are more complicated than for the European options and
in beyond the scope of this lecture notes. For more information of the Monte Carlo methods for American
options, see [19], [7], or the textbook [15].
American call option on discrete dividend paying asset
Unlike continuous dividend problem, the discrete dividend cannot be solved as a single free boundary problem. Consider an asset which pays dividend yield of d P p0, 1q at times t1 ă t2 ă ... ă tn “ T . Proposition
5.2 suggests that at any time t P rti , ti`1 q between the times of dividend payments it is better to wait and
not to exercise. However, at time ti of the dividend payment, the price of the asset decreases by the dividend
and so does the price of the call option. Therefore, if the continuation is not optimal, the option should be
exercises at the moment just before the time of a dividend payment.
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Appendix A
A CONVEX FUNCTIONS
Convex functions
Definition A.1. A set A Ď Rd is called convex if for any λ P p0, 1q and x, y P A, we have
λx ` p1 ´ λqy P A.
Definition A.2. Let A be a convex set. A real function f : A Ď Rd Ñ R is called convex if for any λ P p0, 1q
and x, y P Rd , we have
f pλx ` p1 ´ λqyq ď λf pxq ` p1 ´ λqf pyq.
A function f is called concave if ´f is convex.
Proposition A.1. A set A Ď Rd is called convex. Then, a function f : A Ď Rd Ñ R is convex if and only if
¸
˜
n
n
ÿ
ÿ
f
λi xi ď
λi f pxi q,
i“1
for all x1 , ..., xn P A and λ1 , ..., λn P R` with
i“1
řn
i“1 λi
“ 1.
If a convex function f is twice differentiable, then the Hessian matrix of second derivatives of f , ∇2 f , has
all eigenvalues non-negative. In one dimensional case, this is equivalent to f 2 ě 0.
However, not all convex function are twice differentiable or even differentiable. We actually know that all
convex functions are continuous. In addition, we can show that the one-sided directional derivatives of a
continuous function exits.
Proposition A.2. Let A be a convex set and f : A Ď Rd Ñ R be a convex function. Then, for all x in the
interior of A and all vectors v P Rd , the limit
lim
εÑ0`
f px ` εvq ´ f pxq
ε
exits. In particular, f is continuous at all points of A.
In one dimensional case, for a convex functions f the above Proposition implies that the right and left
derivatives, f 1 p¨`q and f 1 p¨´q, exist at all points. If the function has second derivative, then f 2 ě 0 at all
points. As a consequence, the linear approximation of f is always under-approximating the function, i.e.
f px0 ` ∆xq ě f px0 q ` f 1 px0 q∆x for ∆x sufficiently small.
In addition, the the convexity is strict, meaning f 2 ą 0 near the point x0 , then the above inequality is strict.
One of the results of convexity is Jensen inequality. Consider a probabilistic setting with a probability P
and a random variable X. Then for any convex function f , we have
f pErXsq ď Erf pXqs.
Jensen inequality is reduced to (A.2) when X is a random variable with two values x1 and x2 . More
precisely, if PpX “ x1 q “ λ and PpX “ x2 q “ 1 ´ λ, we have
f pErXsq “ f pλx1 ` p1 ´ λx2 qq ď λf px1 q ` p1 ´ λqf px2 q “ Erf pXqs.
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B PROBABILITY
A review of probability theory
B.1 Basic concepts and definitions of discrete probability
A (finite, countably infinite) sample space is a collection of possibles outcome of a random experiment. Any
subset A of the sample space Ω is called an event.
Example B.1 (Flipping a coin). The sample space is thead, tailu.
Example B.2 (Arrow-Debreu model in Section 3). The sample space can be chosen to be t1, ..., M u, i.e.
the collection of all the possible states of the system.
Example B.3 (T -period binomial model). The sample space of the T -period binomial model in Section 2
can be chosen to be the collection of all T -sequences of the form pa1 , ..., aT q where each ai is either u or l.
Each sample addresses the complete movements of the asset price over time.
A probability over
ř a finite sample space Ω “ tω1 , ..., ωL u is a vector π “ pπ1 , ..., πL q of nonnegative
values such that L
i“1 πi “ 1. For countably infinite sample space probability can be defined similarly.
Notation-wise, we write Pπ pωi q “ πi or if no confusion occurs Ppωi q “ πi .
Example B.4 (Flipping a coin). As for the example of flipping a fair coin, the probability divided in half
between head and tail. In two consecutive flips of a coin, the probability of having pH, Hq, pH, T q, pT, Hq,
or pT, T q is equally 1{4. If the coin is not fair then the sample space is not changed. But, the probability is
changed to πH “ p and πT “ 1 ´ p where p P r0, 1s. In two consecutive flips, we have πH,H “ p1 ´ pq2 ,
πH,T “ πT,H “ pp1 ´ pq, and πT,T “ p2 .
Example B.5 (Single asset T -period binomial model). In binomial model with T -periods, a risk-neutral
probability assigns probability πuk πlT ´k to an outcome pa1 , ..., aT q in which k of the entities at u and the
T ´ k remaining are l.
A random variable X is a function from sample space to Rd , X : ω P Ω ÞÑ Xpωq P Rd . The values
that X takes with positive probability are called the values of the random variable, i.e. x P Rd such that
Ppω : Xpωq “ xq ą 0. To simplify the notation, we often write PpX “ xq for Ppω : Xpωq “ xq. When
the sample space is finite or countably infinite, the random variables can only take finitely of countably
infinitely many values. Random variables with at most countably infinitely many values are called discrete
random variables.
Remark B.1. Notice that the values of a random variable is a relative to the choice of probability measure.
For example, a random variable X : t0, 1, 2u Ñ R defined by Xpxq “ x has values t0, 1, 2u relative to
probability Pp0q “ Pp1q “ Pp2q “ 1{3. However, relative to a new probability Qp0q “ Qp1q “ 1{2 and
Qp2q “ 0, the set of values is given by t0, 1u.
Example B.6 (Bernoulli random variable). Flipping a coin creates a Bernoulli random variable by assigning values to the outcomes head and tail. Bernoulli random variable X takes value `1 if the coin turns
head and 0 otherwise. If the coin has probability of tail equal to p, then X “ 1 has probability p and X “ 0
has probability 1 ´ p.
Example B.7 (Binomial random variable). In flipping a coin n times, binomial random variable X takes
value of the number of heads. The set of values of X is t0, ..., nu. If the coin has probability of tail equal to
p, then for x in the set of values, the probability X “ x is given by
ˆ ˙
n x
p p1 ´ pqn´x .
x
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Example B.8 (Random walk). In a game of chance, each round the player flips a coin. If it turns tail, he
gains $1, and otherwise he loses $1. Technically, each round has an outcome given by 2X ´ 1 where X
is a brand new Bernoulli random variable. The player’s accumulated reward after 2 rounds is a random
variable W2 with takes values ´2, 0, and 2 with probabilities 1{4, 1{2, and 1{4 respectively. If the coin has
probability of tail equal p, then PpW2 “ ´2q “ p2 , PpW2 “ 2q “ p1 ´ pq2 , and PpW2 “ 0q “ 2p1 ´ pqp.
Here 1 is not a value of W2 because PpW2 “ 1q “ 0.
At time t the position Wt of random walk takes values t´t, ´t`2, ..., tu for t “ 1, ..., T . Under risk-neutral
probability, the probability of Wt “ x is given by
ˆ ˙
t k
t`x
p p1 ´ pqt`1´k with k “
.
k
2
Example B.9 (Negative binomial random variable). In filliping a coin, the negative binomial random variable X counts the number of heads before r number of tails appear. The set of values of X is t0, 1, ...u. If
the coin has probability of head equal to p, then for x in the set of values, the probability X “ x is given by
ˆ
˙
x`r´1 x
p p1 ´ pqr .
x
Exercise B.1. In Examples B.8 and B.9, find an appropriate sample space and an accurate probability on
this sample space.
Example B.10 (T -period binomial model). Recall from Section 2 that in binomial model with T -periods,
the value St of the asset at time t is a random variable which takes values tS0 uk lt´k
..., tu. Thus,
` t ˘ : k k“ 0,t´k
π
k
t´k
one can say that under risk-neutral probability we have P̂ pSt “ S0 u l q “ k pπ̂u q pπ̂l q . This is
` ˘
because exactly kt numbers of outcomes in the sample space lead to St “ S0 uk lt´k and each outcome has
pπ̂u qk pπ̂l qt´k . Under physical probability, see 3.1, the probability of St “ S0 uk lt´k changes to
`probability
˘ k
t
t´k .
k ppq p1 ´ pq
Remark B.2 (Random walk as a corner stone of financial models). A binomial model is related to the
random walk in Example B.8 through taking logarithm. If Vt “ lnpSt q then Vt takes values lnpS0 q `
k lnpuq ` pt ´ kq lnplq for k “ 0, ..., t. In other words, Vt is the position of a generalized random walk after
t rounds, starting at lnpS0 q which moves to lnpuq or lnplq in each round, respective to the outcomes of a
coin. If S0 “ 1, u “ e and l “ e´1 , then the random walk Vt is the standard random walk Wt described
in Example B.8. Otherwise, Vt “ lnpS0 q ` µt ` σWt where µ “ lnpuq`lnplq
and σ “ lnpuq´lnplq
. In other
2
2
words,
St “ S0 exp pµt ` σWt q
For an event A, the indicator of A is a random variable which takes value 1 if A occurs and 0 otherwise.
