Download Current Topics in Risk Management

Chapter 20
Current Topics in Risk Management
• Value at Risk (VaR)
• Credit Derivatives
• Exotic Options
©David Dubofsky and 20-1
Thomas W. Miller, Jr.
Value at Risk (VAR)
(http://www.gloriamundi.org)
•
Value At Risk (VaR): A risk management concept wherein senior
management can be informed, via a single number, of the firm’s
short-term price risk.
•
VaR is an estimated dollar decline with a stated probability during a
stated period of time.
•
The dollar loss may be estimated for the firm’s cash flows or for the
value of its assets and/or liabilities.
•
Note Bene: No finance theory exists to show that VaR is the
appropriate measure upon which to build optimal decision rules.
©David Dubofsky and 20-2
Thomas W. Miller, Jr.
Background
• One goal of active risk management: Reduce the
variability of uncertain cash flows.
• Usually, risk management cannot eliminate cash
flow variability.
• The modern corporation could have hundreds, or
thousands, of sources of uncertain cash flows that
are being hedged (or not being hedged).
• Thus, there exists a portfolio of risks.
©David Dubofsky and 20-3
Thomas W. Miller, Jr.
VaR Origins
• VaR stems from the request by J.P Morgan’s Chairman, Dennis
Weatherstone.
• Mr. Weatherstone requested a simple report be made available
to him every day concerning the firm’s risk exposure.
• Since then, VaR has become pervasive.
• VaR has become the financial industry’s standard for measuring
exposure to financial price risks.
• Most financial firms make VaR part of their daily reporting to
senior management.
©David Dubofsky and 20-4
Thomas W. Miller, Jr.
Sources of VaR Pervasiveness
• Effective 1/1/1998, The Bank for International Settlements (BIS),
required major banks to determine their capital adequacy using
VaR.
• The U.S. Federal Reserve and the International Swaps and
Derivatives Association (ISDA) have endorsed the BIS’s
recommendations about VaR.
• In December 1995, the U.S. Securities and Exchange
Commission (SEC) proposed rules that would require
corporations to disclose information concerning their use of
derivatives.
©David Dubofsky and 20-5
Thomas W. Miller, Jr.
Sources of VaR Pervasiveness, Cont.
•
Risk Standards Working Group (1996) self-imposed charge: “Create a
set of risk standards for institutional investment managers and
institutional investors.”
•
Risk Standard 12 states that money managers “should regularly
measure relevant risks and quantify the key drivers of risk and return.”
•
Risk Standard 12 proceeds to suggest VaR as one possible method for
measurement of risk.
•
For a voluminous source of regulatory and other documents concerning
risk management, see http://riskinstitute.ch.
©David Dubofsky and 20-6
Thomas W. Miller, Jr.
The VaR Concept
• VaR is an attempt to encapsulate an estimate of the price risk
possessed by a portfolio of derivatives and other financial
assets.
• The statistic that VaR provides is the dollar amount by which the
value of a portfolio might change with a stated probability
during a stated time horizon, say one day, one week, or
longer.
• For example, a financial institution might estimate that there is a
1% chance that its portfolio will decline by $15 million during the
next week.
©David Dubofsky and 20-7
Thomas W. Miller, Jr.
VaR Concept, Cont.
•
Most likely, the decline in value will be caused by changes in the prices
of fundamental risk factors.
•
Examples of these are changes in foreign exchange rates, interest rate
changes, commodity price fluctuations, changes in stock prices, and/or
increases in volatility.
©David Dubofsky and 20-8
Thomas W. Miller, Jr.
Important!
• The price risk number obtained from a VaR model summarizes
risk exposure into one dollar figure.
• This dollar figure purportedly represents the estimated
maximum loss over an interval of time.
• So, a dollar loss greater than the VaR estimate can occur, but
with a smaller probability than the VAR estimate.
©David Dubofsky and 20-9
Thomas W. Miller, Jr.
