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Chapter 1: Introduction
Financial markets teach you humility. Two years ago, I made these
related forecasts. First I forecast the euro would strengthen as
European economic recovery picked up and the US economy
slowed. Second, I forecast euro strength would be augmented over the
next five years by a reduction of Europe's $100 billion to $150 billion
of excess dollar reserves. Third, I forecast that the authorities would be
less concerned over exchange rates, would only intervene after bigger
exchange rate moves, and so exchange rate volatility would
rise. Interestingly, people still ask for my opinion.
Avinash Persaud, Managing director,
Global Markets Analysis, State Street Bank
Risk, October, 2000, p. 29
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Ch. 1: 1
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Important Concepts in Chapter 1
Different types of derivatives
Presuppositions for financial markets, risk preferences,
risk-return tradeoff, and market efficiency
Theoretical fair value
Arbitrage, storage, and delivery
The role of derivative markets
Criticisms of derivatives
Ethics
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Ch. 1: 2
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Business risk vs. financial risk
Derivatives
A derivative is a financial instrument whose return is
derived from the return on another instrument.
Size of the OTC derivatives market at year-end 2010
$601 trillion notional principal
GDP is only $15 trillion
See Figure 1.1 and Figure 1.2
Real vs. financial assets
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Ch. 1: 3
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Derivative Markets and Instruments
Derivative Markets
Over-the-counter and exchange traded
Exchange traded derivatives volume in 2010 was over
22 billion contracts on at least 78 derivatives
exchanges, according to Futures Industry magazine (a
leading source of derivatives industry information
Derivatives trade all over the world
See Table 1.1 for the top ten derivatives exchanges
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Ch. 1: 4
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Derivative Markets and Instruments
Options
Definition: a contract between two parties that gives
one party, the buyer, the right to buy or sell something
from or to the other party, the seller, at a later date at a
price agreed upon today
Option terminology
price/premium
call/put
exchange-listed vs. over-the-counter options
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Ch. 1: 5
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Derivative Markets and Instruments
(continued)
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Forward Contracts
Definition: a contract between two parties for one
party to buy something from the other at a later date at
a price agreed upon today
Exclusively over-the-counter
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Ch. 1: 6
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Derivative Markets and Instruments
(continued)
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Futures Contracts
Definition: a contract between two parties for one
party to buy something from the other at a later date at
a price agreed upon today; subject to a daily settlement
of gains and losses and guaranteed against the risk that
either party might default
Exclusively traded on a futures exchange
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Ch. 1: 7
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Derivative Markets and Instruments
(continued)
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Options on Futures (also known as commodity options or
futures options)
Definition: a contract between two parties giving one
party the right to buy or sell a futures contract from the
other at a later date at a price agreed upon today
Exclusively traded on a futures exchange
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Ch. 1: 8
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Swaps and Other Derivatives
Definition of a swap: a contract in which two parties
agree to exchange a series of cash flows
Exclusively over-the-counter
Other types of derivatives include swaptions and
hybrids. Their creation is a process called financial
engineering.
The Underlying Asset
Called the underlying
A derivative derives its value from the underlying.
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Ch. 1: 9
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Some Important Concepts in Financial and
Derivative Markets
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Presuppositions – rule of law, property rights, culture of
trust
Risk Preference
Risk aversion vs. risk neutrality
Risk premium
Short Selling
Repurchase agreements (repos)
Return and Risk
Risk defined
The risk-return tradeoff (see Figure 1.3)
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Ch. 1: 10
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Some Important Concepts in Financial and
Derivative Markets (continued)
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Market Efficiency and Theoretical Fair Value
Efficient market defined: A market in which the price of
an asset equals its true economic value.
An efficient market is a consequence of rational and
knowledgeable investor behavior
The concept of theoretical fair value
The true economic value
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Ch. 1: 11
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Fundamental Linkages Between Spot and
Derivative Markets
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Arbitrage and the Law of One Price
Arbitrage defined: A type of profit-seeking transaction
where the same good trades at two prices.
Example: See Figure 1.4
The concept of states of the world
The Law of One Price
The Storage Mechanism: Spreading Consumption across
Time
Delivery and Settlement
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Ch. 1: 12
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The Role of Derivative Markets
Risk Management
Hedging vs. speculation
Setting risk to an acceptable level
Example: Southwest Airlines
Price Discovery
Operational Advantages
Transaction costs
Liquidity
Ease of short selling
Market Efficiency
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Criticisms of Derivative Markets
Speculation
Comparison to gambling
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Ch. 1: 14
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Misuses of Derivatives
High leverage
Inappropriate use
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Ch. 1: 15
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Derivatives and Ethics
Codes of ethics and standards of professional conduct are
vital components of the derivatives profession
Examples
CFA Institute
Professional Risk Managers International Association
Global Association of Risk Professionals
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Derivatives and Your Career
Financial management in a business
Small businesses ownership
Investment management
Public service
Source of Information on Derivatives
http://www.cengage.com/finance/chance
Summary
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Ch. 1: 17
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(Return to text slide)
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Ch. 1: 18
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Ch. 1: 22
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Chapter 2: Structure of Options Markets
There weren't many traders at the sharp end over thirty.
Eyes flitting between flickering lines of information on four
different screens, one ear on the phone, the other on the
cries of the colleagues, twelve hours of split-second
calculations, judging yourself and being judged on the score
at the end of every day. These men and women lived and
breathed the market.
Linda Davies
Into the Fire, 1999, p. 34
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Important Concepts in Chapter 2
Definitions and examples of call and put options
Institutional characteristics of options markets
Options available for trading
Placing an options order
The clearinghouse
Accessing option price quotations
Transaction costs
Regulation of options markets
Margins and taxes in option transactions
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Ch. 2: 2
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Option terminology
price/premium
call vs. put
exercise price/strike price/striking price
expiration date
Everyday examples of options
rain check
discount coupon
airline ticket with cancellation right
right to drop a course
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Ch. 2: 3
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Development of Options Markets
Early origins
Put and Call Brokers and Dealers Association
Chicago Board Options Exchange, 1973
Resurgence of over-the-counter market
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Ch. 2: 4
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Call Options
Current example
Objective of a call buyer
Moneyness concepts
In-the-money
Out-of-the-money
At-the-money
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Ch. 2: 5
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Put Options
Current example
Objective of a put buyer
Moneyness concepts
In-the-money
Out-of-the-money
At-the-money
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Ch. 2: 6
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Options Trading Activity
In 2010, exchange-traded option volume (number of
contracts) approximately 11.1 billion contracts (Futures
Industry magazine)
In 2010, over-the-counter option volume approximately
$64 trillion notional principal and $2.2 trillion market
value (Bank of International Settlements)
OTC options notional amount outstanding fell dramatically
during the Financial Crisis of 2008 (see Figure 2.1)
OTC options market value outstanding rose sharply during
initial phase of Financial Crisis of 2008 (see Figure 2.2)
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Ch. 2: 7
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Over-the-Counter Options Market
Worldwide
Credit risk
Customized terms
Private transactions
Unregulated
Options on stocks and stock indices, bonds, interest rates,
commodities, swaps & currencies
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Ch. 2: 8
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Organized Options Trading
The concept of an options exchange
Listing Requirements
Contract Size
Exercise Prices
Expiration Dates
Position and Exercise Limits
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Ch. 2: 9
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Option Traders
Liquidity Providers
Provide bid and ask prices to facilitate trading
Scalpers, position traders, spreaders
Lead market makers, designated primary market
makers
Floor Broker – acts as agent for customers
Order Book Official
Limit orders
Electronic order processing
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Ch. 2: 10
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Options Traders (continued)
Other Option Trading Systems
Specialists
Registered options traders
Electronic trading systems
Off-Floor Option Traders
Option brokers
Proprietary options traders
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Ch. 2: 11
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Mechanics of Trading
Placing an Opening Order
Types of orders
Role of the Clearinghouse
Options Clearing Corporation (OCC)
Clearing firms
See Figure 2.3
Margin (see Appendix 2.A)
Open interest
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Ch. 2: 12
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Mechanics of Trading (continued)
Placing an Offsetting Order
In the exchange-listed options market
In the over-the-counter options market
Exercising an Option
European vs. American style
Assignment
Cash settlement
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Ch. 2: 13
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Option Price Quotations
See Web sites of newspapers and options exchanges
Problems
Delayed information
Non-synchronized prices
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Ch. 2: 14
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Types of Options
Stock Options
Index Options
Currency Options
Other Types of Options
interest rate options
currency options
options attached to bonds
exotic options
warrants, callable bonds, convertible bonds
non-traded executive options
Real Options
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Ch. 2: 15
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Transaction Costs in Option Trading
Floor Trading and Clearing Fees
Commissions
Bid-Ask Spread
Other Transaction Costs
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Ch. 2: 16
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The Regulation of Options Markets
Federal regulation
Industry regulation
Over-the-counter market regulation
The issue of which agency has regulatory responsibility
has occasionally arisen.
