Download Geometric Brownian Motion

Geometric Brownian Motion
Note that as a model for the rate of return, dS(t)/S(t) geometric
Brownian motion is similar to other common statistical models:
dS(t)
S(t)
=
µdt
+
σdW (t)
or
response = systematic component + random error.
Without the stochastic component, the differential equation has
the simple solution
S(t) = ceµt,
from which we get the formula for continuous compounding for
a rate µ.
1
An Intuitive Examination of Geometric
Brownian Motion in Prices
What rate of growth do we expect for S in the geometric Brownian motion
model
dS(t)
= µdt + σdW (t)?
S(t)
Should it be µ because that is the rate for the systematic component, and
the expected value of the random component is 0?
Consider a rate of change σ, that is equally likely to be positive or negative.
What is the effect on a given quantity if there is an uptick of σ followed by
a downtick of equal magnitude (or a downtick followed by an uptick)? The
result for the two periods is
−σ 2 .
(This comes from the multiplication of the given quantity by (1 + σ)(1 − σ).)
The average over the two periods is −σ 2/2.
The stochastic component reduces the expected rate of µ by −σ 2 /2.
This is the price of risk.
2
Ito’s Lemma
We can formalize the preceding discussion using Ito’s formula.
Ito’s lemma:
Suppose X follows an Ito process,
dX(t) = a(X, t)dt + b(X, t)dW (t),
where dW is a Wiener process. Let G be an infinitely differentiable function of X and t. Then G follows the process
dG(t) =
1 ∂ 2G
!
∂G
∂G
2 dt + ∂G b(X, t)dW (t). (1)
a(X, t) +
+
b
∂X
∂t
2 ∂X 2
∂X
3
Ito’s Lemma
Thus, Ito’s lemma provides a formula that tells us that G also
follows an Ito process.
The drift rate is
∂G
∂G
1 ∂ 2G 2
a(X, t) +
+
b
2
∂X
∂t
2 ∂X
and the volatility is
∂G
b(X, t).
∂X
This allows us to work out expected values and standard deviations of G over time.
4
Derivation of Ito’s Formula
First, suppose that G is an infinitely differentiable function of X and an
unrelated variable y, and consider a Taylor series expansion for ∆G:
2G
2G
∂G
1 ∂ 2G
∂
∂
∂G
2
2
∆X +
∆y +
(∆X)
+
(∆y)
+2
∆X∆y + · · ·
∆G =
∂X
∂y
2 ∂X 2
∂y2
∂X∂y
(2)
In the limit as ∆X and ∆y tend to zero, this is the usual “total derivative”
dG =
∂G
∂G
dX +
dy,
∂X
∂y
(3)
in which the terms in ∆X and ∆y have dominated and effectively those in
(∆X)2 and (∆y)2 and higher powers have disappeared.
Now consider an X that follows an Ito process,
dX(t) = a(X, t)dt + b(X, t)dW (t),
or
√
∆X(t) = a(X, t)∆t + b(X, t)Z ∆t.
Now let G be a function of both X and t, and consider the analogue to
equation (2). The factor (∆X)2 , which could be ignored in moving to equation (3), now contains a term with the factor ∆t, which cannot be ignored.
We have
(∆X(t))2 = b(X, t)2 Z 2∆t + terms of higher degree in ∆t.
5
Consider the Taylor series expansion
∂G
∂G
1
∆G =
∆X +
∆t +
∂X
∂t
2
2
2
∂ 2G
∂
G
∂
G
2
2
(∆X)
+
(∆t)
+2
∆X∆t + · · ·
∂X 2
∂t2
∂X∂t
(4)
Now under the assumptions of Brownian motion, (∆X(t))2 or,
equivalently, Z 2 ∆t is nonstochastic; that is, we can treat Z 2 ∆t
as equal to its expected value as ∆t tends to zero. Therefore,
when we substitute for ∆X(t), and take limits in equation (4)
as ∆X and ∆t tend to zero, we get
∂G
∂G
1 ∂ 2G 2
dG(t) =
dX +
dt +
b dt
(5)
∂X
∂t
2 ∂X 2
or, after substituting for dX and rearranging, we have Ito’s formula
dG(t) =
1 ∂ 2G
!