Indicator of A is denoted by 1A 38 .
If we know that an event has occurred, it can potentially change the probabilities of other events. For
example, knowing that a coin flip turned tail, the probability of the same flip turns hear is now 0!39 The new
probabilities can be described by properly defining conditional probability. If the occurrence of event A is
observed, then the conditional probability of event B is defined by
PpB | Aq :“
PpA X Bq
.
PpAq
Exercise B.2. In Example B.8, calculate PpY2 “ 2 | Y1 “ 1q and PpY3 “ ´1 | Y1 “ 1q.
38
39
Indicator is also denoted by χA in the literature.
This is an extreme case.
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The conditional probability gives birth to the important notion of independence. Two events A and B are
called independent if
PpB | Aq “ PpBq,
or equivalently
PpA | Bq “ PpAq.
It is easier to write independence as
PpB X Aq “ PpAqPpBq.
Two random variables X and Y are called independent if each event related to X is independent of each
event related to Y , i.e.
For all x in values of X and all y is values of Y , we have
PpX “ x, Y “ yq “ PpX “ xqPpY “ yq.
Exercise B.3. Show that two events A and B are independent iff the indicator random variables 1A and 1B
are independent.
In modeling random experiments, independence comes and an assumptions which either is a common
belief or a part of the model. However, sometimes independence comes as a result of previous independency
assumptions. For example in a random walk, the outcomes of two different rounds are independent by
assumption that two flips of a coin are independent trials. However, the independence of random variables
S5 and S8 ´ S5 is a result.
Exercise B.4. Show that in a random walk, S5 and S8 ´ S5 provided that the outcome of each round is
independent of other rounds.
Defining independence for more than two events (equivalently random variables) is a little tricky. We call
X independent of the sequence of random variables X1 , X2 , ..., Xn if
PpX “ x, X1 “ x1 , X2 “ x2 , ..., Xn “ xn q “ PpX “ xqPpX1 “ x1 , X2 “ x2 , ..., Xn “ xn q.
This indicates that any event about X is independent of any event about X1 , X2 , ..., Xn , i.e.
PpX P A, pX1 , X2 , ..., Xn q P Bq “ PpX P AqPppX1 , X2 , ..., Xn q P Bq.
A finite sequence X1 , X2 , ..., Xn is called independent if for each i Xi and tXj : j ‰ iu are independent.
As observed in the following exercise, to show the independency of several random variables, it is not
enough to check that each pair of random variables Xi and Xj are independent.
Exercise B.5. In two consecutive flips of a fair coin, let A be the event that the first flip turns head, B be
the event that the second flip turns head, and C be the event that only one of the flips turn head. Show that
A, B and C are not independent but pairwise independent.
Another equivalent definition of independency is as follows. A finite sequence X1 , X2 , ..., Xn is called
independent if for all subsets A1 , ..., An of values of X1 , ..., Xn respectively, we have
PpX1 P A1 , ..., Xn P An q “ PpX1 P A1 q...PpXn P An q.
This can also be extended to an infinite sequence of random variables. A sequence X1 , X2 , ... is called independent if each finite subset tXi1 , ..., Xin u makes a set of independent random variables. Having defined
the notion on independence for a sequence of random variables, we can now properly define a random walk,
as the previous definition in Example B.8 is more heuristic than rigorous.
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Definition B.1. Let ξ1 , ξ2 , ... be a sequence of independent and identically distributed (i.i.d.) random variables such that Ppξi “ 1q “ p and Ppξi “ ´1q “ 1 ´ p for some p P p0, 1q. For x P Z, the sequence
W0 “ x, W1 , W2 , ... with
n
ÿ
ξi for n ě 1.
Wn :“ x `
i“1
is called a random walk.
When p “ 21 , we call it a symmetric random walk and otherwise it is called a bias random walk.
Let X be a random variable on a discrete sample space Ω with probability vector πpωq for each ω P Ω.
Then, the expectation of X is defined by
ÿ
ErXs :“
Xpωqπpωq.
(B.1)
ωPΩ
If a random variable X takes values x1 , x2 , ... with probabilities p1 , p2 , ..., respectively, then the expectation
of X can equivalently be given by
8
ÿ
ErXs :“
xi pi .
i“1
In particular, if values of X are finitely many x1 , x2 , ..., xn with probabilities p1 , p2 , ..., pn , respectively,
then the expectation of X is equivalently given by
ErXs :“
n
ÿ
xi pi .
(B.2)
i“1
The advantage of (B.2) over (B.1) is that some finite sample spaces can be very large while the random
variable over them only takes small number of values. For example in Example B.5, single asset T -period
binomial model generates a sample space of all paths of the asset price of size 2T while the values of random
variable St is only T ` 1.
By straightforward calculations, the expectation of a function f pxq of X is given by
Erf pXqs :“
8
ÿ
f pxi qpi .
i“1
Let Y be another random variable with values y1 , ..., ym . To write the expectation of a function f px, yq of
two random variables X and Y , we need to know the joint (mutual) probabilities of the pair pX, Y q, i.e.
pi,j :“ PpX “ xi , Y “ yj q for i “ 1, ..., n, and j “ 1, ..., m..
Notice that although PpX “ xi q and PpY “ yj q are positive, pi,j can be zero, which means if in an event
xi is realized for X, then yj cannot be realized for Y and vice versa. Then, we define the expected value of
f pX, Y q by
8 ÿ
8
ÿ
Erf pX, Y qs :“
f pxi , yj qpi,j .
i“1 j“1
We can define conditional expectation similarly by replacing the probabilities pi,j with the conditional
probabilities
pi,j
pi|j :“ PpX “ xi | Y “ yj q “ Y ,
pj
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with pYj “ PpY “ yj q. More precisely,
Erf pX, Y q | Y “ yj s :“
8
ÿ
f pxi , yj qpi|j “
i“1
8
1 ÿ
f pxi , yj qpi,j .
pYj i“1
(B.3)
Notice that if PpY “ yq “ 0, then Erf pX, Y q | Y “ ys in (B.3) is not defined. However, we can define
function h : y ÞÑ Erf pX, Y q | Y “ ys on the set of values of the random variable Y . This in particular
helps us to define the random variable
Erf pX, Y q | Y s :“ hpY q.
Remark B.3. Notice the difference between Erf pX, Y q | Y “ yj s, Erf pX, Y q | Y “ ys, and Erf pX, Y q |
Y s. Erf pX, Y q | Y “ yj s is a real number, Erf pX, Y q | Y “ ys is a real function on variable y, and finally
Erf pX, Y q | Y s is a random variable.
As a particular case, when f px, yq “ x, we have
hpyq “ ErX | Y “ ys
and
ErX | Y s “ hpY q.
Corollary B.1. If X and Y are random variables and f is a real function, then we have
Erf pY qX | Y s “ f pY qErX | Y s.
The following proposition, which is a direct result of (B.3), explains a very important property of independent random variables.
Proposition B.1. X and Y are independent if and only if for any real function f px, yq of X and Y we have
Erf pX, Y q | Y “ ys “ Erf pX, yqs for all y in the set of values of Y.
Corollary B.2. If X and Y are independent and f is a real function, then we have
Erf pXq | Y s “ Erf pXqs.
One of the important properties of conditional expectation is the tower property which is presented in the
next proposition.
Proposition B.2 (Tower property of conditional expectation). Let X, Y and Z be random variables. Then,
ErErX | Y, Zs | Y s “ ErX | Y s.
In particular,
ErErX | Y ss “ ErXs.
B.2 General probability spaces
Some random experiments can generate uncountable outcomes, e.g. choosing a point in the unit interval
r0, 1s, or choosing a chord of a unit circle. In such cases, definitions in Section B.1 does not make sense;
e.g. the summation in (B.1) relies on the countability of the sample space. In Section B.4, we consider the
continuous random variables in which the summations can be replaced by an integral. However, in general,
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there can be random variables which are neither continuous nor discrete. In that case, the definitions such
as expectation and conditional expectation should be treated differently and requires advanced techniques
from measure theory.
Measure theory is gradually shaped as a theory for integration by several mathematicians such as Émile
Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet. However, Andrey Kolmogorov was the first
who noticed that this theory can be used as a foundation of probability theory; a probability is a non-negative
finite measure (normalized to mass one) and expectation is an integral with respect to that measure.
One other difficulty in the uncountable sample spaces in the notion of random variable. In discrete space
any function from sample space to Rd is a random variable and therefore we can define the expected value by
means of summation. However, in uncountable sample spaces, there are even some bounded functions for
which the integral of those functions cannot be defined, i.e. non-measurable functions. Therefore, a random
variable should be defined by as a proper way. Here, we address the issue briefly. For more information on
the concept of probability measures and measurable function see [4] or [1].