Computing VaR
• As yet, there is no standard method to compute VaR.
• However, several accepted methods for computing VaR have
emerged:
– The variance-covariance approach
– The historical simulation method
– The Monte Carlo simulation method
• The VaR estimate is sensitive to the assumptions made and to
the method used.
©David Dubofsky and 20-10
Thomas W. Miller, Jr.
The Variance-Covariance Approach
•
Also know as the “Delta-Normal Method.”
•
The variance-covariance approach is used by JP Morgan’s RiskMetrics
model.
•
www.riskmetrics.com
•
The important assumption of this approach: The returns for each of
the institution’s assets are normally distributed.
©David Dubofsky and 20-11
Thomas W. Miller, Jr.
Calculating VaR using the V-C Approach
•
Recall the variance of a portfolio’s returns are computed using the
following formula:
~
Var(R) 
 w w σ   w w σ σ ρ
i
i
j
j
ij
i
i
j
i
j ij
j
Where: wi is the fraction of the total portfolio value consisting of asset i
ij is the covariance of asset i’s returns with asset j’s returns
I is the standard deviation of asset i’s returns
ij is the correlation of asset i’s returns with asset j’s returns
©David Dubofsky and 20-12
Thomas W. Miller, Jr.
Shortcomings of the V-C Approach
• Return distributions may not be normal.
• The returns distribution might be skewed, or it may possess “fat
tails.”
• A “fat-tailed” distribution has leptokurtosis.
– Characterized by “too many” observations in the tails.
– So, “unusual” events occur more frequently than a normal
distribution would predict.
– There is considerable evidence that the returns distributions of
many risk factors are leptokurtic.
– Options, and assets with option-like characteristics, possess nonnormal return distributions.
©David Dubofsky and 20-13
Thomas W. Miller, Jr.
•
Suppose a risk manager assumes that the return distribution of all
assets in the portfolio is normal with a mean of zero.
•
The first step to obtain a VaR estimate is to obtain an estimate of the
variance of the portfolio’s periodic returns.
•
For three assets, the variance equation is:
Var(R) = w12 12 + w22 22 + w32 32
+ 2w1w21212
+ 2w1w31313
+ 2w2w32323
©David Dubofsky and 20-14
Thomas W. Miller, Jr.
After obtaining a portfolio’s mean return
and return variance:
• A simple statistical procedure is used to estimate what the loss
in value of the portfolio will be during that period.
• Further note that this estimate can be calculated with any
desired probability.
– For example, the value at risk, VaR, will equal 1.645 times the
portfolio’s standard deviation with a probability of 5%.
– The maximum loss with a 1% probability will equal 2.327 times the
portfolio’s standard deviation.
©David Dubofsky and 20-15
Thomas W. Miller, Jr.
Using V-C to Calculate VaR: Example
• Consider a portfolio consists of these three assets:
– A currency swap. Because of changes in the exchange rate since
the swap was first entered into, the swap now has a value of $2
million, or 8.7% of the portfolio’s total value.
– A bond. The market value of the bond is $17 million, which is 73.9%
of the portfolio’s total value.
– 10,000 shares of a stock that are worth $4 million, or 17.4% of the
portfolio’s total value.
©David Dubofsky and 20-16
Thomas W. Miller, Jr.
Assume the variance-covariance matrix of
the assets’ daily returns is:
Swap
Bond
Stock
Swap Bond
Stock
0.00090
-0.000080 0.00007
0.000400 -0.00010
0.00300
©David Dubofsky and 20-17
Thomas W. Miller, Jr.
The variance of this portfolio’s daily
returns distribution is:
 
~
Var R  w 12 σ12 + w 22 σ 22 + w 32 σ 32 + 2w 1 w 2 σ12 + 2w 1 w 3 σ13 + 2w 2 w 3 σ 23
 0.087  0.00090 + 0.739 0.00040 + 0.174  0.00300
+ 2 0.087 0.739  0.00008
+ 2 0.087 0.174 0.00007 
+ 2 0.1740.739  0.00010
2
2
2
 0.0002822.