∂G
∂G
2 dt + ∂G b(X, t)dW (t).
a(X, t) +
+
b
∂X
∂t
2 ∂X 2
∂X
Equation (5) is also called Ito’s formula. Compare equation (5)
with equation (3).
6
Applications of Ito’s Formula
Ito’s formula has applications in many stochastic differential equations used
as models in finance.
In the differential equation for geometric Brownian motion for S,
dS(t) = µS(t)dt + σS(t)dW (t),
we can let G = log S, and so substituting in Ito’s formula we have
σ2
dt + σdW (t).
dG(t) = µ −
2
Using previous results we have that the difference in G at time zero and time
T is normally distributed with mean
2
σ
T
µ−
2
(recognize this term from our intuitive discussion) and variance
σ 2 T.
7
The Distribution of Stock Prices
The geometric Brownian motion model is the simplest model
for stock prices that is somewhat realistic. Looking at it somewhat critically, we can see certain problems. First, is the form
of the differential equation reasonable? Next, we have the big
questions: are µ and σ constant? We will explore these issues
later.
The basic distributional assumption in the geometric Brownian
motion model is that the rates of change of stock prices in very
small increments of time are identically and independently normally distributed.
8
The Distribution of Stock Prices
The compounded rate of return is
1
log(S(t + ∆t)/S(t)).
∆t
From t0 to T we can write this as
1
1
log(ST ) −
log(St0 ).
T − t0
T − t0
If this has a normal distribution, then log(ST ) has a normal distribution; that is, ST has a lognormal distribution.
*** properties
9
Solution of Stochastic Differential Equations
The solution of a differential equation is obtained by integrating
both sides and allowing for constant terms. Constant terms
are evaluated by satisfying known boundary conditions, or initial
values.
In a stochastic differential equation (SDE), we must be careful
in how the integration is performed, although different interpretations may be equally appropriate.
For example, the SDE that defines an Ito process
dX(t) = a(X, t)dt + b(X, t)dW (t),
when integrated from time t0 to T yields
X(T ) − X(t0) =
Z T
t0
a(X, t)dt +
Z T
t0
b(X, t)dW (t).
The second integral is a stochastic integral. We will interpret it
as an Ito integral.
10
The nature of a(X, t) and b(X, t) determine the complexity of the
solution to the SDE.
In the Ito process
dS(t) = µ(t)S(t)dt + σ(t)S(t)dW (t),
using Ito’s formula for the log as before, we get the solution
S(T ) = S(t0) exp
Z T
t0
!
Z T
1
µ(t) − σ(t)2 dt +
σ(t)dW (t) .
2
t0
In the simpler version of a geometric Brownian motion model, in
which µ and σ are constants, we have
S(T ) = S(t0) exp
1 2
µ − σ ∆t + σ∆W .
2
11
Expected Values of Solutions of Stochastic
Differential Equations
Given a solution of a differential equation we may determine the
mean, variance and so on by taking expectations of the random
component in the solution.
Sometimes, however, it is easier just to develop an ordinary (nonstochastic) differential equation for the moments. We do this
from an Ito process
dX(t) = a(X, t)dt + b(X, t)dW (t),
by using Ito’s formula on the powers of the variable. So we have
1
dX p (t) = pX(t)p−1a(X, t) + p(p − 1)X(t)p−2b(X, t)2 dt +
2
pX(t)p−1 b(X, t)dW (t).
** exercise
Taking expectations of both sides, we have an ordinary differential equation in the expected values.
12
Geometric Brownian Motion in Prices
Although the geometric Brownian motion model for rates of returns is quite useful,
dS(t)
= µdt + σdW (t),
S(t)
it has limitations. As we have mentioned, one problem is the
assumption of constancy of µ and σ.
problem of stochastic volatility
There are other considerations also.
shocks, market and stock
13
Derivatives: Basics
A derivative is a financial instrument whose value depends on
values of other financial instruments or on some measure of the
state of the economy or of nature. The instrument or measure
on whose value the derivative depends is called the “underlying”.
A derivative is an agreement with two sides.
One of the most important questions in finance is how to price
a derivative.
14
Forward Contracts
One of the simplest kinds of derivatives is a forward contract,
which is an agreement to buy or to sell an asset at a specified
time at a specified price.