Summary
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Ch. 2: 17
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Appendix 2.A: Margin Requirements
Definitions
Margin
Initial margin
Maintenance margin
Margin Requirements on Stock Transactions
Margin Requirements on Option Purchases
Margin Requirements on the Uncovered Sale of Options
Margin Requirements on Covered Calls
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Ch. 2: 18
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Appendix 2.B: Taxation of Option Transactions
Taxation of Long Call Transactions
Taxation of Short Call Transactions
Taxation of Long Put Transactions
Taxation of Short Put Transactions
Taxation of Non-Equity Options
Wash and Constructive Sales
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 2: 19
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(Return to text slide)
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Ch. 2: 20
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(Return to text slide)
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Ch. 2: 21
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(Return to text slide)
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Ch. 2: 22
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Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them
one way and you have heat and energy - useful stuff. Split
them another way and you have a bomb. You have to
understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
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Ch. 3: 1
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Important Concepts in Chapter 3
Role of arbitrage in pricing options
Minimum value, maximum value, value at expiration and
lower bound of an option price
Effect of exercise price, time to expiration, risk-free rate
and volatility on an option price
Difference between prices of European and American
options
Put-call parity
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Ch. 3: 2
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Basic Notation and Terminology
Symbols
S0 (stock price)
X (exercise price)
T (time to expiration = (days until expiration)/365)
r (see below)
ST (stock price at expiration)
C(S0,T,X), P(S0,T,X)
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Ch. 3: 3
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Basic Notation and Terminology (continued)
Computation of risk-free rate (r)
Date: May 14. Option expiration: May 21
T-bill bid discount = 4.45, ask discount = 4.37
Average T-bill discount = (4.45+4.37)/2 = 4.41
T-bill price = 100 - 4.41(7/360) = 99.91425
T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457
So 4.57 % is risk-free rate for options expiring May 21
Other risk-free rates: 4.56 (June 18), 4.63 (July 16)
See Table 3.1 for prices of DCRB options
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Ch. 3: 4
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Principles of Call Option Pricing
Minimum Value of a Call
C(S0,T,X)  0 (for any call)
For American calls:
Ca(S0,T,X)  Max(0,S0 - X)
Concept of intrinsic value: Max(0,S0 - X)
Proof of intrinsic value rule for DCRB calls
Concept of time value
See Table 3.2 for time values of DCRB calls
See Figure 3.1 for minimum values of calls
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Ch. 3: 5
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Principles of Call Option Pricing (continued)
Maximum Value of a Call
C(S0,T,X)  S0
Intuition
See Figure 3.2, which adds this to Figure 3.1
Value of a Call at Expiration
C(ST,0,X) = Max(0,ST - X)
Proof/intuition
For American and European options
See Figure 3.3
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Ch. 3: 6
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Principles of Call Option Pricing (continued)
Effect of Time to Expiration
Two American calls differing only by time to
expiration, T1 and T2 where T1 < T2.
Ca(S0,T2,X)  Ca(S0,T1,X)
Proof/intuition
Deep in- and out-of-the-money
Time value maximized when at-the-money
Concept of time value decay
See Figure 3.4 and Table 3.2
Cannot be proven (yet) for European calls
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Ch. 3: 7
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Principles of Call Option Pricing (continued)
Effect of Exercise Price
Effect on Option Value
Two European calls differing only by strikes of X1
and X2. Which is greater, Ce(S0,T,X1) or
Ce(S0,T,X2)?
Construct portfolios A and B. See Table 3.3.
Portfolio A has non-negative payoff; therefore,
• Ce(S0,T,X1)  Ce(S0,T,X2)
• Intuition: show what happens if not true
Prices of DCRB options conform
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Ch. 3: 8
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Principles of Call Option Pricing (continued)
Effect of Exercise Price (continued)
Limits on the Difference in Premiums
Again, note Table 3.3. We must have
• (X2 - X1)(1+r)-T  Ce(S0,T,X1) - Ce(S0,T,X2)
• X2 - X1  Ce(S0,T,X1) - Ce(S0,T,X2)
• X2 - X1  Ca(S0,T,X1) - Ca(S0,T,X2)
• Implications
See Table 3.4. Prices of DCRB options conform
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Ch. 3: 9
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Principles of Call Option Pricing (continued)
Lower Bound of a European Call
Construct portfolios A and B. See Table 3.5.
B dominates A. This implies that (after rearranging)
Ce(S0,T,X)  Max[0,S0 - X(1+r)-T]
This is the lower bound for a European call
See Figure 3.5 for the price curve for European calls
Dividend adjustment: subtract present value of
dividends from S0; adjusted stock price is S0´
For foreign currency calls,
Ce(S0,T,X)  Max[0,S0(1+)-T - X(1+r)-T]
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Ch. 3: 10
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Principles of Call Option Pricing (continued)
American Call Versus European Call
Ca(S0,T,X)  Ce(S0,T,X)
But S0 - X(1+r)-T > S0 - X prior to expiration so
Ca(S0,T,X)  Max(0,S0 - X(1+r)-T)
Look at Table 3.6 for lower bounds of DCRB calls
If there are no dividends on the stock, an American call
will never be exercised early. It will always be better to
sell the call in the market.
Intuition
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Ch. 3: 11
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Principles of Call Option Pricing (continued)
Early Exercise of American Calls on Dividend-Paying
Stocks
If a stock pays a dividend, it is possible that an
American call will be exercised as close as possible to
the ex-dividend date. (For a currency, the foreign
interest can induce early exercise.)
Intuition
Effect of Interest Rates
Effect of Stock Volatility
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Ch. 3: 12
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Principles of Put Option Pricing
Minimum Value of a Put
P(S0,T,X)  0 (for any put)
For American puts:
Pa(S0,T,X)  Max(0,X - S0)
Concept of intrinsic value: Max(0,X - S0)
Proof of intrinsic value rule for DCRB puts
See Figure 3.6 for minimum values of puts
Concept of time value
See Table 3.7 for time values of DCRB puts
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Ch. 3: 13
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Principles of Put Option Pricing (continued)
Maximum Value of a Put
Pe(S0,T,X)  X(1+r)-T
Pa(S0,T,X)  X
Intuition
See Figure 3.7, which adds this to Figure 3.6
Value of a Put at Expiration
P(ST,0,X) = Max(0,X - ST)
Proof/intuition
For American and European options
See Figure 3.8
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Ch. 3: 14
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Principles of Put Option Pricing (continued)
Effect of Time to Expiration
Two American puts differing only by time to
expiration, T1 and T2 where T1 < T2.
Pa(S0,T2,X)  Pa(S0,T1,X)
Proof/intuition
See Figure 3.9 and Table 3.7
Cannot be proven for European puts
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Ch. 3: 15
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Principles of Put Option Pricing (continued)
Effect of Exercise Price
Effect on Option Value
Two European puts differing only by X1 and X2.
Which is greater, Pe(S0,T,X1) or Pe(S0,T,X2)?
Construct portfolios A and B. See Table 3.8.