©David Dubofsky and 20-18
Thomas W. Miller, Jr.
•
The standard deviation of the daily returns distribution is (0.0002822)1/2
= 0.0168, or 1.68%.
•
One standard deviation of dollar loss from the portfolio value of
$23,000,000 is $386,375 (1.68% of $23 million is $386,375).
•
To calculate the VaR, multiply the number of standard deviations for a
stated probability level by one standard deviation of dollar loss.
•
Thus, there is a 5% probability that a one day loss of (1.645)($386,375)
= $635,587 will be realized.
•
There is a 1% probability that a one day loss of (2.327)($386,375) =
$899,095 will be realized.
©David Dubofsky and 20-19
Thomas W. Miller, Jr.
Caveats:
• VaR calculated using historical data will differ,
depending on the time period chosen.
• If one year of daily historical data were gathered on
January 1, 1989, then that data would not include the
very unusual financial events of October, 1987.
• Example: Consider a $23,000,000 portfolio that
mimics the S&P 500 index.
©David Dubofsky and 20-20
Thomas W. Miller, Jr.
The mean daily return and standard deviation of
daily returns for the S&P 500 Index was:
1988 Only 1987 and 1988
Mean
Std Deviation
5% VAR
1% VAR
0.000284
0.010613
$401,554
$568,035
0.000237
0.016858
$637,833
$902,272
•
Assuming a mean of zero, if one calculates VaR using 1988 data only, the
1% VaR is $568,035.
•
However, if one calculates VaR using 1987 and 1988 data, the 1% VAR is
$902,272, about 60% higher.
©David Dubofsky and 20-21
Thomas W. Miller, Jr.
Using the Historical Simulation Approach
to Calculate VaR
• Here, the risk manager determines how each relevant price has
changed during each of the past ‘N’ time periods.
•
For example, the relevant prices might be the dollar price of the
Japanese yen and Deutsche mark, short-term U.S. interest
rates, and the dollar price of oil.
•
Using a historical database, it is possible to find the price
changes for each of these four variables on each of the last 300
days.
©David Dubofsky and 20-22
Thomas W. Miller, Jr.
•
The risk manager then estimates how each of the 300 sets of price
changes would affect the value of the current portfolio of spot assets
and derivatives.
•
Assuming that each of these outcomes is equally likely, rank the
resulting estimated value changes from most positive to most negative.
•
If the risk manager is interested in the VAR at the 95% confidence
level, then use the 15th worst outcome out of the 300. Why?
•
The VAR at the 99% confidence level is the 3rd worst outcome.
•
These are the losses that will be exceeded only 5% of the time and 1%
of the time, respectively.
©David Dubofsky and 20-23
Thomas W. Miller, Jr.
Using the Monte Carlo Simulation
Approach to Calculate VaR, I.
•
Here, returns are not assumed to be equally likely.
•
Instead, the risk manager specifies probability distributions or
stochastic processes for prices in the future, based on economic
scenarios.
•
A different probability distribution can be assumed for each
fundamental economic factor. For example:
– changes in interest rates might be skewed right
– changes in oil prices might be drawn from a uniform distribution
– changes in the price of some particular currency might be normally
distributed.
©David Dubofsky and 20-24
Thomas W. Miller, Jr.
Using the Monte Carlo Simulation
Approach to Calculate VaR, II.
•
After the distributions or processes are defined, random realizations of
outcomes can be simulated.
•
Each randomly chosen outcome is a set of prices. Correlations are
explicitly incorporated in the simulation.
•
The change in value of each asset in the portfolio is estimated for each
randomly selected set of prices, thereby producing a probability
distribution of future value changes of the portfolio.
•
The risk manager can then determine what the worst outcome will be
with a desired confidence level.
©David Dubofsky and 20-25
Thomas W. Miller, Jr.
Stress Testing
• Stress testing attempts to answer this question: How well would
the portfolio have performed under extreme market moves?