The agreement to buy is a long position and the agreement to
sell is a short position. The agreed upon price is the delivery
price. The consumation of the agreement is an execution.
Forward contracts are relatively simple to price, and their analysis
is important for developing pricing methods for other derivatives.
15
Analysis of Forward Contracts
Using standard notation, let K be the delivery price or “strike price”, at time
T , and let Xt be the value of the underlying.
The result at settlement is shown below.
payoff
payoff
6
0
6
@
-
K
long position
XT
0
@
@
@
@
@
@
@
@
K
@
@
XT
@
@
@
@
@
short position @
@
The important question is what is the price of the derivative as a function of
XT and of time to settlement.
For a forward contract it is easy. The answer is
F0 = X0erT − K,
where r is the (annual) riskfree rate of growth, and T is the time (in years).
16
Derivatives: Basics
There is a variety of modifications to the basic forward contract
that involve
• nature of the underlying
– asset
∗ investment
∗ consumable
∗ income producing
– nonasset (e.g., index, price of electricity, weather)
• negociability of the instrument (market, possibility of short
positions, intermediate party, etc.)
17
Derivatives: Basics
Other modifications to the basic forward contract involve
• nature of the agreement (right, that is, “contingent claim”,
or obligation)
• flexibility of time of execution (at a specified time or up to
a specified time)
• method of settlement (cash or delivery)
• dependence of the derivative value on the path of the value
of the underlying
18
Types of Derivatives
These variations on the basic forward contract are all interesting. Only a few
of them are actually available.
Some variations are much easier to analyze than others. The simple ones are
interesting for classroom analyses and they may provide useful approximations
for derivatives that are actually available.
For individual investors, there is a relatively small set of derivatives available
(realistically).
For all of them there is a ready market (always) through a third party, and
short sales are possible.
Most of the readily traded derivatives are options, or contingent claims. That
kind of derivative is a right; not an obligation.
Therefore, a long position is a right and a short position (in the derivative)
is an obligation. The right “expires” at the settlement date.
19
Derivatives That Have Markets
The common types of derivatives are
• Stock options
• Index options
• Commodity futures
• Rate futures
20
Uses
Stock options are used by individual investors and by investment
companies for leverage, hedging, and income.
Index options are used by individual investors and by investment
companies for hedging and speculative income.
Commodity futures are used by individual investors for speculative income, by investment companies for income, and by producers and traders for hedging.
Rate futures are used by individual investors for speculative income, by investment companies for income and for hedgins, and
by traders for hedging.
21
Types of Common Derivatives
The variations depend on the nature of the underlying.
• Stock (investment asset, possibly income-producing). The buy side is a
“call” and the sell side is a “put”.
– Settlement (exercise) is by delivery.
– Exercise can be any time prior to expiration date (“American style”).
• Index (investment asset, not income-producing). The buy side is a “call”
and the sell side is a “put”.
– Settlement is by cash.
– Exercise can only be at expiration date.
• Commodity (consumable asset, not income-producing).
– Settlement is by delivery.
– Exercise can only be at expiration date.
22
U.S. Stock Options
A market for stock options in the U.S. is one of the national
security exchanges: Amex, CBOE, NYSE, Pacific Exchange, and
Philadelphia Exchange.
All (almost all) options are initiated with and through the Options Clearing Corporation, owned by the exchanges and headquartered on LaSalle St.
Options (and also futures) are regulated by the Commodity Futures Trading Commission (the analogue of the SEC). The SEC
also regulates options through its regulatory oversight of the
exchanges.
23
Analysis of Stock Options
Another important difference between stock options and forward
contracts is that stock options are rights, not obligations. The
payoff therefore cannot be negative.
Because the payoff cannot be negative, there must be a cost to
obtain a stock option.
The profit is the difference between the payoff and the price paid.
Another difference in stock options and forward contracts is that
(real-world) stock options can be exercised at any time (during
trading hours) prior to expiration. We will, however, often consider a modification, the “European option”, which can only be
exercised at a specified time. (There are some European options that are actually traded, but they are generally for large
amounts, and they are rarely traded by individuals.)
24
Analysis of Stock Options
The two sides of a forward contract result in the two types of stock options.