For a (possibly uncountable) sample space Ω, the set of events should use the notion of σ-field form measure
theory. A σ-field F is a collection of subsets of Ω which satisfies
a) H and Ω P F.
b) If A P F then AC P F.
c) For a sequence tAn u8
n“0 Ď F,
Ť8
n“0 An
P F.
On the specific choice of probability setting in Chapter II, it is necessary to consider the Borel σ-field B; the
smallest σ-field which includes all open subsets of Ω “ pRd` qT or .
In the remaining of this section, we show how to define expectation for general random variables by means
of the distribution function, i.e. without appealing to the notion of integration with respect to a measure.
Definition B.2. For a random variable X with real values, the cumulative distribution function (or cdf or
simply distribution function) FX pxq is defined by
FX pxq :“ PpX ď xq.
The definition of distribution function implies that PpX P pa, bsq “ FX pbq ´ FX paq.
For a discrete random, FX is a step function. However, for non-discrete random variables FX pxq satisfies
the following properties.
a) F p´8q “ 0 and F p8q “ 1.
b) F is increasing right-continuous function with left-limit at all points.
Exercise B.6. Show that PpX P ra, bsq “ FX pbq ´ FX pa´q, where FX pa´q is the left limit of FX at a.
It is also well-known that if a function F satisfies the properties (a)-(b) of distribution function, then there
is always a random variable X in a proper sample space such that FX “ F .
Notice that F is an increasing function and therefore one can define Riemann-Steilijes integral with respect
to F .
Proposition B.3. For a random variable X the expectation of X satisfies
ż8
ErXs “
xF pdxq,
´8
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given Er|X|s ă 8. In particular, for any function g,
ż8
ErgpXqs “
gpxqF pdxq,
´8
given Er|gpXq|s ă 8.
Similarly, for a random vector X “ pX1 , ..., Xd q, the distribution function is defined by
FX pxq :“ PpX1 ď x1 , ..., Xd ď xd q,
and the expectation of gpX1 , ..., Xd q is given by
ż8
ErgpX1 , ..., Xd qs “
gpx1 , ..., xd qF pdx1 , ..., dxd q,
´8
given Er|gpX1 , ..., Xd q|s ă 8.
However, conditional expectation cannot simply be defined in terms of cdf and more advanced methods
are needed. Unlike the stepwise definition in Section sec:A review of probability theory, we can define
ErX | Y s directly through a prominent theorem in analysis, i.e. Radon-Nikodym theorem. To give you a
glimpse of the definition, ErX | Y s is the unique random variable hpY q which satisfies
ErhpY qgpY qs “ ErXgpY qs,
for all real bounded functions g with domain containing the set of values of Y . The existence and uniqueness
of such random variable is guaranteed by Radon-Nikodym theorem. After this definition, all the results of
Remark B.3, Corollary B.1, Proposition B.1, and Proposition B.2 hold for general random variables.
In this approach, function ErX | Y “ ys is given the existence of a function h which gives ErX | Y s as a
function of the random variable Y .
B.3 Martingales
The conditional expectation is required for the concept of martingale in probability theory. Consider a
probability space, i.e. sample space Ω and a probability P on Ω, and let X :“ tXt uTt“0 (possibly T “ 8)
be a stochastic process which generates the information.
Definition B.3. A discrete-time stochastic process tMt uTt“0 on probability space pΩ, Pq the is called a
martingale with respect to X if
a) the expected value of |Mt | is finite for all t “ 0, ..., T , i.e. Er|Mt |s ă 8.
b) the conditional expectation Mt given Xs , Xs´1 , ..., X0 is equal to Ms , i.e. ErMt |Xs , Xs´1 , ..., X0 s “
Ms , for s ă t.
Condition (a) in the definition of martingale is technical and guarantees the existence of the conditional
expectation in condition (b).
Condition (b) can also be given equivalently by
b’) ErMt |Xt´1 , ..., X0 s “ Xs , for t ě 1.
Notice that as a result of tower property of conditional expectations, if (b’) holds we can write
ErErMt |Xt´1 , ..., X0 s|Xs , ..., X0 s “ ErMt´1 |Xs , ..., X0 s.
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By applying the tower property inductively, we obtain
ErErMt |Xt´1 , ..., X0 s|Xs , ..., X0 s “ ... “ ErMs`1 |Xs , ..., X0 s “ Ms .
Example B.11. Let Y be an arbitrary random variable. Then, Mt :“ ErY | Xt , ..., X0 s is a martingale
with respect to X.
Process X carries the dynamics of the information as time passes. At time t, the occurrence or nonoccurrence of all events related to Xt , ..., X0 are known. The martingale represents a fair game based on the
public information.
Example B.12 (Standard symmetric random walk). The symmetric random walk in Definition B.1 is a
martingale. From (b’), we have to calculate ErWt`1 |Wt , ..., W0 s. Since Wi ´ Wi´1 “ ξi , we have
ErWt`1 |Wt , ..., W0 s “ ErWt ` ξt`1 |ξt , ..., ξ1 , W0 s
“ ErWt |ξt , ..., ξ1 , W0 s ` Erξt`1 |ξt , ..., ξ1 , W0 s.
ErWt |ξt , ..., ξ1 , W0 s is simply ErWt |Wt , ..., W0 s “ Wt . On the other hand, since ξi ’s are independent, it
follows from Corollary B.2 that
Erξt`1 |ξt , ..., ξ1 , W0 s “ Erξt`1 s “ 0.
Thus, ErWt`1 |Wt , ..., W0 s “ Wt .
Example B.13 (Multi-period binomial model). Under the risk neutral probability, the discounted price of
St
the asset, Ŝt :“ p1`Rq
t is a martingale. Recall from Section II.2 that
St “ St´1 Ht ,
where Hi is a sequence of i.i.d. random variables with risk-neutral distribution
Pπ pHi “ uq “
1`R´l
u´1´R
and Pπ pHi “ lq “
.
u´l
u´l
Thus, Ht is independent of St´1 “ S0 Ht´1 ¨ ¨ ¨ Hq1 and we have
ErSt |St´1 , ..., S0 s “ ErSt´1 Ht |Ht´1 , ..., H1 , S0 s
“ St´1 ErHt |Ht´1 , ..., H1 , S0 s
“ St´1 ErHt s “ p1 ` RqSt´1 .
In the second equality above, we used Corollary B.1, and the third equality is the result of Corollary B.2.
Remark B.4. As a result of tower property of conditional expectation, the expectation of a martingale
remain constant with time, i.e.
ErMt s “ ErM0 s.
Definition B.4. Consider a probability space, i.e. sample space Ω and a probability P on Ω, and let X :“
tXt uTt“0 be a stochastic process. A discrete-time stochastic process tMt uTt“0 on probability space pΩ, Pq
the is called a martingale with respect to X if
a) the expected value of |Xt | is finite for all t “ 0, ..., T , i.e. Er|Mt |s ă 8.
b) the conditional expectation Mt`1 given Xt , ..., X0 is equal to Mt , i.e. ErMt`1 |Xt , ..., X0 s “ Mt .
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Example B.14. Let X be a symmetric random walk in Example B.1 and define Mt :“ Xt2 ´ t. Then, Mt is
a martingale with respect to X. To see this, we have to show
2
ErXt`1
´ pt ` 1q|Xt , ..., X0 s “ Xt2 ´ t.
Recall that Xt`1 “ Xt ` ξt`1 . Thus,
2
ErXt`1
|Xt , ..., X0 s “ ErpXt ` ξt`1 q2 |ξt , ..., ξ1 , X0 s
2
“ ErXt2 |ξt , ..., ξ1 , X0 s ` Erξt`1
|ξt , ..., ξ1 , X0 s ` 2ErMt ξt`1 |ξt , ..., ξ1 , X0 s.
It follows from Corollary B.1 that
ErXt2 |ξt , ..., ξ1 , X0 s “ Xt2
ErXt ξt`1 |ξt , ..., ξ1 , X0 s “ Xt Erξt`1 |ξt , ..., ξ1 , X0 s.
On the other hand, by Corollary B.2, we have
2
2
Erξt`1
|ξt , ..., ξ1 , X0 s “ Erξt`1
s“1
Erξt`1 |ξt , ..., ξ1 , X0 s “ Erξt`1 s “ 0
Thus,
2
ErXt`1
|Xt , ..., X0 s “ Xt2 ` 1,
and
ErMt`1 |ξt , ..., ξ1 , X0 s “ Xt2 ` 1 ´ pt ` 1q “ Mt .
Example B.15. In example B.11, if Y0 , Y1 , ... is given and X be a random variable, then Mt :“ ErX |
Yt , ..., Y0 s is a martingale with respect to Y0 , Y1 , ....
In general, if M1 , M2 , ..., MT is a martingale with respect to
B.4 Continuous random variables
In this section, we review basic concepts of continuous random variables without referring to measurability
issues. Therefore, a gap on the rigorousness of the concepts remains throughout this section.
Definition B.5. A random variable X is called continuous if there exits a non-negative function f : Rd Ñ R
such that
ż x1
ż xd
FX pxq :“
¨¨¨
f pyqdy.