– Suppose all equities in the portfolio fall 15% at once.
– Suppose all interest rates change by 200bp at once.
• Stress testing takes into account extreme events (i.e., jumps)
that cannot be accounted for by probability distributions.
• It is important to run stress tests on your portfolio.
©David Dubofsky and 20-26
Thomas W. Miller, Jr.
Back Testing
• Regardless of the VaR method employed, it is important to back
test the VaR estimates.
• That is, how well would the VaR estimate have performed in the
past?
• Suppose you are interested in the 5% VaR.
– How often did the portfolio lose more than the number predicted by
the 5% VaR?
– Was it about 5% of the time?
– Or, was it, say, 12% of the time?
– If so, then it is important to adjust the VaR method.
• Remember, there is no finance theory that suggests VaR should
be used for decision making.
©David Dubofsky and 20-27
Thomas W. Miller, Jr.
What are Credit Derivatives?
• A derivative is a contract whose value is derived from some
underlying asset or variable.
• Credit risk is the underlying variable of a credit derivative.
• Credit risk is an asset class unto itself.
• A credit derivative permits institutions to trade credit risk (credit (yield)
spreads, rating downgrades, defaults, any credit event). Credit risk is
traded in isolation from all other attributes that define the reference
asset. The reference asset itself is not transferred.
©David Dubofsky and 20-28
Thomas W. Miller, Jr.
Credit Risk Management
• 1971: Municipal bond insurance
• Letters of credit
• Collateralization
• Marking to market
• Netting
• Early termination or assignment
©David Dubofsky and 20-29
Thomas W. Miller, Jr.
History
• 1991: birth of the OTC credit derivatives market.
• 1992: ISDA coins the term “credit derivatives.”
• By May 1996, the size of the market was $39 billion (CIBC Wood
Gundy survey).
• Recent surveys:
– BBA, July 2000: $893 billion
– Risk, Feb. 2001: $810 billion
– OCC: June 2001: $351 billion
– ISDA: June 2001: $631 billion
©David Dubofsky and 20-30
Thomas W. Miller, Jr.
Types of Credit Derivatives
• Forward contracts
• Swaps
– Total return swap
• Options
– Default puts (credit default swaps)
– Credit spread options
©David Dubofsky and 20-31
Thomas W. Miller, Jr.
Credit Forward Contracts
• The buyer commits to buy a specified bond (reference asset) on
a specified date at a contractually agreed upon price.
• Buyer is buying credit risk. If the credit worthiness of the bond
issuer deteriorates, the buyer loses.
• Payoff can also be based on the change in the bond’s yield
spread, say, relative to a Treasury security
– E.g., pay (spread at maturity – contract spread) X duration X
notional principal
©David Dubofsky and 20-32
Thomas W. Miller, Jr.
Swaps
• Total return swap:
–
–
–
–
C = coupon payment on a reference bond
P(0) = price of reference bond at origination
P(T) = price of reference bond on the swap’s maturity date
R(t) = rate of return on the reference bond during period t =
[P(t)+C– P(t-1)]/P(t-1)
R1
R2
R3
RT
C
C
C
C + P(T)
LIBOR+
LIBOR+
LIBOR+
1
2
3
LIBOR+ + P(0)
T
©David Dubofsky and 20-33
Thomas W. Miller, Jr.
Total Return Swaps
A
R(t)
B
R(t)
Portfolio
Manager
C
t=1,2,…T-1
Time T
C+P(T)
LIBOR+
Portfolio
Manager
Portfolio
Manager
Dealer
(protection seller)
C
Dealer
LIBOR+
C+P(T)
Dealer
LIBOR+ +P(0)
©David Dubofsky and 20-34
Thomas W. Miller, Jr.
Options
• Buyer of a default put (credit default swap) pays a premium (either up
front, or amortized into a stream of payments). If there is a “credit
event”, the seller pays the buyer a default payment.