Portfolio A has non-negative payoff; therefore,
• Pe(S0,T,X2)  Pe(S0,T,X1)
• Intuition: show what happens if not true
Prices of DCRB options conform
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Ch. 3: 16
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Principles of Put Option Pricing (continued)
Effect of Exercise Price (continued)
Limits on the Difference in Premiums
Again, note Table 3.8. We must have
• (X2 - X1)(1+r)-T  Pe(S0,T,X2) - Pe(S0,T,X1)
• X2 - X1  Pe(S0,T,X2) - Pe(S0,T,X1)
• X2 - X1  Pa(S0,T,X2) - Pa(S0,T,X1)
• Implications
See Table 3.9. Prices of DCRB options conform
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Ch. 3: 17
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Principles of Put Option Pricing (continued)
Lower Bound of a European Put
Construct portfolios A and B. See Table 3.10.
A dominates B. This implies that (after rearranging)
Pe(S0,T,X)  Max(0,X(1+r)-T - S0)
This is the lower bound for a European put
See Figure 3.10 for the price curve for European
puts
Dividend adjustment: subtract present value of
dividends from S to obtain S´
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Ch. 3: 18
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Principles of Put Option Pricing (continued)
American Put Versus European Put
Pa(S0,T,X)  Pe(S0,T,X)
Early Exercise of American Puts
There is always a sufficiently low stock price that will
make it optimal to exercise an American put early.
Dividends on the stock reduce the likelihood of early
exercise.
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Ch. 3: 19
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Principles of Put Option Pricing (continued)
Put-Call Parity
Form portfolios A and B where the options are
European. See Table 3.11.
The portfolios have the same outcomes at the options’
expiration. Thus, it must be true that
S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T
This is called put-call parity.
It is important to see the alternative ways the
equation can be arranged and their interpretations.
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Ch. 3: 20
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Principles of Put Option Pricing (continued)
Put-Call parity for American options can be stated only
as inequalities:
N
C a (S'0 , T, X)  X   D j (1  r)
t j
j1
 S0  Pa (S'0 , T, X)
 C a (S'0 , T, X)  X(1  r) T
See Table 3.12 for put-call parity for DCRB options
See Figure 3.11 for linkages between underlying asset,
risk-free bond, call, and put through put-call parity.
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Ch. 3: 21
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Principles of Put Option Pricing (continued)
The Effect of Interest Rates
The Effect of Stock Volatility
Summary
See Table 3.13.
Appendix 3: The Dynamics of Option
Boundary Conditions: A Learning Exercise
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Ch. 3: 40
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Ch. 3: 42
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Ch. 3: 43
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Ch. 3: 45
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Ch. 3: 46
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Chapter 5: Option Pricing Models:
The Black-Scholes-Merton Model
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Good theories, like Black-Scholes-Merton, provide a
theoretical laboratory in which you can explore the likely
effect of possible causes. They give you a common language
with which to quantify and communicate your feelings about
value.
Emanuel Derman
The Journal of Derivatives, Winter, 2000, p. 64
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Ch. 5: 1
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Important Concepts in Chapter 5
The Black-Scholes-Merton option pricing model
The relationship of the model’s inputs to the option price
How to adjust the model to accommodate dividends and
put options
The concepts of historical and implied volatility
Hedging an option position
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Ch. 5: 2
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Origins of the Black-Scholes-Merton Formula
Brownian motion and the works of Einstein, Bachelier,
Wiener, Itô
Black, Scholes, Merton and the 1997 Nobel Prize
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Ch. 5: 3
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Black-Scholes-Merton Model as the Limit of
the Binomial Model
Recall the binomial model and the notion of a dynamic
risk-free hedge in which no arbitrage opportunities are
available.
Consider the DCRB June 125 call option.
Figure 5.1 shows the model price for an increasing number
of time steps.
The binomial model is in discrete time. As you decrease
the length of each time step, it converges to continuous
time.
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Ch. 5: 4
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Assumptions of the Model
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Stock prices behave randomly and evolve according to a
lognormal distribution.
See Figure 5.2a, 5.2b and 5.3 for a look at the notion of
randomness.
A lognormal distribution means that the log
(continuously compounded) return is normally
distributed. See Figure 5.4.
The risk-free rate and volatility of the log return on the
stock are constant throughout the option’s life
There are no taxes or transaction costs
The stock pays no dividends
The options are European
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Ch. 5: 5
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The Black-Scholes-Merton model gives the correct
formula for a European call under these assumptions.
The model is derived with complex mathematics but is
easily understandable. The formula is
C  S0 N(d1 )  Xe  rcT N(d 2 )
where
ln(S 0 /X)  (rc  σ 2 /2)T
d1 
σ T
d 2  d1  σ T
Chance/Brooks
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A Nobel Formula
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Ch. 5: 6
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A Nobel Formula (continued)
where
N(d1), N(d2) = cumulative normal probability
 = annualized standard deviation (volatility) of the
continuously compounded return on the stock
rc = continuously compounded risk-free rate
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Ch. 5: 7
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A Nobel Formula (continued)
A Digression on Using the Normal Distribution
The familiar normal, bell-shaped curve
(Figure 5.5)
See Table 5.1 for determining the normal probability
for d1 and d2. This gives you N(d1) and N(d2).
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Ch. 5: 8
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A Nobel Formula (continued)
A Numerical Example
Price the DCRB June 125 call
S0 = 125.94, X = 125, rc = ln(1.0456) = 0.0446,
T = 0.0959,  = 0.83.
See Table 5.2 for calculations. C = $13.21.
Familiarize yourself with the accompanying software
BSMbin8e.xls. Note the use of Excel’s =normsdist()
function.
BSMImpVol8e.xls. See Appendix.
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Ch. 5: 9
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A Nobel Formula (continued)
Characteristics of the Black-Scholes-Merton Formula
Interpretation of the Formula
The concept of risk neutrality, risk neutral
probability, and its role in pricing options
The option price is the discounted expected payoff,
Max(0,ST - X). We need the expected value of
ST - X for those cases where ST > X.
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Ch. 5: 10
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A Nobel Formula (continued)
Characteristics of the Black-Scholes-Merton Formula
(continued)
Interpretation of the Formula (continued)
The first term of the formula is the expected value of
the stock price given that it exceeds the exercise
price times the probability of the stock price
exceeding the exercise price, discounted to the
present.
The second term is the expected value of the
payment of the exercise price at expiration.
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Ch. 5: 11
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A Nobel Formula (continued)
Characteristics of the Black-Scholes-Merton Formula
(continued)
The Black-Scholes-Merton Formula and the Lower
Bound of a European Call
Recall from Chapter 3 that the lower bound would
be
Max(0, S0  Xe  rcT )
The Black-Scholes-Merton formula always exceeds
this value as seen by letting S0 be very high and then
let it approach zero.
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Ch. 5: 12
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A Nobel Formula (continued)
Characteristics of the Black-Scholes-Merton Formula
(continued)
The Formula When T = 0
At expiration, the formula must converge to the
intrinsic value.
It does but requires taking limits since otherwise it
would be division by zero.
Must consider the separate cases of ST  X and
ST < X.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 5: 13
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A Nobel Formula (continued)
Characteristics of the Black-Scholes-Merton Formula
(continued)
The Formula When S0 = 0
Here the company is bankrupt so the formula must
converge to zero.
It requires taking the log of zero, but by taking
limits we obtain the correct result.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 5: 14
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A Nobel Formula (continued)
Characteristics of the Black-Scholes-Merton Formula
(continued)
The Formula When  = 0
Again, this requires dividing by zero, but we can
take limits and obtain the right answer
If the option is in-the-money as defined by the stock
price exceeding the present value of the exercise
price, the formula converges to the stock price
minus the present value of the exercise price.
Otherwise, it converges to zero.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 5: 15
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A Nobel Formula (continued)
Characteristics of the Black-Scholes-Merton Formula
(continued)
The Formula When X = 0
From Chapter 3, the call price should converge to
the stock price.
Here both N(d1) and N(d2) approach 1.0 so by
taking limits, the formula converges to S0.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 5: 16
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A Nobel Formula (continued)
Characteristics of the Black-Scholes-Merton Formula
(continued)
The Formula When rc = 0
A zero interest rate is not a special case and no
special result is obtained.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 5: 17
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Variables in the Black-Scholes-Merton Model
The Stock Price
Let S  then C . See Figure 5.6.