´8
´8
In this case, the function f is called probability density function (pdf) of the continuous random variable X.
Working with continuous random variables is often easier since one can easily approximate the integrals to
estimate relevant quantities. For example, when X in a univariate continuous random variable with pdf f ,
expectation of X is given by
ż
ErXs “
8
xf pxqdx.
´8
More generally, for a function h : R Ñ R, we have
ż8
ErhpXqs “
hpxqf pxqdx.
´8
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If I Ď R, then
PpX P Iq “
B PROBABILITY
ż
f pxqdx.
I
Example B.16 (Normal distribution). A continuous random variable X with density
x2
1
fX pxq “ ? e´ 2 for all x P R
2π
is called standard normal random variable. By using integration techniques, one can see that ErXs “ 0
and varpXq :“ ErX 2 s “ 1.
If Y “ σX ` µ for σ ą 0 and µ P R, then Y is also a continuous random variable with density
fY pxq “ ?
px´µq2
1
e´ 2σ2 for all x P R.
2πσ
Then, Y is called a normal random variable with mean µ and variance σ 2 and is denoted by Y „ N pµ, σ 2 q.
Exercise B.7. Show that when X „ N p0, 1q, we have ErXs “ 0 and varpXq :“ ErX 2 s “ 1.
When X “ pX1 , ..., Xd q is a jointly continuous random vector, we refer to pdf as the joint probability
distribution of X1 , ..., Xd . For simplicity, let’s focus on two random variables. Let the joint pdf of pX, Y q
be f px, yq. Then, it is easy to see that
a) pdf of X, fX satisfies
ż8
fX pxq “
f px, yqdy.
´8
Similar formula holds for the pdf of Y .
b) for a function h : R2 Ñ R, we have
ErhpXqs “
ż8 ż8
hpx, yqf px, yqdxdy
´8 ´8
c) if K Ď R2 , then
PppX, Y q P Kq “
ij
f px, yqdA.
K
Example B.17 (Bivariate normal distribution). Let C be a symmetric positive-definite matrix and µ “
pµ1 , ..., µd q P Rd . A jointly continuous random vector X “ pX1 , ..., Xd q with density
f pxq “
´1
T
1
´ px´µqC 2 px´µq
e
for all x P Rd
p2πdetpCqqd{2
is called multi-variate normal random vector. For each j “ 1, ..., d, ErXj s “ µj . The matrix C is called
the covariance matrix of X, because its entities correspond to the covariance of components of X, i.e.
Ci,j “ ErpXi ´ µi qpXj ´ µj qs.
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In particular, for d “ 2, for a positive definite matrix40
ȷ
„ 2
σ1 σ12
C“
σ12 σ22
and µ “ 0, we have
f px, yq “
1
2π
a
2
σ12 σ22 ´ σ12
e
´
2 x2 ´2σ xy`σ 2 y 2
σ2
12
1
2 σ 2 ´σ 2 q
2pσ1
2
12
for all x P R.
Here by evaluating double integrals, we can see that σ12 and σ22 are variance of X and Y respectively, and
σ12 is the covariance of X and Y .
Exercise B.8. In Example B.17, show that covpX, Y q :““ ErpXi ´ µi qpXj ´ µj qs “ Ci,j .
Exercise B.9. In Example B.17, show that σ12 and σ22 are variance of X and Y respectively, and σ12 is the
covariance of X and Y .
Defining and calculating conditional probability and conditional expectation is also done through integral
definition for continuous random variables. Let pX, Y q be a jointly continuous random variable with density
f px, yq. Then, the conditional density of X given Y “ y is defined by
fX|Y px | yq :“
f px, yq
,
f pyq
provided that y be in the set of values of Y , i.e. ty : f pyq ‰ 0u. Using the above definition, the conditional
probability of X P I given Y “ y is given by
ş
f px, yqdx
PpX P I | Y “ yq :“ I
(B.4)
f pyq
Similarly, if h : R Ñ R be a function, then the conditional expectation of hpXq given Y “ y is given by
ş8
hpxqf px, yqdx
ErhpXq | Y “ ys :“ ´8
.
(B.5)
f pyq
In particular, the conditional expectation of X given Y “ y is given by
ş8
xf px, yqdx
ErX | Y “ ys :“ ´8
.
f pyq
(B.6)
Notice that conditional expectation and conditional probability in (B.4), (B.5), and(B.6) are functions of
the variable y. The domain of all these is the set of values of Y , i.e. ty : f pyq ‰ 0u. This, in particular,
can be useful in defining conditional distribution and conditional probability given Y . Let’s first make the
definition for conditional
probability of X P I given Y . Define the function Y of y which maps y onto
ş
I f px,yqdx
PpX P I | Y “ yq :“
. Then, one can define
f pyq
ş
PpX P I | Y q :“ YpY q “
40
I
f px, Y qdx
.
f pY q
2
In order for C to be positive definite, is necessary and sufficient to have σ1 , σ2 ‰ 0 and σ12 σ22 ´ σ12
ą 0.
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Notice that unlike PpX P I | Y “ yq which is a real function, PpX P I | Y q is a random variable which is
completely dependent on random variable Y . Similarly we have
ş8
ş8
xf px, Y qdx
hpxqf
px,
Y
qdx
and ErX | Y s :“ ´8
.
ErhpXq | Y s :“ ´8
f pY q
f pY q
For continuous random variables independence can be defined in terms of the joint pdf f px, yq; let X and
Y be jointly continuously. Then, X and Y are called independent if f is a separable function, i.e.
f px, yq “ gpxqhpyq.
Notice that the choice of h and g is not unique and varies by multiplying or dividing constants. In this case,
one can write the separation in a standard form
f px, yq “ fX pxqfY pyq,
(B.7)
where fX pxq and fY pyq are respectively the pdf of x and pdf of Y .
Example B.18. From Example B.17 and (B.7), one can see bivariate normal random variables are independent if and only if they are uncorrelated, i.e. they have zero correlation or simply σ12 “ 0.
Remark B.5. Discrete and continuous random variables are two restricted categories of random variables.
However, there are many simple examples which does not fit into these categories. For example consider a
random variable X “ Y Z, where Y is continuous with pdf f pyq “ e´y for y ě 0 and Z is independent of
Y and takes values 0 and 1 with probabilities p and 1 ´ p where p P p0, 1q, respectively. One can easily see
that X is neither continuous nor discrete.
To generalize the concepts of this section to general random variables, one needs to use measure theory
which is an advanced subject in mathematical analysis. For example, defining expectation of a random
variable uses the theory of integration and defining conditional expectation uses Radon-Nikodym derivative
theorem. X and Y are called independent if for any two real functions h1 pxq and h2 pyq we have
Erh1 pXqh2 pY qs “ Erh1 pXqsErh2 pY qs.
(B.8)
Exercise B.10. Show that (B.8) and (B.7) are equivalent.
The Proposition B.1 is not only restricted to discrete random variables and holds for any two random
variables X and Y .
Proposition B.4. X and Y are independent if and only if for any function f px, yq of X and Y we have
Erf pX, Y q | Y “ ys “ Erf pX, yqs for y in the set of values of Y.
Exercise B.11. If tXi u8
of independent Bernoulli random variables with equally likely
i“1
ř is aXsequence
i
values 0 and 1, then U “ 8
is
uniformly
distributed in r0, 1s.
i
i“1 2
B.5 Characteristic function
The characteristic function of a (univariate) random variable is a complex function defined by
χX pθq :“ EreiθX s.
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ř
iθxi p , and when X is continuous with pdf f ,
When X is a discrete
random variable, then χX pθq “ 8
1
i“1 e
ş8 iθx
then χX pθq “ ´8 e f pxqdx. In the latter case, the characteristic function is the Fourier transform of the
pdf f .
If F is the distribution function of X, the the characteristic function can be equivalently given by
ż
χX pθq “
eiθx dF pxq,
R
where the above integral should be interpreted as Reimann-Stieltjes integral. Especially for continuous
random variables we have
ż
χX pθq “
eiθx f pxqdx.
R
Therefore, for a continuous random variable, the characteristic function is the Fourier transform of the pdf.
Therefore, if we know the characteristic function of a distribution, then one can find the distribution function
by inverse Fourier transform.
Example B.19. We want to find the characteristic function of Y „ N pµ, σq. First tale the standard case of
X „ N p0, 1q.
θ2 ż
ż
px´iθq2
x2
θ2
1
e´ 2
iθx
´
χX pθq “ ?
e e 2 dx “ ?
e´ 2 dx “ e´ 2 .
2π R
2π R
Now, since Y “ µ ` σX, we have
χY pθq “ Ereiθpµ`σXq s “ eiµθ χX pσθq “ eiµθ´
σ2 θ2
2
.
For the inversion theorem, see [11, Theorem 3.3.4], the characteristic function uniquely determines the
distribution of the random variable. Therefore, all the information
For a random vector pX, Y q, the characteristic function is defined as
χpθ1 , θ2 q :“ Ereipθ1 X`θ2 Y q s.