Put seller
(protection
seller)
‘X’ basis points
per year
No Credit Event: 0
Credit Event: Default
Payment
Put buyer
(protection
buyer)
Loan to ABC
Co., or ABC
bond
• Default payment might be the price of the reference asset on a
specific date, a specified % of the notional principal, or payment of par
by the seller, in exchange for the “defaulted” bond.
©David Dubofsky and 20-35
Thomas W. Miller, Jr.
First-to-Default Put
(Basket Default Swap)
• Bank owns several loans. It receives a swap default payment
whenever one of the loans (but only the first one) defaults.
Otherwise, it pays ‘X’ basis points per period on the total value
of the loan portfolio.
• If the default correlations are very low, the probability of there
being more than one default is tiny.
– E.g., if the probability is 1% that one of four loans defaults,
the probability of there being more than one default is
0.06%.
©David Dubofsky and 20-36
Thomas W. Miller, Jr.
Credit-linked Note, CLN
• A CLN is an actual security, with a credit derivative (typically a credit
default swap) embedded in its structure.
CLN Principal at Time 0
Investor
Bank
(Protection
Buyer)
Interest on CLN
(Protection
Seller)
Recovery Value
(Y)
(N)
Credit Event?
Principal
Loans
©David Dubofsky and 20-37
Thomas W. Miller, Jr.
Credit Spread Option
• Strike price might be the time 0 yield spread between a corporate
bond and a Treasury with the same time to maturity.
• Payoff at maturity =
MAX[0, (spread at maturity – strike) X duration X NP]
• Thus, the spread option buyer gets paid if there is an increase in the
market price of default risk and/or an increase in the probability of the
corporate bond issuer defaulting.
©David Dubofsky and 20-38
Thomas W. Miller, Jr.
How Credit Derivatives Are Used
• A fund can buy Treasuries and also go long credit risk using a credit
derivative in order to create a synthetic high-yield bond.
• A bank can own risky loans, and can sell the credit risk in order to
have a less risky portfolio of loans.
– This is a hedge.
– The bank need not disturb its existing relationships with its customers.
– Lending is increased.
• An institution might wish to assume a form of credit risk because it is
uncorrelated with the other risks it faces.
• Credit derivatives based on sovereign debt are popular.
• Some institutions are prohibited from investing in a particular market
segment, but they can gain exposure using credit derivatives.
©David Dubofsky and 20-39
Thomas W. Miller, Jr.
Regulatory Capital Arbitrage
• Bank is required by Basel Accord, to commit 8% of a loan made
to any corporation, regardless of its creditworthiness.
• E.g., $50 million one year loan requires $4 million in regulatory
capital.
• If it is earning 25 basis points over its funding costs, then it will
earn $125,000 profit, which is 3.125% on regulatory capital
• BUT, suppose it uses a CDS to purchase protection, from an
OECD bank. The bank pays 15 bp in the CDS, leaving it with
only a 10bp spread over its funding costs, or $50,000 on the $50
million loan.
• But the rules permit the bank using the CDS to supply only 20%
of the regulatory capital, or $800,000 (rather than $4 million).
• The return on regulatory capital is increased to 6.25%.
©David Dubofsky and 20-40
Thomas W. Miller, Jr.
How Credit Derivatives are Used
• Credit derivatives permit diversification:
Return on
loans made
to energy
companies
A
Houston bank
swap
Lender to
airlines or
aircraft
manufacturers
Return
on loans
©David Dubofsky and 20-41
Thomas W. Miller, Jr.
Product Structuring
• Suppose an investor wants to invest in a Brazilian government
bond, denominated in euros, but no such instrument exists. The
investor can
– Buy a German bond denominated in euros,
– Sell default protection on the government of Brazil, in exchange for
Euribor+
• Result: Investor gets coupon payments on the German bond in
euros, and also receives a periodic payment from the sale of
default protection. Effectively, she owns a Brazilian government
bond denominated in euros. She is receiving a premium over
the German interest rate, paid in euros, and is exposed to
Brazilian default risk.