The results at expiration are
profit
profit
6
0
6
@
K
-
X
@
@
0
call option
@
@
@
@
K
@
@@
-
X
put option
For short positions, just flip the graphs.
25
Market Models for Derivative Pricing
A simple model of the market assumes two assets:
a riskless asset with price at time t of βt ,
and a risky asset with price at time t of Xt .
The price of a derivative can be determined based on trading
strategies involving these two assets.
The price of the riskless asset follows the deterministic ordinary
differential equation
dβt = rβt dt,
where r is the instantaneous riskfree interest rate.
The price of the risky asset follows the stochastic differential
equation
dXt = µXtdt + σXt dWt.
26
Pricing Derivatives: Basics
Start with a European call option. How do we value it?
Current time: t0
Expiration: T
Strike price: K
Price of underlying: S(t0), . . . , S(t), . . . , S(T )
Value of the call: C(t0), . . . , C(t), . . . , C(T )
P for put; V for either
We have been (and in this course will continue to be) vague
about the pricing unit. In general, we call the price, or the
pricing unit, a numeraire. A more careful development of this
concept rests on the idea of a pricing kernel.
We will usually call them dollars.
27
The Price of a European Call Option
A European call option is a contract that gives the owner the
right to buy a specified amount of an underlying for a fixed strike
price, K on the expiration or maturity date T .
The owner of the option does not have any obligations in the
contract.
The payoff, h, of the option at time T is either 0 or the excess
of the price of the underlying S(T ) over the strike price K.
Once the parameters K and T are set, it is a function of S(T ):
h(S(T )) =
(
S(T ) − K
0
if S(T ) > K
otherwise
28
The Price of Call Options
The price of the option at any time is a function of the time t,
and the price of the underlying s. We denote it as P (t, s).
What is the price at time t = 0?
It seems natural that the price of the European call option should
be the expected value of the payoff of the option at expiration,
discounted back to t = 0:
P (0, s) = e−rT E(h(S(T ))).
Likewise, for an American option, we could maximize the expected value over all stopping times, 0 < τ < T :
P (0, s) = sup e−rτ E(h(S(τ ))).
τ ≤T
29
Principles
Basic principle of Black-Scholes: We seek a portfolio with zero
expected value, that consists of short and/or long positions in
the option, the underlying, and a risk-free bond.
There are two key ideas in developing pricing formulas for derivatives:
1. no-arbitrage principle
2. replicating, or hedging, portfolio
30
An arbitrage is a trading strategy with a guaranteed rate of return
that exceeds the riskless rate of return.
In financial analysis, we assume that arbitrages do not exist.
31
Example of the No-Arbitrage Principle
Consider a forward contract that obligates one to pay K at T for the underlying. At time t, with t < T , the price of the underlying is S(t). What should
the price of the contract be or, equivalently,
What should K be so that the price of the contract is 0?
Its value at expiry is S(T )− K, and of course we do not know S(T ). If we have
a riskless rate of return r, we can use the no-arbitrage principle to determine
the correct price of the contract.
To apply the no-arbitrage principle, consider the following strategy:
take a long position in the forward contract and
sell the underlying short.
With this strategy, the investor immediately receives S(t). At time T this
amount can be guaranteed to be
S(t)er(T −t) .
32
The No-Arbitrage Principle
If
K < S(t)er(T −t) ,
a long position in the forward contract and
a short position in the underlying
is an arbitrage.
Conversely, if
K > S(t)er(T −t) ,
a short position in the forward contract and
a long position in the underlying
is an arbitrage.
Therefore, under the no-arbitrage assumption, the correct value
of K is S(t)er(T −t).
33
The replication approach is to determine a portfolio and an associated trading strategy that will provide a payout that is identical
to that of the underlying. This portfolio and trading strategy
replicates the derivative.
A replicating strategy involves both long and short positions.
34
Pricing Derivatives
If every derivative can be replicated by positions in the underlying
(and cash), the economy or market is said to be complete.
We will generally assume complete markets.
The Black-Scholes approach leads to the idea of a self-financing
replicating hedging strategy.
The approach yields the interesting fact that the price of the call
does not depend on the expected value of the underlying.
It does depend on its volatility, however.
35
Self-Financing Replicating Hedging Strategy
Neil Chriss’s example of a casino.