This effect is called the delta, which is given by N(d1).
Measures the change in call price over the change in
stock price for a very small change in the stock price.
Delta ranges from zero to one. See Figure 5.7 for how
delta varies with the stock price.
The delta changes throughout the option’s life. See
Figure 5.8.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 5: 18
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Variables in the Black-Scholes-Merton Model
(continued)
The Stock Price (continued)
Delta hedging/delta neutral: holding shares of stock
and selling calls to maintain a risk-free position
The number of shares held per option sold is the
delta, N(d1).
As the stock goes up/down by $1, the option goes
up/down by N(d1). By holding N(d1) shares per call,
the effects offset.
The position must be adjusted as the delta changes.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 5: 19
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The Stock Price (continued)
Delta hedging works only for small stock price
changes. For larger changes, the delta does not
accurately reflect the option price change. This risk is
captured by the gamma:
 d12 /2
e
Call Gamma 
S0σ 2T
For our DCRB June 125 call,
Call Gamma 
Chance/Brooks
e
 ( 0.1742) 2 /2
125.94(0.8 3) 2(3.14159) 0.0959
An Introduction to Derivatives and Risk Management, 9th ed.
 0.0123
Ch. 5: 20
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Variables in the Black-Scholes-Merton Model
(continued)
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The Stock Price (continued)
If the stock goes from 125.94 to 130, the delta is
predicted to change from 0.569 to 0.569 + (130 125.94)(0.0123) = 0.6189. The actual delta at a price of
130 is 0.6171. So gamma captures most of the change
in delta.
The larger is the gamma, the more sensitive is the
option price to large stock price moves, the more
sensitive is the delta, and the faster the delta changes.
This makes it more difficult to hedge.
See Figure 5.9 for gamma vs. the stock price
See Figure 5.10 for gamma vs. time
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 5: 21
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Variables in the Black-Scholes-Merton Model
(continued)
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The Exercise Price
Let X , then C 
The exercise price does not change in most options so
this is useful only for comparing options differing only
by a small change in the exercise price.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 5: 22
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Variables in the Black-Scholes-Merton Model
(continued)
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The Risk-Free Rate
Take ln(1 + discrete risk-free rate from Chapter 3).
Let rc  then C  See Figure 5.11. The effect is called
rho
Call Rho  TXe  rcT N(d 2 )
In our example,
Call Rho  (0.0959)125e -0.0446(0.0959)(0.4670 )  5.57
If the risk-free rate goes to 0.12, the rho estimates that
the call price will go to (0.12 - 0.0446)(5.57) = 0.42.
The actual change is 0.43.
See Figure 5.12 for rho vs. stock price.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 5: 23
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Variables in the Black-Scholes-Merton Model
(continued)
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The Volatility or Standard Deviation
The most critical variable in the Black-Scholes-Merton
model because the option price is very sensitive to the
volatility and it is the only unobservable variable.
Let  , then C  See Figure 5.13.
This effect is known as vega.
S0 Te
Call vega 
2
-d12 /2
In our problem this is
Call vega 
Chance/Brooks
125.94 0.0959 e
-0.17422 /2
2(3.14159)
 15.32
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 5: 24
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Variables in the Black-Scholes-Merton Model
(continued)
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The Volatility or Standard Deviation (continued)
Thus if volatility changes by 0.01, the call price is
estimated to change by 15.32(0.01) = 0.15
If we increase volatility to, say, 0.95, the estimated
change would be 15.32(0.12) = 1.84. The actual call
price at a volatility of 0.95 would be 15.39, which is an
increase of 1.84. The accuracy is due to the near
linearity of the call price with respect to the volatility.
See Figure 5.14 for the vega vs. the stock price. Notice
how it is highest when the call is approximately at-themoney.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 5: 25
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Variables in the Black-Scholes-Merton Model
(continued)
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The Time to Expiration
Calculated as (days to expiration)/365
Let T , then C . See Figure 5.15. This effect is known
as theta:
2
S0 e d1 /2
 rc Xe rcT N(d 2 )
Call theta  2 2 T
In our problem, this would be
 (0.1742)2 /2
125.94(0.8 3)e
Call theta  2 2(3.14159) (0.0959)
 (0.0446)12 5e 0.0446(0.0959) (0.4670)  - 68.91
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 5: 26
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Variables in the Black-Scholes-Merton Model
(continued)
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The Time to Expiration (continued)
If one week elapsed, the call price would be expected to
change to (0.0959 - 0.0767)(-68.91) = -1.32. The actual
call price with T = 0.0767 is 12.16, a decrease of 1.39.
See Figure 5.16 for theta vs. the stock price
Note that your spreadsheet BSMbin8e.xls calculate the
delta, gamma, vega, theta, and rho for calls and puts.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 5: 27
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Variables in the Black-Scholes-Merton Model
(continued)
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Known Discrete Dividends
Assume a single dividend of Dt where the ex-dividend date is time
t during the option’s life.
Subtract present value of dividends from stock price.
Adjusted stock price, S, is inserted into the B-S-M model:
S0  S0  D t e
 rc t
See Table 5.3 for example.
The Excel spreadsheet BSMbin8e.xls allows up to 50 discrete
dividends.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 5: 28
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Black-Scholes-Merton Model When the Stock
Pays Dividends
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Continuous Dividend Yield
Assume the stock pays dividends continuously at the rate of .
Subtract present value of dividends from stock price. Adjusted
stock price, S, is inserted into the B-S model.
 c T

S0  S 0 e
See Table 5.4 for example.
This approach could also be used if the underlying is a foreign
currency, where the yield is replaced by the continuously
compounded foreign risk-free rate.
The Excel spreadsheet BSMbin8e.xls permit you to enter a
continuous dividend yield.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 5: 29
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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Black-Scholes-Merton Model When the Stock
Pays Dividends (continued)
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Black-Scholes-Merton Model and Some
Insights into American Call Options
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Table 5.5 illustrates how the early exercise decision is
made when the dividend is the only one during the option’s
life
The value obtained upon exercise is compared to the exdividend value of the option.
High dividends and low time value lead to early exercise.
Your Excel spreadsheet BSMbin8e.xls will calculate the
American call price using the binomial model.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 5: 30
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Historical Volatility
This is the volatility over a recent time period.
Collect daily, weekly, or monthly returns on the stock.
Convert each return to its continuously compounded
equivalent by taking ln(1 + return). Calculate variance.
Annualize by multiplying by 250 (daily returns), 52
(weekly returns) or 12 (monthly returns). Take square
root. See Table 5.6 for example with DCRB.
Your Excel spreadsheet Hisv8e.xls will do these
calculations. See Software Demonstration 5.2.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Estimating the Volatility
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Ch. 5: 31
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Estimating the Volatility (continued)
Implied Volatility
This is the volatility implied when the market price of
the option is set to the model price.
Figure 5.17 illustrates the procedure.
Substitute estimates of the volatility into the B-S-M
formula until the market price converges to the model
price. See Table 5.7 for the implied volatilities of the
DCRB calls.
A short-cut for at-the-money options is

C
(0.398)S 0 T
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 5: 32
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Estimating the Volatility (continued)
Implied Volatility (continued)
For our DCRB June 125 call, this gives
 
13.50
(0.398)125 .94 0.0959
 0.8697
This is quite close; the actual implied volatility is 0.83.
Appendix 5.A shows a method to produce faster
convergence.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 5: 33
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Estimating the Volatility (continued)
Implied Volatility (continued)
Interpreting the Implied Volatility
The relationship between the implied volatility and the time to
expiration is called the term structure of implied volatility. See
Figure 5.18.
The relationship between the implied volatility and the
exercise price is called the volatility smile or volatility skew.
Figure 5.19. These volatilities are actually supposed to be the
same. This effect is puzzling and has not been adequately
explained.
The CBOE has constructed indices of implied volatility of onemonth at-the-money options based on the S&P 100 (VIX) and
Nasdaq (VXN). See Figure 5.20.