One of the important implications of definition of independent random variable in Remark B.5 is that if X
and Y are independent, then for any pθ1 , θ2 q we have
χpθ1 , θ2 q “ MX pθ1 qMY pθ2 q.
(B.9)
The inverse is also true; see [11, Theorem 3.3.2]. This provides an easy way to formulate and verify the
independence of random variable in theory.
Proposition B.5. X and Y are independent if and only if (B.9) holds true.
B.6 Weak convergence
The most well-known place where the weak convergence comes to play is the central limit theorem (CLT).
Theorem B.1. Let X1 , X2 , ... be a sequence of i.i.d. random
řn variables with expectation µ “ ErX1 s and
a
pX ´µq
2
2
standard deviation σ :“ ErX1 s ´ µ and define Wn :“ i“1σ?ni
. Then,
P pWn ď xq Ñ
żx
´8
as n Ñ 8.
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1
? e´ 2 dy,
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The appellation “weak” is originated from the fact that this convergence is weaker that the concept of
pointwise or almost sure (shortly a.s.) convergence. The pointwise convergence indicates that the sequence
of random variablestWn pωqun converges for all ω P Ω; while almost sure convergence means the probability
of the event
An :“ tω : Wn pωqun does not convergeu
converges to zero as n Ñ 8. Almost sure convergence, for instance, appears in the law of large numbers
(LLN).
Theorem B.2. Let X1 , X
a2 , ... be a sequence of i.i.d. random variables with expectation µ “ ErX1 s and
standard deviation σ :“ ErX12 s ´ µ2 . Then,
řn
i“1 Xi ´ nµ
Ñ 0 almost surely,
n
as n Ñ 8. Here almost surely means that the probability that this convergence does not happen is zero.
řn
X
i
In other words, the statistical average i“1
converges to the expectation (mean), when n Ñ 8, except
n
for a set of outcomes with zero total probability. For example in the context of flipping a fair coin, the
fraction of flips that the coin turns head converges to 12 exclusively. However, one can simply construct
infinite sequences of head and tail such as H, T, T, H, T, T, ... with statistical average converging to some
value other than 12 .
In the weak sense of convergence in CLT, the sequence of random variablestWn pωqun do not actually
converge to a normal random variable over a significantly large part of the sample space, as a result of law
of iterated logarithms, i.e.
lim sup a
nÑ8
Wn
“ 1 almost surely.
2 logplogpnqq
Instead, the probability distribution
řnfunction of Wn can be approximated by normal distribution function for
X ´nµ
converges weakly or in distribution to standard normal.
large n. In this case, we say that i“1σ?ni
Definition B.6. We say the sequence tYn u8
n“1 of random variables converges weakly (or in distribution) to
random variable Y if for the distribution functions we have
FYn pyq Ñ FY pyq for any y such that FY is continuous at y.
We denote the weak convergence by Yn ñ Y .
The following example reveals a different aspect of weak convergence in regard to comparison with a.s.
convergence.
Example B.20. On a probability space pΩ, Pq, let the random variable Yn pωq “ yn for all ω (a.s.), i.e. Yn
is a constant random variable equal to yn . If yn Ñ y, then Yn Ñ Y ” y pointwise (a.s., respectively).
Now, consider a sequence of possibly different probability space pΩ “ t0, 1u, Pn q such that Pn pt0uq “ 1
if n is odd and Pn pt0uq “ 0 if n is even. For each n, let the random variable Zn : Ω Ñ R be defined by
#
1
pω “ 0 and n is oddq or pω “ 1 and n is evenq
Zn pωq “ n
1 otherwise.
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In particular, the distribution of Zn is a Dirac distribution located at n1 ;
#
1 x ě n1
Fn pxq :“ Pn pZn ď xq “
0 otherwise.
Zn does not converge pointwise, since Zn p0q alternates between n1 and 1 as n increases successively. However, the distribution function Fn pxq of Zn converges to the distribution function
#
1 xě0
F pxq :“
0 otherwise.
The following proposition is one of the equivalent conditions of weak convergence.
Proposition B.6. Yn ñ Y if and only if for any bounded continuous function f : R Ñ R we have
lim Erf pYn qs “ Erf pY qs
nÑ8
Notice that the expectation in ErYn s and ErY s are to be interpreted in different sample spaces with different
probabilities.
One of the ways to establish weak convergence results is to use characteristic functions; see [11, Theorem
3.3.6]
Theorem B.3. Let tXn u be a sequence of random variables such that for any θ, χXn pθq converges to a
function χpθq which is continuous at θ “ 0. Then, Xn converges weakly to a random variable X with
characteristic function χ.
As a consequence
of the this Theorem, one can easily provide a formal derivation for central limit theorem.
řn
Xi ´nµ
i“1 ?
Let Sn :“
. Thanks to the properties of characteristic function and (B.9), the characteristic
σ n
function of Sn is given by
χSn pθq “
n
ź
`
? ˘
?
χXi ´µ pθ{pσ nqq “ χX1 ´µ pθ{pσ nqq n .
i“1
Here we used identical distribution of sequence tXn u to write the last equality. Since eix “ 1 ` ix ´
opx2 q, we can write
σ 2 θ2
χXi ´µ pθq “ 1 ´
` opθ2 q.
2
Therefore,
ˆ
˙n
`
? ˘
θ2
θ2
´1
χSn pθq “ χpXi ´µq pθ{pσ nqq n “ 1 ´
` opn q
Ñ e´ 2 ,
2n
as n Ñ 8. This finished the argument since e´
θ2
2
x2
2
`
is the characteristic function of standard normal.
Remark B.6. In the Definition B.6 of weak convergence, only the distribution of random variables matters,
and not the sample space of each random variable. Therefore, in weak convergence, the random variables in
sequence tYn u8
n“1 can live in different sample spaces. However, one can make one universal sample space
for all Yn ’s andś
Y ; more precisely, if Yn and Y are defined on probability space pΩn , Fn , Pn q and pΩ, F, Pq,
then Ω̃ “ Ω ˆ n Ωn is a universal sample space. The random variables Yn and Y are redefined on Ω̃ by
Ỹ pω, ω1 , ω2 , ...q “ Y pωq, and Ỹn pω, ω1 , ω2 , ...q “ Yn pωn q.
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B PROBABILITY
W
Time
Time
Figure B.1: Left: Sample path of a random walk. Right: Interpolated sample path of a random walk
ś
The distribution of random variables on Ω̃ is determined by the probability P̃ :“ P b n Pn .
Therefore, the weak convergence of random variables can be reduced to weak convergence of probabilities
on a single sample space.
Weak convergence of probabilities
If the sample space is a Polish space (complete metrizable topological space), then one can define weak
convergence of probabilities (or even measures). Sample spaces with a topology contribute to the richness
of the probabilistic structure; the concept of convergence of probabilities can be defined.
Definition B.7. Consider a sequence of probabilities tPn un on a Polish probability space pΩ, Fq. We say
Pn converges weakly to a probability P on pΩ, Fq, denoted by Pn ñ P, if for any bounded continuous
function f : Ω Ñ R we have
lim En rf s “ Erf s
nÑ8
Notice that in the above definition topology of Ω has been used in the continuity of function f .
One can always reduce the weak convergence of random variables to weak convergence of probabilities in
the Polish space Rd . Let tYn u8
n“1 be a sequence of random variables and Y be a random variable all with
values in Rd . Then, the distributions of Yn ’s and Y defined probability measures in Rd as follows.
P̃n pAq “ Pn pY P Aq and P̃pAq “ PpY P Aq.
Notice that, as emphasized in Remark B.6, Yn and Y live in different probability spaces pΩn , Fn , Pn q and
pΩ, F, Pq, respectively. However, the probabilities P̃n and P̃ are defined on the same sample space Rd .
Corollary B.3. Yn ñ Y if and only if P̃n ñ P̃.
B.7 Donsker Invariance Principle and Brownian motion
In this section, we heuristically construct Brownian motion (or Wiener process) as the weak limit of symmetric discrete-time random walk in Definition B.1. First, we make the sample paths of random walk
continuous by linear interpolation. For t P r0, 8q, we define the interpolated random walk by Wt :“
Wrts ` pt ´ rtsqWrts`1 ; see Figure B.1. Then, tWt : t ě 0u becomes a continuous-time stochastic process
with continuous sample paths, i.e. for each t ě 0, Wt is a random variable and for any realization ω of
random walk, Wt pωq is continuous in t.
Motivated by central limit theorem, we defined normalized random walk by
pnq
Xt
1
:“ ? Wnt for all t P r0, 8q.
n
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W
?1
n
Time
n´1
A sample path of a normalized random walk X pnq
pnq
Then, we define Brownian motion is the weak limit of Xt as n Ñ 8. Indeed, a rigorous definition of
the Brownian motion is way more technical and requires advanced techniques from analysis and measure
theory. Here we only need the properties which characterize a Brownian motion.
Remark B.7. In the above construction, one can take any sequence of i.i.d. random variables tξn uně0 with
pnq
1
finite variance σ 2 and define Xt :“ σ?