Game is to flip a fair coin 3 times.
Casino pays $1 if heads occurs 3 times in a row, HHH.
How much should it cost to play? (Casino will then add operating
and profit margin.)
Relation to options; who’s short and who’s long
Big player: $100,000.
Bet broker: casino bets $12,500 on H on first toss; casino is
even
If H occurs, casino has $25,000 and bets on H on second toss
36
Analysis of Casino Hedging Example
Hedging – in general
Special properties:
Self-financing
Replicating.
Roles of three participants; who’s short and who’s long
bid-ask spread (will broker charge the expected value of the bet?)
other transaction costs...
Assumptions: market impact, complete market
37
Expected Rate of Return on Stock
XYZ selling at S(t0); no dividends.
What is its expected value at time T > t0?
It is merely the forward price for what it could be bought now.
Forward price: er(T −t0 )S(t0),
where r is the risk-free rate of return, S(t0) is the spot price,
and T − t0 is the time interval.
This is the no-arbitrage principle.
The expected value of the stock does not depend on the
rate of return of the stock
(that’s µ in some of the models we’ve used).
This is true because of the cost of a forward contract.
38
Call Option
Holder of the forward contract (long position) on XYZ must buy
stock at time T for er(T −t0 ) S(t0).
Holder of a call option buys stock only if S(T ) > K.
Role of volatility
on forward contract holder (volatility not good)
on call option holder (volatility good) – enhances the value of
the option
Assumptions ...
Conclusion under assumptions: expected volatility of the underlying affects the value of an option, but expected rate of return
of the underlying does not.
39
Hedging
Hedging risk: defray the risk of one investment by making an
offsetting investment.
Cost of hedge.
Return of hedge (reduced risk).
Perfect hedge: returns exactly the amount needed to cover any
loss, and no more.
Example: write call (i.e., go short). what is a hedge? owning enough stock to cover is a hedge, but not perfect (it costs
too much, and writer is not protected against drop in price of
underlying).
40
Hedging
Black-Scholes approach constructs a portfolio consisting of some
underlying and some risk-free (zero-coupon) bonds to offset the
short call.
(discuss dividends, coupons, etc.)
41
Hedging
In a perfect hedge, the hedging instrument behaves exactly the
way the hedged instrument does.
Call option vs. hedging instrument: value at T .
they’re equal ... called “payoff replication”.
42
Dynamic Hedging
process of managing the risk of options
hedging portfolio’s value at any time is equal to the value of the
option at that point in time.
1. replicates the payoff
2. has fixed and known total cost
weighted portfolio, balancing
hedging strategy produces a synthetic version of the option.
43
Self-Financing Dynamic Hedging
Cost of hedge:
1. infusion of funds cost
2. transaction costs (bid-ask, friction, inability to execute trades
at exactly the price specified by the strategy, etc.)
A hedging strategy is self-financing if its total to-date cost at
any time (excluding transaction costs) is equal to the setup cost.
Setup cost: initial outlay; e.q. strategy requires going long
$1,000 and short $700; setup is $300
To do this, we need a formula for the relative rates of change of
the price of the call and that of the underlying,
∆=
dC
.
dS
44
Delta of an Option
The delta of an option is the rate of change of the option’s value
with respect to the change in the underlying’s price.
Consider times t0 and t1. If the value of an option at time t is
V (t), and the price of the underlying is S(t), the delta at t0 is
approximated by
V (t0) − e−r(t1 −t0) V (t1)
.
∆t0 =
−r(t
−t
)
1
0 S(t1 )
S(t0) − e
We essentially neutralize the change in time by the risk-free rate.
comments: zero in denominator; time value
45
Market Models for Derivative Pricing
A simple model of the market assumes two assets:
a riskless asset with price at time t of βt ,
and a risky asset with price at time t of S(t).
The price of a derivative can be determined based on trading
strategies involving these two assets.
The price of the riskless asset follows the deterministic ordinary
differential equation
dβt = rβt dt,
where r is the instantaneous riskfree interest rate.
The price of the risky asset follows the stochastic differential
equation
dS(t) = µS(t)dt + σS(t)dWt .
46
Preliminary Formula
C(t) = ∆t S(t) − er(T −t) B(t),
where B(t) is the current value of a riskless bond.