©David Dubofsky and 20-42
Thomas W. Miller, Jr.
Corporations can use Credit Derivatives
• Corporations can hedge their receivables.
• They can hedge sovereign credit-related project risk.
• They can use a credit derivative as an alternative to
buying back their own bonds. (There may be adverse
tax consequences to a bond repurchase.)
©David Dubofsky and 20-43
Thomas W. Miller, Jr.
Pricing Credit Derivatives
• In theory, a credit derivative should be priced like any other
derivative. The underlying asset is a variable (just like a stock
price), which changes according to an assumed process (e.g.,
Geometric Brownian Motion).
• But, credit risk typically does not follow normal probability
distribution processes
– Many credit events are “jumps”
– Many credit processes are unobservable (what is the probability of
default?)
– Credit markets are often illiquid and have high transactions costs
©David Dubofsky and 20-44
Thomas W. Miller, Jr.
Other Issues with Pricing Credit Derivatives
• One also needs to estimate recovery rates on
defaulted securities.
• How do dealers hedge their own unwanted credit risk
exposure?
– Could trade the issuer’s bonds (but, they often trade in
illiquid markets)
– Could trade the issuer’s stock
• Does something cause the default, or is it random?
©David Dubofsky and 20-45
Thomas W. Miller, Jr.
The Elements of Credit Risk Models
• Asset Credit Risk Model
– What is the credit risk of an asset today?
– How might that credit risk change over time?
• Migration models (prob. of moving from one credit quality to
another, within a given period of time) are often based on
historic data.
– Should you assume the migration follows a random Markov
process, based on historical probabilities, or
– Should you use macroeconomic data to drive the model?
• Historic default data.
• Stock price can be an indicator of default risk.
• Must assume that interest rates can change; higher interest
rates mean higher default probability.
©David Dubofsky and 20-46
Thomas W. Miller, Jr.
The Elements of Credit Risk Models
• A credit risk model must also model how credit risk is priced.
• Yield spreads over Treasuries.
• Credit risk term structures.
• Account for correlations among assets and over time.
• Measure the credit risk for the portfolio over different holding
periods.
• Produce risk/return profiles. What is the expected return of the
portfolio? What is the portfolio’s risk? Is the portfolio optimal?
©David Dubofsky and 20-47
Thomas W. Miller, Jr.
Exotic Options
•
The recent proliferation of option products has lead to a new class of
options called “exotics.”
•
Where to ordinary options end and exotics begin?
•
It is “common” to refer to almost any option not traded on an exchange
as an exotic.
©David Dubofsky and 20-48
Thomas W. Miller, Jr.
There are Two Basic
Types of Exotic Options
• Path Dependent
• Path Independent (AKA “Free Range”)
• We will start our investigation with Free Range
Options.
©David Dubofsky and 20-49
Thomas W. Miller, Jr.
Cash or Nothing (CON)
• Sometimes, these options are called digital, binary, or strike
or nothing options.
Payoff:
K if ST  K
0 if ST  K
Value Today:
CONc  Ke  rT N (d 2 )
Intuition: This term is the
discounted (risk-neutral)
probability that the option will
finish in the money.
©David Dubofsky and 20-50
Thomas W. Miller, Jr.
Asset or Nothing (AON)
Payoff:
ST if ST  K
0 if ST  K
Value Today:
AONc  SN (d1 )
This term is the PV of: the
expected value of the stock price
at expiration ( conditional on the
stock price exceeding the strike
price) times the risk-neutral
probability that the stock price
exceeds the strike price.
©David Dubofsky and 20-51
Thomas W. Miller, Jr.
So, the holder of a Black-Scholes call
option is long an AON and short a CON
C  SN (d1 )  Ke  rT N (d 2 )
• So, a market maker in digital options, say, can hedge with
ordinary European options:
• Sell an AON; Buy a CON; Sell Ordinary European Call
results in a “Flat” position.