Chance/Brooks
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Put Option Pricing Models
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Restate put-call parity with continuous discounting
Pe ( S0 , T , X )  Ce (S0 , T, X)  S0  Xe  rcT
Substituting the B-S-M formula for C above gives the
B-S-M put option pricing model
P  Xe  rcT [1  N(d 2 )]  S0 [1  N(d1 )]
N(d1) and N(d2) are the same as in the call model.
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Put Option Pricing Models (continued)
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Note calculation of put price:
P  125e  (0.0446)0.0959[1  0 .4670]
 125.94[1  0 .5692]  12.08
The Black-Scholes-Merton price does not reflect early exercise and,
thus, is extremely biased here since the American option price in the
market is 11.50. A binomial model would be necessary to get an
accurate price. With n = 100, we obtained 12.11.
See Table 5.8 for the effect of the input variables on the BlackScholes-Merton put formula.
Your software also calculates put prices and Greeks.
Chance/Brooks
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Ch. 5: 36
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Managing the Risk of Options
Here we talk about how option dealers hedge the risk of
option positions they take.
Assume a dealer sells 1,000 DCRB June 125 calls at the
Black-Scholes-Merton price of 13.5533 with a delta of
0.5692. Dealer will buy 569 shares and adjust the hedge
daily.
To buy 569 shares at $125.94 and sell 1,000 calls at
$13.5533 will require $58,107.
We simulate the daily stock prices for 35 days, at which
time the call expires.
Chance/Brooks
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Ch. 5: 37
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Managing the Risk of Options (continued)
The second day, the stock price is 120.4020. There are
now 34 days left. Using BSMbin8e.xls, we get a call price
of 10.4078 and delta of 0.4981. We have
Stock worth 569($120.4020) = $68,509
Options worth -1,000($10.4078) = -$10,408
Total of $58,101
Had we invested $58,107 in bonds, we would have had
$58,107e0.0446(1/365) = $58,114.
Table 5.9 shows the remaining outcomes. We must adjust
to the new delta of 0.4981. We need 498 shares so sell 71
and invest the money ($8,549) in bonds.
Chance/Brooks
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Ch. 5: 38
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Managing the Risk of Options (continued)
At the end of the second day, the stock goes to 126.2305 and the call to
13.3358. The bonds accrue to a value of $8,550. We have
Stock worth 498($126.2305) = $62,863
Options worth -1,000($13.3358) = -$13,336
Bonds worth $8,550 (includes one days’ interest)
Total of $58,077
Had we invested the original amount in bonds, we would have had
$58,107e0.0446(2/365) = $58,121. We are now short by over $44.
At the end we have $59,762, a excess of $1,406.
Chance/Brooks
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Ch. 5: 39
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Managing the Risk of Options (continued)
What we have seen is the second order or gamma effect.
Large price changes, combined with an inability to trade
continuously result in imperfections in the delta hedge.
To deal with this problem, we must gamma hedge, i.e.,
reduce the gamma to zero. We can do this only by adding
another option. Let us use the June 130 call, selling at
11.3792 with a delta of 0.5087 and gamma of 0.0123. Our
original June 125 call has a gamma of 0.0121. The stock
gamma is zero.
We shall use the symbols 1, 2, 1 and 2. We use hS
shares of stock and hC of the June 130 calls.
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Ch. 5: 40
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Managing the Risk of Options (continued)
The delta hedge condition is
hS(1) - 1,0001 + hC  2 = 0
The gamma hedge condition is
-1,0001 + hC 2 = 0
We can solve the second equation and get hC and then
substitute back into the first to get hS. Solving for hC and
hS, we obtain
hC = 1,000(0.0121/0.0123) = 984
hS = 1,000(0.5692 - (0.0121/0.0123)0.5087) = 68
So buy 68 shares, sell 1,000 June 125s, buy 984 June 130s.
Chance/Brooks
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Ch. 5: 41
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Managing the Risk of Options (continued)
The initial outlay will be
68($125.94) - 1,000($13.5533) + 985($11.3792) =
$6,219
At the end of day one, the stock is at 120.4020, the 125 call
is at 10.4078, the 130 call is at 8.5729. The portfolio is
worth
68($120.4020) - 1,000($10.4078) + 985($8.5729)
= $6,224
It should be worth $6,218e0.0446(1/365) = $6,220.
The new deltas are 0.4981 and 0.4366 and the new
gammas are 0.0131 and 0.0129.
Chance/Brooks
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Ch. 5: 42
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Managing the Risk of Options (continued)
The new values are 1,013 of the 130 calls so we buy 28.
The new number of shares is 56 so we sell 12. Overall,
this generates $1,444, which we invest in bonds.
The next day, the stock is at $126.2305, the 125 call is at
$13.3358 and the 130 call is at $11.1394. The bonds are
worth $1,205. The portfolio is worth
56($126.2305) - 1,000($13.3358) + 1,013($11.1394) +
$1,205 = $6,222.
The portfolio should be worth $6,219e0.0446(2/365) = $6,221.
Continuing this, we end up at $6,267 and should have
$6,246, a difference of $21. We are much closer than
when only delta hedging.
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Liquidity
Short-Selling
Information Asymmetry
Problems with Exotic Options
Performativity and Counter-Performativity
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 5: 44
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
.c
When the Black-Scholes-Merton may or may
not hold
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See Figure 5.21 for the relationship between call, put,
underlying asset, risk-free bond, put-call parity, and BlackScholes-Merton call and put option pricing models.
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Ch. 5: 45
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Appendix 5.A: A Shortcut to the Calculation
of Implied Volatility
This technique developed by Manaster and Koehler gives a
starting point and guarantees convergence. Let a given
volatility be * and the corresponding Black-ScholesMerton price be C(*). The initial guess should be
 S0 
2
  ln    rc T  
T
X
*
1
You then compute C(1*). If it is not close enough, you
make the next guess.
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Ch. 5: 46
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Appendix 5.A: A Shortcut to the Calculation
of Implied Volatility (continued)
Given the ith guess, the next guess should be

*
i 1

C( )  C( )e
  
*
i
*
i
d12 /2
2
S0 T
where d1 is computed using 1*. Let us illustrate using the
DCRB June 125 call. C() = 13.50. The initial guess is
*
1 
Chance/Brooks
 125.9375 
 2 
ln 
  0.0446(0.0 959) 
  0.4950
 0.0959 
 125 
An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 5: 47
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Appendix 5.A: A Shortcut to the Calculation
of Implied Volatility (continued)
At a volatility of 0.4950, the Black-Scholes-Merton value
is 8.41. The next guess should be
*2  0.4950 
8.41  13.50 e
(0.1533)2 /2
(2.5066)
125.9375 0.0959
 0.8260
where 0.1533 is d1 computed from the Black-ScholesMerton-Merton model using 0.4950 as the volatility and
2.5066 is the square root of 2. Now using 0.8260, we
obtain a Black-Scholes-Merton value of 13.49, which is
close enough to 13.50. So 0.83 is the implied volatility.
Chance/Brooks
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Ch. 5: 57
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Ch. 5: 58
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Ch. 5: 59
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Ch. 5: 60
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Ch. 5: 61
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Ch. 5: 62
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Ch. 5: 63
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Ch. 5: 64
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Ch. 5: 65
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Ch. 5: 66
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Ch. 5: 67
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Ch. 5: 68
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Ch. 5: 69
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Ch. 5: 70
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Ch. 5: 71
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Ch. 5: 72
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Ch. 5: 73
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Ch. 5: 74
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Ch. 5: 75
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Ch. 5: 76
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Ch. 5: 77
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Ch. 5: 78
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Ch. 5: 80
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Chapter 6: Basic Option Strategies
A good trader with a bad model can beat a bad trader with a
good model.