47
We speak of a portfolio as a vector p whose elements sum to 1.
The length of the vector is the number of assets in our universe.
Sometimes we limit this to assets whose values are independent
of each other; that is, we may exclude derivatives.
The no-arbitrage principle can be stated as:
There does not exist a p such that for some t > 0, either
• pTs0 < 0 and pTS(t)(ω) ≥ 0 for all ω,
or
• pTs0 ≤ 0 and pTS(t)(ω) ≥ 0 for all ω, and pTS(t)(ω) > 0 for
some ω.
48
A derivative D is said to be attainable (over a universe of assets
S = (S (1) , S (2) , . . . , S (k) )) if there exists a portfolio p such that
for all ω and t,
Dt(ω) = pTS(t)(ω).
Not all derivatives are attainable. The replicating portfolio approach to pricing derivatives applies only to those that are attainable.
49
Dynamic and Self-Financing Portfolios
The value of a derivative changes in time and as a function of the
value of the underlying; therefore, a replicating portfolio must be
changing in time or “dynamic”.
In analyses with replicating portfolios, transaction costs are ignored.
Also, the replicating portfolio must be self-financing; that is,
once the portfolio is initiated, no further capital is required. Every purchase is financed by a sale.
50
A Replicating Strategy
Using our simple market model, with
a riskless asset with price at time t of βt ,
and a risky asset with price at time t of S(t),
(with the usual assumptions on the prices of these assets),
we can construct a portfolio whose value will almost surely the
payoff of a European call option on the risky asset at time T .
At time t, the portfolio consists of at units of the risky asset,
and of bt units of the riskless asset. Therefore, the value of the
portfolio is atS(t) + btβt . If we scale βt so that β0 = 1 and adjust
bt accordingly, the expression simplifies, so that βt = ert.
The portfolio replicates the value of the option at time T if it
has value K − S(T ) if this is positive and zero otherwise.
If the portfolio is self-financing
d(atS(t) + btert) = at dS(t) + rbtertdt.
51
The Black-Scholes Differential Equation
Consider the fair value V of a European call option at time
t < T . At any time this is a function of both t and the price
of the underlying St . We would like to construct a dynamic,
self-financing portfolio (at , bt) that will replicate the derivative at
maturity. If we can, then the no-arbitrage principle requires that
atSt + btert = V (t, St ),
for t < T .
Further, if V (t, St ) is continuously twice-differentiable, we can
use Itô’s formula to develop an expression for V (t, St ).
We assume no-arbitrage and we assume that a risk-free return
is available.
These are big “if”s, but nevertheless, let’s proceed.
52
First, differentiate both sides of the equation that represents a
replicating portfolio with no arbitrage:
atdSt + rbtert dt =
1 ∂ 2V
!
∂V
∂V
2S 2 dt
µSt +
+
σ
t
∂St
∂t
2 ∂St2
∂V
+
(σSt )dBt
∂St
By the market model for dSt the left-hand side is
(atµSt + rbtert )dt + at σSt dBt.
Equating the coefficients of dBt, we have
∂V
.
∂St
From our equation for the replicating portfolio we have
at =
bt = (V (t, St ) − atSt ) e−rt.
53
Now, equating coefficients of dt and substituting for at and bt,
we have the Black-Scholes differential equation,
∂V
r V − St
∂St
!
∂V
1 2 2 ∂ 2V
=
+ σ St
∂t
2
∂St2
Notice that µ is not in the equation.
54
Delta Hedging
The Black-Scholes differential equation can also be derived by
constructing a riskless, self-financing portfolio consisting of a
long position in the underlying and a short position in the call.
Letting At be the amount long in the underlying, and Nt be the
amount short in the derivative, we arrive at the equation
At
∂V
=
.
Nt
∂St
At
The ratio N
is called the Delta, and this approach to riskless
t
portfolio construction is called delta hedging.
55
The Black-Scholes Differential Equation
Instead of European calls, we can consider European puts, and
proceed in the same ways (replicating portfolio or Delta hedging).
We arrive at the same the Black-Scholes differential equation
(rewritten),
∂V
1 2 2 ∂ 2V
∂V
+ σ St
= rV.