©David Dubofsky and 20-52
Thomas W. Miller, Jr.
The Profit Profiles at Expiration:
Profit Profiles: Black-Scholes; Cash-orNothing; Asset-or-Nothing
B-S Profit
CON Profit
AON Profit
80.00
Profit
60.00
40.00
20.00
0.00
-20.00
-40.00
0
10
20
30
40
50
60
70
80
90 100
Stock Price at Expiration
©David Dubofsky and 20-53
Thomas W. Miller, Jr.
Paylater Option
• These are contingent options: The option premium is
paid at exercise, if the option is in-the-money at
expiration.
– NB: The buyer must exercise the option if it is in-the-money
at expiration, regardless whether it is “sufficiently” deep in
the money to pay the premium and make a profit.
– These options insure against a large adverse move, at zero
initial cost.
©David Dubofsky and 20-54
Thomas W. Miller, Jr.
Paylater Payoff:
Payoff:
ST - K - CPL if ST  K
0
if S T  K
Value Today:
SN (d1 )  Ke  rT N (d 2 )
PLc 
e  rT N (d 2 )
Note that an at-themoney paylater option
premium is about twice
that of an ordinary call,
because N(d2) is
“close” to 0.50 when an
option is at-the-money.
©David Dubofsky and 20-55
Thomas W. Miller, Jr.
The Profit Profile at Expiration:
Profit Profiles: Black-Scholes and Paylater
B-S Profit
PL Profit
30.00
Profit
20.00
10.00
0.00
-10.00
-20.00
20
30
40
50
60
70
80
Stock Price at Expiration
©David Dubofsky and 20-56
Thomas W. Miller, Jr.
Chooser, AKA, “As You Like It”
(Or, Options for the Undecided)
• The holder of a chooser option gets to decide, at a designated
future point before expiration (t), whether this option becomes a
call option, or a put option.
• Why buy Choosers? (What else could we do?)
– Straddle Cost: C + P (both with expiration T and strike K.)
– Chooser Cost: Replication Portfolio: Call with expiration T, strike K;
Put with expiration t and strike Ke-r(T-t)
– So, the Chooser is cheaper than the straddle (the put in the
replication portfolio is cheaper.)
©David Dubofsky and 20-57
Thomas W. Miller, Jr.
How does one decide at time t whether
the Chooser should be a call or a put?
If Ct >Pt , we choose call. Recall Put-Call Parity:
Pt + St  Ct + Ke  r (T t )
Pt  Ct + Ke  r (T t )  St
So, if: Ct  Ct + Ke  r (T t )  St
or St  Ke  r (T t ) , then choose call.
©David Dubofsky and 20-58
Thomas W. Miller, Jr.
Path Dependent Exotic Options
• Closed-form formulas are generally not available.
• Must use computer algorithms, in general, to solve
for these prices.
©David Dubofsky and 20-59
Thomas W. Miller, Jr.
Barrier Options, AKA Knocks:
Cheaper than Ordinary Options
• Down-and-Out (Knock Out)
• Down-and-In (behaves as an ordinary option once it is ‘in’)
• Up-and-Out (Knock Out)
• Up-and-In (behaves as an ordinary option once it is ‘in’)
• Parisian
©David Dubofsky and 20-60
Thomas W. Miller, Jr.
Asian (or Average) Call Options
Payoff:
S A if S A  K
0
if S A  K
S A is the average price of the stock
from option initiation to expiration.
These options are cheaper than European
options, because they are cheap er.
( is flattened out.)
©David Dubofsky and 20-61
Thomas W. Miller, Jr.
Lookback Call Options
Payoff:
SMAX if SMAX  K
0
if SMAX  K
SMAX is the maximum price of the stock
from option initiation to expiration.
These options are much more expensive than European
options, because they are like an American option
guaranteed to be exercised at the most favorable price.
©David Dubofsky and 20-62
Thomas W. Miller, Jr.