W . The rest of the arguments in this section can be easily
n nt
modified for this case.
pnq
pnq
k
ℓ
Because tξi u8
´ Xs has mean zero and variance
i“1 are i.i.d., for any t “ n and s “ n with t ą s, Xt
t ´ s. If t ą s are real numbers, then the mean is still zero but the variance is n1 prtns ´ rsnsq which
pnq
pnq
pnq
converges to t ´ s. In addition, Xt ´ Xs is independent of Xu when u ď s. By central limit theorem,
pnq
pnq
Xt ´ Xs ñ N p0, t ´ sq, a normal distribution with mean zero and variance t ´ s. This suggests that
pnq
Brownian motion inherits the following properties in the limit from Xt :
a) B has continuous sample paths,
b) B0 “ 0,
c) when s ă t, the increment Bt ´ Bs is a normally distributed random variables with mean 0 and
variance t ´ s and is independent of Bu ; for all u ď s.
The properties above fully characterize the Brownian motion.
Definition B.8. A stochastic process is a Brownian motion if it satisfies the properties (a)-(c) above.
Property (c) in the definition of Brownian motion also implies some new properties for the Brownian
motion which will be useful in modeling financial asset prices.
- Time-homogeneity. Brownian motion is time homogeneous, i.e. Bt ´ Bs has the same distribution
as Bt´s .
- Markovian. The distribution of Bt given tBu : u ď su has the same distribution as Bt given Bs ,
i.e. the most recent past is the only relevant information. Notice that Bt “ Bs ` Bt ´ Bs . Since by
property (c) Bt ´ Bs is independent of Fs :“ tBu : u ď su, the distribution of Bt “ Bs ` Bt ´ Bs
given Fs “ tBu : u ď su is normal with mean Bs and variance t ´ s, which only depends on the
most recent past Bs . In other word, by Proposition B.4
Erf pBt q | Fs s “ Erf pBt ´ Bs ` Bs q | Fs s “ Erf pBt ´ Bs ` Bs q | Bs s
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- Martingale. Finally, the conditional expectation ErBt | Bu : u ď ss can be shown to be equal to
Bs .
ErBt | Bu : u ď ss “ ErBt | Bs s “ ErBs | Bs s ` ErBt ´ Bs | Bs s “ Bs .
In the above we used property (c) to conclude that ErBt ´ Bs | Bs s “ 0.
A typical sample paths of Brownian motion is shown in Figure B.2.
Figure B.2: A sample paths of Brownian motion
Sample space for Brownian motion
In order to construct a Brownian motion, we need to specify the sample space. In the early work of Kolmogorov, we choose the sample space to be pRd qr0,8q , i.e. space of all functions from r0, 8q to Rd . This is
motivated by the fact that for any ω P Ω, the sample path of Brownian motion associated with sample ω is
given by the function Bt pωq : r0, 8q Ñ Rd . See Figure B.2. Kolmogorov made a theory which in particular
resulted in the existence of Brownian motion. While it is not hard to show property (c), in his theory it
is not easy to show property (a) of Brownian motion, i.e. the sample paths are continuous. By using the
weak convergence result of Yuri Prokhorov, Norbert Wiener take the construction of Brownian motion to a
new level by taking the sample space Ω :“ Cpr0, 8q; Rd q, the space of all continuous functions. This way
property (a) becomes trivial, while, property (c) is more challenging.
Among all the continuous functions in Cpr0, 8q; Rd q only a small set can be a sample paths of a Brownian
motion. In the following, we present some of the characteristics of the paths of Brownian motion.
i) Sample paths of Brownian motion are nowhere differentiable. In addition,
lim sup
δÑ0
Bt`δ ´ Bt
Bt`δ ´ Bt
“ 8, and lim inf
“ ´8.
δÑ0
δ
δ
ii) Sample paths of Brownian motion are of bounded quadratic variation variations. More precisely, the
quadratic variation of the path of Brownian motion until time t, equals t, i.e.
lim
}Π}Ñ0
Nÿ
´1
pBti`1 ´ Bti q2 “ t, a.s.,
i“0
135
(B.10)
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Summary of the lecture
where for the partition Π :“ tt0 “ 0 ă t1 ă ¨ ¨ ¨ ă tN “ tu, }Π} “ maxi“0,...,N ´1 pti`1 ´ ti q
iii) Sample paths of Brownian motion are not of bounded variations almost surely, i.e. for
sup
Π
Nÿ
´1
|Bti`1 ´ Bti | “ 8, a.s.,
i“0
where the supremum is over all partitions Π :“ tt0 “ 0 ă t1 ă ¨ ¨ ¨ ă tN “ tu.
In the above, property (iii) is a result of (ii). More precisely, for a a continuous function g : ra, bs Ñ R, a nonzero quadratic variation implies infinite bounded variation. To show this, let’s assume that g is continuous
and bounded variation B. Then, g is uniformly continuous on ra, bs, i.e. for any ε ą 0, there exists a δ ą 0
such that if }Π} ď δ, then
max t|gpti`1 q ´ gpti q|u ď ε.
i“0,...,N ´1
Thus,
Nÿ
´1
2
pgpti`1 q ´ gpti qq ď
i“0
max
i“0,...,N ´1
t|gpti`1 q ´ gpti q|u
Nÿ
´1
|gpti`1 q ´ gpti q| ď εB.
i“0
By sending ε Ñ 0, we obtain that the the quadratic variation vanishes.
The following exercise shows the relation between the quadratic variation is martingale properties of Brownian motion.
Exercise B.12. Show that Bt2 ´ t is a martingale, i.e.
2
ErBt`s
´ pt ` sq | Bs s “ Bs2 ´ s.
Appendix C
Stochastic analysis
Calculus is the study of derivative (not financial) and integral. Stochastic calculus is therefore the study
of integrals and differentials of stochastic objects such as Brownian motion. In this section, we provide a
brief overview of the stochastic integral and Itô formula (stochastic chain rule). The application to finance
is provided in Part III.
şb
In calculus, the Riemann integral a f ptqdt is defined by the limit of Riemann sums:
lim δ
δÑ0
N
ÿ
f pt˚i q,
i“1
˚
where δ “ b´a
N , t0 “ a, ti “ t0 ` iδ, and ti is an arbitrary point in interval rti´1 , ti s. The Reiman integral
can be defined for a limited class of integrands, i.e. the real functions f is Riemann integrable on ra, bs if
and only if it is bounded and continuous almost everywhere. A natural extension of Riemann integral is
Lebesgue integral, which can be defined on a large clsss of real functions, i.e. bounded measurable function
on ra, bs.
A more general form of Riemann integral, Riemann-Stieltjes41 integral is defined in a similar fashion. For
two real functions f, g : ra, bs Ñ R, the integral of the integrand f with respect to integrator g is defined by
żb
f ptqdgptq lim
a
41
δÑ0
N
ÿ
f pt˚i qpgpti q ´ gpti´1 q.
i“1
Read“Stilchess”.
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For Riemann-Stieltjes integral, and its extension, Lebegue-Stieltjes integral, to be well-defined, we need
some conditions on f and g.
The condition on the integrand f is similar to the those in Riemann and Lebesgue integrals. For example
if f is continuous almost everywhere and at the points of discontinuity of g, then no further condition needs
to be imposed on f in the Riemann-Stieltjes integral. For the Lebesgue-Stieltjes integral, f only needs to be
measurable. However, for g a very crucial condition is needed to make the integral well-defined. g must be
of bounded variation, i.e.
N
ÿ
lim
|gpti`1 q ´ gpti q| ă 8.
δÑ0
i“0
No matter how nice the function f is, if g is unbounded variation, Riemann-Stieltjes or Lebesgue-Stieltjes
integral cannot be defined. As seen in Section B.7, the sample paths of Brownian motion are
şb of unbounded
variation, which makes it impossible to use them as the integrator. Therefore, the integral a f ptqdBt with
respect to Brownian motion cannot be defined pathwise in the sense of the Riemann-Stitljes or LebesgueStieltjes
integrals. In this section, we define a new notion of integral, Itô integral42 , which makes sense of
şb
a f ptqdBt in a useful way for some applications, including finance.
One of the major tools in stochastic analysis is the stochastic chain rule. Recall that the chain rule in the
differential form is written as dhpgptqq “ g 1 ptqh1 pgptqqdt, which can be used in the change of variable in
integral. If vptq “ hpgptqq and h and g are differentiable functions, then
żb
żb
f ptqdvptq “
a
f ptqg 1 ptqh1 pgptqqdt.
a
The right hand side above is a Riemann (Lebesgue) integral. As a matter of fact, change of variable formula for Reimann (Lebesgue) integral is the integral format of the chain rule. For bounded variation but
not necessarily differentiable function g, the chain rule in the change of variable for Riemann-Stieltjes
(Lebesgue-Stieltjes) integral can be written in a slightly different way. More precisely, if vptq “ hpgptqq
and h is a differentiable function and g is of bounded variation, then
żb
żb
f ptqdvptq “
a
f ptqh1 pgptqqdgptq.
a
In the chain rule for Itô stochastic integral, an extra term appears. If vptq “ hpBt q and h is a twice
differentiable function, then
ż
żb
żb
1 b
1
f ptqh2 pBt qdt.
f ptqdvptq “
f ptqh pBt qdBt `
2 a
a
a
The bold term in the right hand side is a simple Riemann integral. In this section, we provide a heuristic
argument why this term should be in the chain rule for Itô stochastic integral.