+ rSt
2
∂t
∂St
2
∂St
56
The Black-Scholes Formula
The solution depends on the boundary conditions. In the case
of European options, these are simple.
For calls, they are
Vc (T, St ) = (St − K)+
For puts, they are
Vp(T, St ) = (K − St )+
57
The Black-Scholes Formula
With these boundary conditions, there are closed form solutions
to the Black-Scholes differential equation.
For the call, it is
CBS(t, St ) = St Φ(d1) − Ke−r(T −t)Φ(d2),
where
log(St /K) + (r + 1
σ 2)(T − t)
2
√
d1 =
,
σ T −t
√
d2 = d1 − σ T − t,
and
s
2 /2
1
−y
Φ(s) = √
e
dy.
2π −∞
Z
58
The Black-Scholes Pricing Formula
It is interesting to look at the Black-Scholes prices as a function
of the price of the underlying.
20
10
0
Call Option Price
30
40
The Black−Scholes Call Pricing Function
60
80
100
120
140
Current Stock Price
59
20
10
0
Put Option Price
30
40
The Black−Scholes Put Pricing Function
60
80
100
120
140
Current Stock Price
60
Assumptions
The Black-Scholes model depends on several assumptions:
• differentiability of stock prices with respect to time
• a dynamic replicating portfolio can be maintained without
transaction costs
• returns are
· independent
· normal
· mean stationary
· variance stationary
61
Stochastic Volatility
A condition in which the variance of the returns is random (rather
than stationary) is called “stochastic volatility”.
Data on returns provide empirical evidence that the volatility is
stochastic.
62
Stochastic Volatility and Implied Volatility
Stochastic volatility can also account for other empirically observed failures of the Black-Scholes model.
For a given option (underlying, type, strike price, and expiry)
and given the price of the underlying and the riskfree rate, the
Black-Scholes formula relates the option price to the volatility
of the underlying; that is, the volatility determines the model
option price.
The actual price at which the option trades can be observed,
however.
If this price is used in the Black-Scholes formula, the volatility
can be computed. This is called the “implied volatility”.
63
Implied Volatility
Recall the Black-Scholes formula:
CBS(t, St ) = St Φ(d1) − Ke−r(T −t)Φ(d2),
where
2)(T − t)
log(St /K) + (r + 1
σ
2
√
d1 =
,
σ T −t
√
d2 = d1 − σ T − t,
as before.
Let c be the observed price of the call. Now, set CBS(t, St ) = c,
and
f (σ) = St Φ(d1) − Ke−r(T −t) Φ(d2),
We have
c = f (σ).
64
Computing the Implied Volatility
Given a value for c, that is, the observed price of the option, we
can solve for σ. There is no closed-form solution, so we solve
iteratively. Beginning with σ (0) , we can use the Newton updates,
σ (k+1) = σ (k) − (f (σ (k) ) − c)/f 0 (σ (k) ).
65
Computing the Implied Volatility
We have
dΦ(d1)
dΦ(d2)
− Ke−r(T −t)
dσ
dσ
dd
dd
= St φ(d1 ) 1 − Ke−r(T −t) φ(d2) 2 ,
dσ
dσ
f 0(σ) = St
where
1 −y2/2
φ(y) = √ e
,
2π
and
σ 2(T − t) − log(St /K) − (r + 1
σ 2)(T − t)
dd1
2
√
=
,
2
dσ
σ T −t
dd2
dd1 √
=
− T − t.
dσ
dσ
66
Discrepancies in the Observed Implied
Volatilities
The implied volatility from the Black-Scholes model should be
the same at all points. It is not.
The implied volatility, for given T and St , depends on the strike
price, K.
In general, the implied volatility is greater than the empirical
volatility, but the implied volatility is even greater for far out-ofthe money calls. It also increases for deep in-the-money calls.
This is called the “smile curve”, or the “volatility smile”.
67
The Smile Curve
0.23
0.22
0.20
0.21
Implied Volatility
0.24
0.25
The Black−Scholes Implied Volatility
96
98
100
102
104
Strike Price
The available strike prices are not continuous, of course. This
curve is a smoothed (and idealized) fit of the observed points.
68
The Smile Curve
The smile curve is not well-understood, although we have a lot
of empirical observations on it.