William Margrabe
Derivatives Strategy, April, 1998, p. 27
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Ch. 6: 1
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Important Concepts in Chapter 6
Profit equations and graphs for buying and selling stock,
buying and selling calls, buying and selling puts, covered
calls, protective puts and conversions/reversals
The effect of choosing different exercise prices
The effect of closing out an option position early versus
holding to expiration
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Ch. 6: 2
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Terminology and Notation
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Note the following standard symbols
C = current call price, P = current put price
S0 = current stock price, ST = stock price at expiration
T = time to expiration
X = exercise price
 = profit from strategy
The number of calls, puts and stock is given as
NC = number of calls
NP = number of puts
NS = number of shares of stock
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Ch. 6: 3
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Terminology and Notation (continued)
These symbols imply the following:
NC, NP, or NS > 0 implies buying (going long)
NC, NP, or NS < 0 implies selling (going short)
The Profit Equations
Profit equation for calls held to expiration
 = NC[Max(0,ST - X) - C]
• For buyer of one call (NC = 1) this implies
 = Max(0,ST - X) - C
• For seller of one call (NC = -1) this implies
 = -Max(0,ST - X) + C
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Ch. 6: 4
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The Profit Equations (continued)
Profit equation for puts held to expiration
 = NP[Max(0,X - ST) - P]
• For buyer of one put (NP = 1) this implies
 = Max(0,X - ST) - P
• For seller of one put (NP = -1) this implies
 = -Max(0,X - ST) + P
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Ch. 6: 5
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Terminology and Notation (continued)
The Profit Equations (continued)
Profit equation for stock
 = NS[ST - S0]
• For buyer of one share (NS = 1) this implies
 = ST - S0
• For short seller of one share (NS = -1) this
implies  = -ST + S0
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Ch. 6: 6
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Different Holding Periods
Three holding periods: T1 < T2 < T
For a given stock price at the end of the holding period, compute
the theoretical value of the option using the Black-Scholes-Merton
or other appropriate model.
Remaining time to expiration will be either T - T1,
T - T2 or T - T = 0 (we have already covered the latter)
For a position closed out at T1, the profit will be
  N c [C(ST1 , T  T1 , X)  C].
where the closeout option price is taken from the BlackScholes-Merton model for a given stock price at T1.
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Ch. 6: 7
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Terminology and Notation (continued)
Different Holding Periods (continued)
Similar calculation done for T2
For T, the profit is determined by the intrinsic value, as
already covered
Assumptions
No dividends
No taxes or transaction costs
We continue with the DCRB options. See
Table 6.1.
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Ch. 6: 8
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Stock Transactions
Buy Stock
Profit equation:  = NS[ST - S0] given that NS > 0
See Figure 6.1 for DCRB, S0 = $125.94
Maximum profit = , minimum = -S0
Sell Short Stock
Profit equation:  = NS[ST - S0] given that NS < 0
See Figure 6.2 for DCRB, S0 = $125.94
Maximum profit = S0, minimum = - 
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Ch. 6: 9
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Call Option Transactions
Buy a Call
Profit equation:  = NC[Max(0,ST - X) - C] given that
NC > 0. Letting NC = 1,
 = ST - X - C if ST > X
 = - C if ST  X
See Figure 6.3 for DCRB June 125, C = $13.50
Maximum profit = , minimum = -C
Breakeven stock price found by setting profit equation
to zero and solving: ST* = X + C
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Ch. 6: 10
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Call Option Transactions (continued)
Buy a Call (continued)
See Figure 6.4 for different exercise prices. Note
differences in maximum loss and breakeven.
For different holding periods, compute profit for range
of stock prices at T1, T2, and T using Black-ScholesMerton model. See Table 6.2 and
Figure 6.5.
Note how time value decay affects profit for given
holding period.
Chance/Brooks
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Ch. 6: 11
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Call Option Transactions (continued)
Write a Call
Profit equation:  = NC[Max(0,ST - X) - C] given that
NC < 0. Letting NC = -1,
 = -ST + X + C if ST > X
 = C if ST  X
See Figure 6.6 for DCRB June 125, C = $13.50
Maximum profit = +C, minimum = - 
Breakeven stock price same as buying call:
ST* = X + C
Chance/Brooks
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Ch. 6: 12
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Call Option Transactions (continued)
Write a Call (continued)
See Figure 6.7 for different exercise prices. Note
differences in maximum loss and breakeven.
For different holding periods, compute profit for range
of stock prices at T1, T2, and T using Black-ScholesMerton model. See Figure 6.8.
Note how time value decay affects profit for given
holding period.
Chance/Brooks
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Ch. 6: 13
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Put Option Transactions
Buy a Put
Profit equation:  = NP[Max(0,X - ST) - P] given that
NP > 0. Letting NP = 1,
 = X - ST - P if ST < X
 = - P if ST  X
See Figure 6.9 for DCRB June 125, P = $11.50
Maximum profit = X - P, minimum = -P
Breakeven stock price found by setting profit equation
to zero and solving: ST* = X - P
Chance/Brooks
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Ch. 6: 14
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Put Option Transactions (continued)
Buy a Put (continued)
See Figure 6.10 for different exercise prices. Note
differences in maximum loss and breakeven.
For different holding periods, compute profit for range
of stock prices at T1, T2, and T using Black-ScholesMerton model. See Figure 6.11.
Note how time value decay affects profit for given
holding period.
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Ch. 6: 15
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Put Option Transactions (continued)
Write a Put
Profit equation:  = NP[Max(0,X - ST)- P] given that
NP < 0. Letting NP = -1
 = -X + ST + P if ST < X
 = P if ST  X
See Figure 6.12 for DCRB June 125, P = $11.50
Maximum profit = +P, minimum = -X + P
Breakeven stock price found by setting profit equation
to zero and solving: ST* = X - P
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Ch. 6: 16
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Put Option Transactions (continued)
Write a Put (continued)
See Figure 6.13 for different exercise prices. Note
differences in maximum loss and breakeven.
For different holding periods, compute profit for range
of stock prices at T1, T2, and T using Black-ScholesMerton model. See Figure 6.14.
Note how time value decay affects profit for given
holding period.
Figure 6.15 summarizes these payoff graphs.
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Ch. 6: 17
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Calls and Stock: the Covered Call
One short call for every share owned
Profit equation:  = NS(ST - S0) + NC[Max(0,ST - X) - C]
given NS > 0, NC < 0, NS = -NC. With NS = 1, NC = -1,
 = ST - S0 + C if ST  X
 = X - S0 + C if ST > X
See Figure 6.16 for DCRB June 125,
S0 = $125.94, C = $13.50
Maximum profit = X - S0 + C, minimum = -S0 + C
Breakeven stock price found by setting profit equation to
zero and solving: ST* = S0 - C
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 6: 18
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See Figure 6.17 for different exercise prices. Note
differences in maximum loss and breakeven.
For different holding periods, compute profit for range
of stock prices at T1, T2, and T using Black-ScholesMerton model. See Figure 6.18.
Note the effect of time value decay.
Some General Considerations for Covered Calls:
alleged attractiveness of the strategy
misconception about picking up income
rolling up to avoid exercise
Opposite is short stock, buy call
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Calls and Stock: the Covered Call
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Ch. 6: 19
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Puts and Stock: the Protective Put
One long put for every share owned
Profit equation:  = NS(ST - S0) + NP[Max(0,X - ST) - P] given
NS > 0, NP > 0, NS = NP. With NS = 1, NP = 1,
 = ST - S0 - P if ST  X
 = X - S0 - P if ST < X
See Figure 6.19 for DCRB June 125, S0 = $125.94,
P = $11.50
Maximum profit = , minimum = X - S0 - P
Breakeven stock price found by setting profit equation to zero and
solving: ST* = P + S0
Like insurance policy
Chance/Brooks
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Ch. 6: 20
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See Figure 6.20 for different exercise prices. Note
differences in maximum loss and breakeven.
For different holding periods, compute profit for range
of stock prices at T1, T2, and T using Black-ScholesMerton model. See Figure 6.21.
Note how time value decay affects profit for given
holding period.
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Ch. 6: 21
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Synthetic Puts and Calls
Rearranging put-call parity to isolate put price
P  C  S0  Xe
 rc T
This implies put = long call, short stock, long risk-free
bond with face value X.
This is a synthetic put.
In practice most synthetic puts are constructed without
risk-free bond, i.e., long call, short stock.
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Ch. 6: 22
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Profit equation:  = NC[Max(0,ST - X) - C]
+ NS(ST - S0) given that NC > 0, NS < 0, NS = NP.
Letting NC = 1, NS = -1,
 = -C - ST + S0 if ST  X
 = S0 - X - C if ST > X
See Figure 6.22 for synthetic put vs. actual put.