42
Named after Japanese mathematician Kiyosi (read Kiyoshi) Itô, 1915-2008.
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C.1 Stochastic integral with respect to Brownian motion and Itô formula
We first introduce a special case of Itô integral, called Wiener integral43 . In Wiener integral, we assume that
the integrand f is simply a real function and is not stochastic. The the partial sums
Sδ :“
N
ÿ
f pti´1 qpBti ´ Bti´1 q,
i“1
is a Gaussian random variable with mean zero and variance
δ
N
ÿ
f 2 pti´1 q.
i“1
Exercise C.1. Show that Sδ is a Gaussian random variable with mean zero and variance δ
řN
i“1 f
2 pt
i´1 q.
Therefore, it follows form
lim δ
δÑ0
N
ÿ
żb
2
f pti´1 q “
f 2 ptqdt,
a
i“1
şb
that Sδ ñ X where X is a normal random variable with mean zero and variance a f 2 ptqdt.
Notice that in the partial sum for Sδ , we choose ti´1 , i.e. the left end point on the interval rti´1 , ti s. This
choice is not crucial to achieve the limit. If we would choose different points on the interval, we still obtain
the same limiting distributions. See Exercise C.2.
Exercise C.2. Calculate that the mean and the variance of partial sums below:
ř
a) N
i“1 f pti qpBti ´ Bti´1 q.
¯
´
ř
ti `ti´1
pBti ´ Bti´1 q.
b) N
i“1 f
2
c)
řN
i“1 f pti´1 qpBti
´ Bti´1 q.
Then, show that in each case the limit of the calculated quantities as δ Ñ 0 is the same.
Itô integral extends Wiener integral to stochastic integrands. The integrand f is now a function of time
t P ra, bs and ω in sample space Ω. For our analysis in this notes, we only need to define Itô integral on the
integrands of the form f pt, Bt q where f : ra, bs ˆ R Ñ R is a measurable function. Similar to the Winer
integral we start with the partial sum
Nÿ
´1
f pti , Bti qpBti`1 ´ Bti q.
i“0
Exercise C.3. Calculate that the mean and the variance of partial sums below:
ř
a) N
i“1 Bti pBti ´ Bti´1 q.
ř
b) N
i“1 B ti `ti´1 pBti ´ Bti´1 q.
2
43
Named after American mathematician Norbert Wiener, 1894-1964.
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c)
řN
C STOCHASTIC CALCULUS
Summary of the lecture
i“1 Bti´1 pBti
´ Bti´1 q.
Then, show that in all cases the limits of the calculated quantities as δ Ñ 0 are different.
żT
f pu, Bu qdBu :“ P ´ lim
δÑ0
0
Nÿ
´1
f pti , Bti qpBti`1 ´ Bti q.
(C.1)
i“0
The notation P ´ lim means the limit is in probability, i.e. for any ε ą 0,
˜
¸
żT
´1
ˇ Nÿ
ˇ
ˇ
ˇ
P ˇ
f pti , Bti qpBti`1 ´ Bti q ´
f pu, Bu qdBu ˇ ą ε Ñ 0 as δ Ñ 0.
0
i“0
The choice of starting point ti in the interval rti , ti`1 s in f pti , Bti q is crucial. This is because choosing other
point in the interval rti , ti`1 s leads to different limits. Fo instance,
Nÿ
´1
f
i“0
´t ` t
¯
i
i`1 Bti `ti`1
,
pBti`1 ´ Bti q
2
2
converges to
żT
żT
f pu, Bu qdBu `
0
Bx f pu, Bu qdu.
0
Martingale property of stochastic integral
Consider the discrete sum which converges to the stochastic integral, i.e.
MT :“
Nÿ
´1
f pti , Bti qpBti`1 ´ Bti q
i“0
Assume that the values of B0 , ..., Btj are given. We want to evaluate the conditional expectation of the
stochastic sum MT , i.e.
ErMT | B0 , ..., Btj s.
Then, we split the stochastic sum into to parts
MT :“
j´1
ÿ
f pti , Bti qpBti`1 ´ Bti q `
Nÿ
´1
f pti , Bti qpBti`1 ´ Bti q.
i“j
i“0
The first summation of the right hand side above is known given B0 , ..., Btj . Thus,
ErMT | B0 , ..., Btj s “
j´1
ÿ
«
f pti , Bti qpBti`1 ´ Bti q ` E
i“0
“
j´1
ÿ
Nÿ
´1
ff
f pti , Bti qpBti`1 ´ Bti q | B0 , ..., Btj
i“j
f pti , Bti qpBti`1 ´ Bti q `
i“0
Nÿ
´1
i“j
139
“
‰
E f pti , Bti qpBti`1 ´ Bti q | B0 , ..., Btj
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C STOCHASTIC CALCULUS
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Each term in the second summation of right hand side above can be calculated by tower property of conditional expectation
“
“
‰
‰
E f pti , Bti qpBti`1 ´ Bti q | B0 , ..., Btj “ E f pti , Bti qErBti`1 ´ Bti | B0 , ..., Bti s | B0 , ..., Btj .
Since Bti`1 ´ Bti is independent of B0 , ..., Bti ,
ErBti`1 ´ Bti | B0 , ..., Bti s “ ErBti`1 ´ Bti s “ 0.
This implies that the second summation vanishes and we have
ErMT | B0 , ..., Btj s “
j´1
ÿ
f pti , Bti qpBti`1 ´ Bti q “: Mtj .
i“0
In other words, given the Brownian motion up to time tj , the expected values of MT is equal to Mtj . In
probability terms, we call this a martingale. By some more technical tools, one can show
şT that given the path
of a Brownian motion until time t ă T , the expected value of the stochastic integral 0 f ps, Bs qdBs is equal
şt
to 0 f ps, Bs qdBs , i.e.
E
„ż T
0
ˇ
ȷ żt
ˇ
ˇ
f ps, Bs qdBs ˇBs for s P r0, ts “
f ps, Bs qdBs .
0
One of the consequence of martingale property is that the expectation of stochastic integral is zero, i.e.
„ż T
ȷ ż0
E
f ps, Bs qdBs “
f ps, Bs qdBs “ 0.
0
0
Remark C.1. The martingale property of the stochastic integral with respect
şt to Brownian motion is basically a result of martingale property of Brownian motion. Riemann integral 0 f ps, Bs qds is a martingale if
şt
and only if f ” 0. Intuitively, if we assume 0 f ps, Bs qds is a martingale, we have
E
” ż t`δ
t
ˇ ı
ˇ
f ps, Bs qdsˇFt “ 0.
By dividing both sides by δ and then sending δ Ñ 0, we obtain
ˇ ı
” 1 ż t`δ
ˇ
lim E
f ps, Bs qdsˇFt “ Erf pt, Bt q | Ft s “ f pt, Bt q.
δÑ0
δ t
C.2 Itô formula
One of the important implications of Itô integral is a very powerful tool called Itô formula. Itô formula is
the the stochastic version of Taylor expansion. To understand this better let’s try to write Taylor expansion
for V pt ` δ, Bt`δ q about the point V pt, Bt q.
1
V pt ` δ, Bt`δ q “ V pt, Bt q ` δBt V pt, Bt q ` Bx V pt, Bt qpBt`δ ´ Bt q ` Bxx V pt, Bt qpBt`δ ´ Bt q2 ` opδq.
2
(C.2)
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?
The remaining term in of order opδq since Bt`δ ´ Bt „ Op δq. Also, we know that
pBt`δ ´ Bt q2 ´ δ „ opδq, pBt`δ ´ Bt qδ „ opδq, and trivially δ 2 „ opδq.
(C.3)
If we take conditional expectation with respect to Bt , we obtain
:0
ErV pt ` δ, Bt`δ q | Bt s “ V pt, Bt q ` δBt V pt, Bt q ` Bx V pt, Bt q ErpBt`δ
´
Bt q | Bt s
1
` Bxx V pt, Bt qδ ` opδq.
2
1
“ V pt, Bt q ` pBt V pt, Bt q ` Bxx V pt, Bt qqδ ` opδq
2
Then, we obtain
«
ff
1
pBt V pti , Bti q ` Bxx V pti , Bti qqδ ` op1q,
ErV pT, BT qs “ V p0, B0 q ` E
2
i“0
Nÿ
´1
which in the limit converges to
ErV pT, BT qs “ V p0, B0 q ` E
„ż T
0
ȷ
1
pBt V pti , Bt q ` Bxx V pt, Bt qqdt .
2
The above formula is called Dynkin formula. If we don’t take conditional expectation, we can write
1
V pt ` δ, Bt`δ q “ V pt, Bt q ` pBt V pt, Bt q ` Bxx V pt, Bt qqδ ` Bx V pt, Bt qpBt`δ ´ Bt q ` opδq.