Interestingly, prior to the 1987 crash, the minimum of the smile
curve was at or near the market price St .
Since then it is generally at a point larger than the market price.
69
Computations of Black-Scholes Implied
Volatility
The Black-Scholes price depends on being able to evaluate the
standard normal CDF, and the implied volatility depends on being
able to evaluate the standard normal PDF.
2 /2
−x
The PDF is easy to compute from the definition: f (x) = e
/
√
2π.
The CDF is more difficult.
In R, the CDF is computed by pnorm and the PDF by dnorm.
In Statistics Toolbox of Matlab, the CDF is computed by normcdf
and the PDF by normpdf.
Without the Statistics Toolbox, in Matlab, the CDF can be computed at the point x by (1 + (erf(x/sqrt(2))))./2.
70
Variation in Volatility over Time
The volatility also varies in time. There are periods of high
volatility and other periods of low volatility. This is called “volatility clustering”.
The volatility of an index is somewhat similar to that of an
individual stock. The volatility of an index is a reflection of
market sentiment. (There are various ways of interpreting this!)
In general, a declining market is viewed as “more risky” than a
rising market, and hence, it is generally true that the volatility
in a declining market is higher.
Contrarians believe high volatility is bullish because it lags market
trends.
71
Measuring the Volatility of the Market
A standard measure of the overall volatility of the market is the
CBOE Volatility Index, VIX, which CBOE introduced in 1993 as
a weighted average of the Black-Scholes-implied volatilities of
the S&P 100 Index from at-the-money near-term call and put
options.
(“At-the-money” is defined as the strike price with the smallest
difference between the call price and the put price.)
In 2004, futures on the VIX began trading on the CBOE Futures
Exchange (CFE), and in 2006, CBOE listed European-style calls
and puts on the VIX.
Another measure is the CBOE Nasdaq Volatility Index, VXN,
which CBOE computes from the Nasdaq-100 Index, NDX, similarly to the VIX. (Note that the more widely-watched Nasdaq
Index is the Composite, IXIC.)
72
The VIX
In 2006, CBOE changed the way the VIX is computed. It is
now based on the volatilities of the S&P 500 Index implied by
several call and put options, not just those at the money, and it
uses near-term and next-term options (where “near-term” is the
earliest expiry mover than 8 days away).
It is no longer computed from the Black-Scholes formula.
It uses the prices of calls with strikes above the current price of
the underlying, starting with the first out-of-the money call and
sequentially including all with higher strikes until two consecutive
such calls have no bids. It uses the prices of puts with strikes
below the current price of the underlying in a similar manner.
The price of an option is the “mid-quote” price, i.e. the average
of the bid and ask prices.
73
Technical Details: Computing the VIX
Let K1 = K2 < K3 < · · · < Kn−1 < Kn = Kn+1 be the strike
prices of the options that are to be used.
The VIX is defined as 100 × σ, where
rT
σ 2 = 2eT
Pn
∆Kj ∆Ki
i=2;i6=j K 2 Q(Ki ) + K 2 Q(Kj put) + Q(Kj call) /2
i
j
2
F −1
,
− T1 K
j
•T is the time to expiry (in our usual notation, we would use
T − t, but we can let t = 0),
•F , called the “forward index level”, is the at-the-money strike
plus erT times the difference in the call and put prices for that
strike,
•Ki is the strike price of the ith out-of-the-money strike price
(that is, of a put if Ki < F and of a call if F < Ki),
•∆Ki = (Ki+1 − Ki−1)/2,
•Q(Ki) is the mid-quote price of the option,
•r is the risk-free interest rate, and
•Kj is the largest strike price less than F .
74
!
Technical Details: Computing the VIX
Time is measured in minutes, and converted to years.
Months are considered to have 30 days and years are considered
to have 365 days.
There are N1 = 1, 440 minutes in a day.
There are N30 = 43, 200 minutes in a month.
There are N365 = 525, 600 minutes in a year.
A value σ12 is computed for the near-term options with expiry T1,
and a value σ22 is computed for the next-term options with expiry
T2, and then σ is computed as
v
!
u
u
NT2 − N30
N30 − NT1 N365
2
2
t
σ=
T1σ1
+ T2 σ1
.
NT2 − NT1
NT2 − NT1 N30
75