Table 6.3 shows payoffs from reverse conversion (long
call, short stock, short put), used when actual put is
overpriced. Like risk-free borrowing.
Similar strategy for conversion, used when actual call
overpriced.
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Ch. 6: 23
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Summary
Chance/Brooks
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Ch. 6: 24
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Ch. 6: 25
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Ch. 6: 26
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Ch. 6: 27
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Ch. 6: 28
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Ch. 6: 29
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Ch. 6: 30
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Ch. 6: 31
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Ch. 6: 32
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Ch. 6: 33
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Ch. 6: 34
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Ch. 6: 35
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Ch. 6: 36
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Ch. 6: 37
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Ch. 6: 38
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Ch. 6: 39
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Ch. 6: 40
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Ch. 6: 41
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Ch. 6: 42
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Ch. 6: 43
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Ch. 6: 44
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Ch. 6: 45
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Ch. 6: 46
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Ch. 6: 47
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Ch. 6: 48
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Ch. 6: 49
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Chapter 7: Advanced Option Strategies
Read every book by traders to study where they lost money.
You will learn nothing relevant from their profits (the
markets adjust). You will learn from their losses.
Nassim Taleb
Derivatives Strategy, April, 1997, p. 25
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Ch. 7: 1
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Important Concepts in Chapter 7
Profit equations and graphs for option spread strategies,
including money spreads, collars, calendar spreads and
ratio spreads
Profit equations and graphs for option combination
strategies including straddles and box spreads
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Ch. 7: 2
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Option Spreads: Basic Concepts
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Definitions
spread
• vertical, strike, money spread
• horizontal, time, calendar spread
spread notation
• June 120/125
• June/July 120
long or short
• long, buying, debit spread
• short, selling, credit spread
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Ch. 7: 3
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Option Spreads: Basic Concepts (continued)
Why Investors Use Option Spreads
Risk reduction
To lower the cost of a long position
Types of spreads
bull spread
bear spread
time spread is based on volatility
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Ch. 7: 4
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Notation
For money spreads
X1 < X2 < X3
C1, C2, C3
N1, N2, N3
For time spreads
T1 < T2
C1, C2
N1, N2
See Table 7.1 for DCRB option data
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 7: 5
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.c
Option Spreads: Basic Concepts (continued)
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Bull Spreads
Buy call with strike X1, sell call with strike X2. Let N1
= 1, N2 = -1
Profit equation:  = Max(0,ST - X1) - C1 - Max(0,ST X2) + C2
 = -C1 + C2 if ST  X1 < X2
 = ST - X1 - C1 + C2 if X1 < ST  X2
 = X2 - X1 - C1 + C2 if X1 < X2 < ST
See Figure 7.1 for DCRB June 125/130, C1 =
$13.50, C2 = $11.35.
Maximum profit = X2 - X1 - C1 + C2, Minimum = - C1
+ C2
Breakeven: ST* = X1 + C1 - C2
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Ch. 7: 6
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Money Spreads (continued)
Bull Spreads (continued)
For different holding periods, compute profit for range
of stock prices at T1, T2, and T using Black-ScholesMerton model. See Figure 7.2.
Note how time value decay affects profit for given
holding period.
Early exercise not a problem.
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Ch. 7: 7
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Bear Spreads
Buy put with strike X2, sell put with strike X1. Let
N1 = -1, N2 = 1
Profit equation:  = -Max(0,X1 - ST) + P1
+ Max(0,X2 - ST) - P2
 = X2 - X1 + P1 - P2 if ST  X1 < X2
 = P1 + X2 - ST - P2 if X1 < ST < X2
 = P1 - P2 if X1 < X2  ST
See Figure 7.3 for DCRB June 130/125,
P1 = $11.50, P2 = $14.25.
Maximum profit = X2 - X1 + P1 - P2.
Minimum = P1 - P2.
Breakeven: ST* = X2 + P1 - P2.
Chance/Brooks
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Money Spreads (continued)
Bear Spreads (continued)
For different holding periods, compute profit for range
of stock prices at T1, T2, and T using Black-ScholesMerton model. See Figure 7.4.
Note how time value decay affects profit for given
holding period.
Note early exercise problem.
A Note About Put Money Spreads
Can construct call bear and put bull spreads.
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Ch. 7: 9
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Collars
Buy stock, buy put with strike X1, sell call with strike
X2. NS = 1, NP = 1, NC = -1.
Profit equation:  = ST - S0 + Max(0,X1 - ST) - P1 Max(0,ST - X2) + C2
 = X1 - S0 - P1 + C2 if ST  X1 < X2
 = ST - S0 - P1 + C2 if X1 < ST < X2
 = X2 - S0 - P1 + C2 if X1 < X2  ST
A common type of collar is what is often referred to as
a zero-cost collar. The call strike is set such that the
call premium offsets the put premium so that there is no
initial outlay for the options.
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Ch. 7: 10
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Collars (continued)
See Figure 7.5 for DCRB July 120/136.165,
P1 = $13.65, C2 = $13.65. That is, a call strike of
136.165 generates the same premium as a put with
strike of 120. This result can be obtained only by
using an option pricing model and plugging in
exercise prices until you find the one that makes the
call premium the same as the put premium.
This will nearly always require the use of OTC
options.
Maximum profit = X2 - S0. Minimum = X1 - S0.
Breakeven: ST* = S0.
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Ch. 7: 11
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Collars (continued)
The collar is a lot like a bull spread
(compare Figure 7.5 to Figure 7.1).
The collar payoff exceeds the bull spread payoff by
the difference between X1 and the interest on X1.
Thus, the collar is equivalent to a bull spread plus a
risk-free bond paying X1 at expiration.
For different holding periods, compute profit for range
of stock prices at T1, T2, and T using Black-ScholesMerton model. See Figure 7.6.
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Ch. 7: 12
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Butterfly Spreads
Buy call with strike X1, buy call with strike X3, sell two
calls with strike X2. Let N1 = 1, N2 = -2, N3 = 1.
Profit equation:  = Max(0,ST - X1) - C1
- 2Max(0,ST - X2) + 2C2 + Max(0,ST - X3) - C3
 = -C1 + 2C2 - C3 if ST  X1 < X2 < X3
 = ST - X1 - C1 + 2C2 - C3 if X1 < ST  X2 < X3
 = -ST +2X2 - X1 - C1 + 2C2 - C3
if X1 < X2 < ST  X3
 = -X1 + 2X2 - X3 - C1 + 2C2 - C3
if X1 < X2 < X3 < ST
See Figure 7.7 for DCRB July 120/125/130, C1 =
$16.00, C2 = $13.50, C3 = $11.35.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Butterfly Spreads (continued)
Maximum profit = X2 - X1 - C1 + 2C2 - C3,
minimum = -C1 + 2C2 - C3
Breakeven: ST* = X1 + C1 - 2C2 + C3 and
ST* = 2X2 - X1 - C1 + 2C2 - C3
For different holding periods, compute profit for range
of stock prices at T1, T2, and T using Black-ScholesMerton model. See Figure 7.8.
Note how time value decay affects profit for given
holding period.
Note early exercise problem.
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Ch. 7: 14
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Buy call with longer time to expiration, sell call with
shorter time to expiration.
Note how this strategy cannot be held to expiration
because there are two different expirations.
Profitability depends on volatility and time value decay.
Use Black-Scholes-Merton model to value options at
end of holding period if prior to expiration.
See Figure 7.9.
Note time value decay. See Table 7.2 and Figure 7.10.
Early exercise can be problem.
Can be constructed with puts as well.
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Ch. 7: 15
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Long one option, short another based on deltas of two
options. Designed to be delta-neutral. Can use any two
options on same stock.
Portfolio value
V = N1C1 + N2C2
Set to zero and solve for N1/N2 = -2/1, which is
ratio of their deltas (recall that  = N(d1) from
Black-Scholes-Merton model).
Buy June 120s, sell June 125s. Delta of 120 is 0.630;
delta of 125 is 0.569. Ratio is –(0.569/0.630) = -0.903.