2
Then, we obtain
V pT, BT q “ V p0, B0 q `
Nÿ
´1
Nÿ
´1
1
pBt V pti , Bti q ` Bxx V pti , Bti qqδ `
Bx V pti , Bti qpBti`1 ´ Bti q ` op1q.
2
i“0
i“0
which in the limit converges to
żT
V pT, BT q “ V p0, B0 q `
0
1
pBt V pt, Bt q ` Bxx V pt, Bt qqdt `
2
żT
Bx V pt, Bt qdBt .
0
In the above, (C.4) is referred to as Itô formula.
In a less formal way, Utô formula is given by
1
dV pt, Bt q “ pBt V pt, Bt q ` Bxx V pt, Bt qqdt ` Bx V pt, Bt qdBt .
2
However, it has to be interpreted as (C.4).
Exercise C.4. Use Itô formula to calculate dV pt, Bt q in the following cases.
a. V pt, xq “ eax
b. V pt, xq “ e´t eax
c. V pt, xq “ e´t cospxq
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C STOCHASTIC CALCULUS
d. V pt, xq “ e´t xa
where a is a given constant.
C.3 Martingale property of stochastic integral and partial differential equations
Why martingale property of stochastic integral is important? Recall the Itô formula
1
dV pt, Bt q “ pBt V pt, Bt q ` Bxx V pt, Bt qqdt ` Bx V pt, Bt qdBt .
2
Assume that V pt, xq satisfies the PDE
1
Bt V pt, xq ` Bxx V pt, xq “ 0.
2
(C.5)
Then,
dV pt, Bt q “ Bx V pt, Bt qdBt .
and thus V pt, Bt q is a martingale, i.e.
V pt, Bt q “ ErV pT, BT q | Bs for s P r0, tss.
Conversely, if V pt, Bt q is a martingale, then V pt, xq must satisfy the PDE (C.5).
C.4 Stochastic integral and Stochastic differential equation
Riemann integral allows us to write a differential equation dx
dt “ f pt, xptqq, xp0q “ x0 as an integral
equation
żt
xptq “ x0 ` f ps, xpsqqds.
(C.6)
0
Integral equations are more general because the solution does not necessarily need to be differentiable. For
instance if
#
1 tě1
f pt, xq “
.
0 tă1
The solution to the integral equation is xptq “ x0 ` pt ´ 1q` , which is not differentiable at t “ 1. While
dx “ f pt, xptqqdt should be interpreted as (C.6).
Itô integral allows us to define stochastic differential equations (SDE for short) in integral form. For
example, the Black-Scholes differential equation is given by
dSt
“ rdt ` σdBt .
St
The true meaning of this term is
żt
St “ S0 ` r
żt
Su du ` σ
0
Su dBu .
0
The solution is a stochastic process St which satisfies the SDE. In the above case, it is easy to verify, by
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means of Itô formula, that the geometric Brownian motion
ˆ
˙
1 2
St “ S0 exp pr ´ σ qt ` σBt .
2
`
˘
Take V pt, xq “ S0 exp pr ´ 21 σ 2 qt ` σx . It follows from Itô formula that
żt
1
V pt, Bt q “ V p0, B0 q ` pBt V ps, Bs q ` Bxx V ps, Bs qqdt `
2
0
żt
Bx V ps, Bs qdBs .
0
Since V p0, xq “ S0 , Bt V ps, xq “ pr ´ 21 σ 2 qV ps, xq, Bx V ps, xq “ σV ps, xq, and Bxx V ps, xq “ σ 2 V ps, xq,
we have
żt
żt
V pt, Bt q “ S0 ` r V ps, Bs qds ` σ V ps, Bs qdBs .
0
0
In general for a pair of given functions µpt, xq and σpt, xq, an equation of the form
dSt “ µpt, St qdt ` σpt, St qdBt
(C.7)
is called a stochastic differential equation (SDE). A solution St is a process such that the Lebesgue integral
żt
µps, Ss qds ă 8 P´a.s.,
0
the Itô integral
żt
σps, Ss qdBs
0
is well-defined, and the following is satisfied:
żt
żt
St “ S0 ` µps, Ss qds ` σps, Ss qdBs .
0
0
For CEV model in Section 4.1, SDE is written as
dSt
“ rdt ` σStβ dBt
St
or in integral form
żt
St “ S0 ` r
żt
Su du ` σ
0
Su1`β dBu .
0
One of the most important applications of the Itôs formula is the chain rule in the stochastic form. Consider
SDE
dSt “ µpt, St qdt ` σpt, St qdBt
(C.8)
and V pt, St q where V pt, xq is a function one time continuously differentiable in t and twice continuously
differentiable on x. Then we can write
1
V pt ` δ, St`δ q “ V pt, St q ` Bt V pt, St qδ ` Bx V pt, St qpSt`δ ´ St q ` Bxx V pt, St qpSt`δ ´ St q2 ` opδq.
2
Notice that by (C.8), we have
St`δ ´ St « µpt, St qδ ` σpt, St qpBt`δ ´ Bt q.
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Therefore, we have
pSt`δ ´ St q2 “ σ 2 pt, St qpBt`δ ´ Bt q2 ` opδq.
and
V pt ` δ, St`δ q “ V pt, St q ` Bt V pt, St qδ ` Bx V pt, St qpµpt, St qδ ` σpt, St qpBt`δ ´ Bt qq
1
` Bxx V pt, St qpµpt, St qδ ` σpt, St qpBt`δ ´ Bt qq2 ` opδq
2
“ V pt, St q ` Bt V pt, St qδ ` Bx V pt, St qpµpt, St qδ ` σpt, St qpBt`δ ´ Bt qq
1
` Bxx V pt, St qσ 2 pt, St qδ ` opδq
2
1
“ V pt, St q ` pBt V pt, St q ` Bx V pt, St qµpt, St q ` Bxx V pt, St qσ 2 pt, St qqδ
2
` σpt, St qBx V pt, St qpBt`δ ´ Bt qq ` opδq.
Or in the integral form we have
V pT, ST q “ V p0, S0 q `
żT´
0
¯
1
Bt V pt, St q ` Bx V pt, St qµpt, St q ` Bxx V pt, St qσ 2 pt, St q dt
2
żT
`
σpt, St qBx V pt, St qdBt .
0
C.5 Itô calculus
The calculations in the previous section can be obtained from a formal calculus. First, we formally write
(C.2) as
1
dV pt, Bt q “ Bt V pt, Bt qdt ` Bx V pt, Bt qdBt ` Bxx V pt, Bt qpdBt q2 .
2
Then, we present (C.3) in formal form of
pdBt q2 “ dt, dBt dt “ dtdBt “ 0 , and pdtq2 “ 0.
(C.9)
which implies the Itô formula for Brownian motion.
1
dV pt, Bt q “ pBt V pt, Bt q ` Bxx V pt, Bt qqdt ` Bx V pt, Bt qdBt .
2
For the Itô formula for process St in (C.8), we can formally write
1
dV pt, St q “ Bt V pt, St qdt ` Bx V pt, St qdSt ` Bxx V pt, St qpdSt q2 .
2
Then, we use (C.9) to obtain
pdSt q2 “ µ2 pdtq2 ` 2µσdtdBt ` σ 2 pdBt q2 “ σ 2 dt.
Thus,
dV pt, St q “ pBt V pt, St q `
σ 2 pt, St q
Bxx V pt, St qqdt ` Bx V pt, St qdSt .
2
Theorem C.1 (Itô formula). Consider St given by (C.8) and assume that function V is once continuously
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differentiable in t and twice continuously differentiable in x. Then, we have
dV pt, St q “ pBt V pt, St q ` µpt, St qBx V pt, St q `
σ 2 pt, St q
Bxx V pt, St qqdt ` σpt, St qBx V pt, St qdBt .
2
Similar to Section C.3, we can use martingale property of stochastic integral to obtain a PDE. More precisely, V pt, St q is a martingale if and only if V pt, xq satisfies
1
Bt V pt, xq ` µpt, xqBx V pt, xq ` σ 2 pt, xq Bxx V pt, xq “ 0.
2
Exercise C.5. Let
dSt “ St dt ` 2St dBt .
Calculate dV pt, St q in the following cases.
a. V pt, xq “ eax
b. V pt, xq “ e´t eax
c. V pt, xq “ e´t cospxq
d. V pt, xq “ e´t xa
where a is a constant.
Exercise C.6. In each of the following SDE, find the PDE for the function V pt, xq such that V pt, St q is a
martingale.
a. dSt “ σdBt where σ is constants.
b. dSt “ κpm ´ St qdt ` σdBt where κ, m and σ are constants.
?
c. dSt “ κpm ´ St qdt ` σ St dBt where κ, m and σ are constants.
d. dSt “ rSt dt ` σSt2 dBt where r and σ are constants.
Exercise C.7. Consider the SDE
dSt “ rSt dt ` σSt dBt where r and σ are constants.
a. Find the ODE for the function V pxq such that e´rt V pSt q is a martingale.
b. Find all the solutions to the ODE in (a).
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