For example, buy 903 June 120s, sell 1,000 June 125s
Note why this works and that delta will change.
Why do this? Hedging mispriced option
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Ch. 7: 16
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Straddle: long an equal number of puts and calls
Profit equation:  = Max(0,ST - X) - C
+ Max(0,X - ST) - P (assuming Nc = 1, Np = 1)
 = ST - X - C - P if ST  X
 = X - ST - C - P if ST < X
Either call or put will be exercised (unless ST = X).
See Figure 7.11 for DCRB June 125,
C = $13.50, P = $11.50.
Breakeven: ST* = X - C - P and ST* = X + C + P
Maximum profit: , minimum = - C - P
See Figure 7.12 for different holding periods. Note
time value decay.
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Ch. 7: 17
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Straddles (continued)
Applications of Straddles
Based on perception of volatility greater than priced by
market
A Short Straddle
Unlimited loss potential
Based on perception of volatility less than priced by
market
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Ch. 7: 18
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Box Spreads
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Definition: bull call money spread plus bear put money spread.
Risk-free payoff if options are European
Construction:
Buy call with strike X1, sell call with strike X2
Buy put with strike X2, sell put with strike X1
Profit equation:  = Max(0,ST - X1) - C1
- Max(0,ST - X2) + C2 + Max(0,X2 - ST) - P2 - Max(0,X1 - ST) + P1
 = X2 - X1 - C1 + C2 - P2 + P1 if ST  X1 < X2
 = X2 - X1 - C1 + C2 - P2 + P1 if X1 < ST  X2
 = X2 - X1 - C1 + C2 - P2 + P1 if X1 < X2  ST
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Ch. 7: 19
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Box Spreads (continued)
Evaluate by determining net present value (NPV)
NPV = (X2 - X1)(1 + r)-T - C1 + C2 - P2 + P1
This determines whether present value of risk-free
payoff exceeds initial value of transaction.
If NPV > 0, do it. If NPV < 0, do the reverse.
See Figure 7.13.
Box spread is also difference between two put-call
parities.
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Ch. 7: 20
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Box Spreads (continued)
Evaluate June 125/130 box spread
Buy 125 call at $13.50, sell 130 call at $11.35
Buy 130 put at $14.25, sell 125 put at $11.50
Initial outlay = $4.90, $490 for 100 each
NPV = 100[(130 - 125)(1.0456)-0.0959 - 4.90]
= 7.85
NPV > 0 so do it
Early exercise a problem only on short box spread
Transaction costs high
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Ch. 7: 21
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Summary
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Ch. 7: 22
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Ch. 7: 34
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Ch. 7: 35
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Ch. 7: 36
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Ch. 7: 37
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Chapter 8: The Structure of Forward and
Futures Markets
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Futures traders tend to be superstitious—when on a good
run they are reluctant to change their mojo, this includes
washing their jackets. Traders will wear their lucky jackets
until they fall apart or their luck runs out. Some traders have
even been buried in their lucky jackets, reflecting a hope that
the good luck their jackets provided in the trading pits on
Earth could be retained for eternity in that Great Trading
Pit in the sky.
Jim Overdahl
Futures Fall Special Issue 2005, p. 14
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Ch. 8: 1
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Important Concepts in Chapter 8
Definitions and examples of forward and futures contracts
Institutional characteristics of forward and futures markets
Futures contracts available for trading
Placing an order, margins, daily settlement
The role of the clearinghouse
Accessing futures price quotations
Magnitude and effects of transaction costs
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Ch. 8: 2
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Development of Forward and Futures Markets
Chicago Futures Markets
Development of Financial Futures
Development of Options on Futures Markets
Parallel Development of Over-the-Counter Markets
interbank market
growth of forward markets
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Ch. 8: 3
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Over-the-Counter Forward Market
customized
private
essentially unregulated
credit risk
market size: $84 trillion face value, $1.3 trillion market
value at year-end 2010
See Figure 8.1 for notional amount of forward market
See Figure 8.2 for market value of forward market
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Ch. 8: 4
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Contract Development (See Figure 8.3 for the daily
volume of the VIX futures contract)
Contract Terms and Conditions
contract size
quotation unit
minimum price fluctuation
contract grade
trading hours
Delivery Terms
delivery date and time
delivery or cash settlement
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Ch. 8: 5
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Organized Futures Trading (continued)
Daily Price Limits and Trading Halts
limit moves
circuit breakers
Other Exchange Responsibilities
minimum financial responsibility requirements
position limits
rules governing the trading floor
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Ch. 8: 6
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Derivatives Exchanges
Global and after-hours trading
Estimated world-wide volume in 2010 was 11.2 billion
contracts
43% Asia Pacific Region
13% North America
3.7 billion at Korea Exchange
3.1 billion at CME Group
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Ch. 8: 7
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Futures Traders
General Classes of Futures Traders
futures commission merchants
locals
dual trading
Classification by Trading Strategy
hedger/speculator
spreader
arbitrageur
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Ch. 8: 8
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Classification by Trading Style
scalpers
day traders
position traders
Off-Floor Futures Traders
individuals
institutions
Others: Introducing Broker (IB), Commodity Trading
Advisor (CTA), Commodity Pool Operator (CPO),
Associated Person (AP)
Forward Market Traders
over-the-counter
primarily institutions
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Ch. 8: 9
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Mechanics of Futures Trading
Placing an Order
pit
open outcry
electronic systems
Role of the Clearinghouse
See Figure 8.4.
margin deposits
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Ch. 8: 10
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Mechanics of Futures Trading (continued)
Daily Settlement
initial margin
maintenance margin
concept of “margin” vs. performance bond
settlement price
variation margin
See Table 8.1 for example.
open interest
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Ch. 8: 11
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Mechanics of Futures Trading (continued)
Delivery and Cash Settlement
three-day delivery process
alternative deliverable grades
offsetting
exchange for physicals
forward market procedures
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Ch. 8: 12
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Futures Price Quotations
Newspapers (such as The Wall Street Journal)
Web sites
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Ch. 8: 13
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Types of Futures Contracts
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Agricultural Commodities
Natural Resources
Miscellaneous Commodities
Foreign Currencies
Federal funds and Eurodollars
Treasury Notes and Bonds
Swap Futures
Equities
Managed Funds
Hedge Funds
Options on Futures
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Ch. 8: 14
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Transaction Costs in Forward and Futures
Trading
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Commissions
Bid-Ask Spread
Delivery Costs
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Ch. 8: 15
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Regulation of Futures Markets
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Regulation is nearly always at the federal level; e.g.,
Commodity Futures Trading Commission (U.S.)
Financial Services Authority (U.K.)
Financial Services Agency (Japan)
Objective of most federal regulation
ensuring public information available
authorization and licensing of contracts and
exchanges
contract approval
market surveillance
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Ch. 8: 16
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Regulation of Futures Markets (continued)
Arbitration of disputes is sometimes done through the
federal government and the courts but often through selfregulatory organizations such as the National Futures
Association in the U. S.
Note: Forward markets are regulated only indirectly and,
thus, are largely unregulated.
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Ch. 8: 17
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OTC Central Clearing
Dodd-Frank Act of 2010 further motivated efforts in the
OTC derivatives markets for central clearing
OTC central clearing should provide more transparency to
this opaque market and more accountability
Several clearing corporations are competing for OTC
derivatives central clearing
OTC central clearing is like the spoke and hub system used
by some airlines
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Summary
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Ch. 8: 19
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Appendix 8: Taxation of Futures Contracts
Treated as 60 % capital gains and 40 % ordinary income.
Capital gains subject to 28 % maximum.
Must be marked to market at year end.
New single stock futures are taxed the same as individual
stocks.
Hedge transactions covered in Chapter 11.
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Chapter 9: Principles of Pricing Forwards,
Futures, and Options on Futures
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Futures markets are an accurate representation of
consensus opinion, but if we pool all our ignorance, we do
not get wisdom from it.
Jim Bianco
The Wall Street Journal, March 11, 2006, Page B3.
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Ch. 9: 1
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Important Concepts in Chapter 9
Price and value of forward and futures contracts
Relationship between forward and futures prices
Determination of the spot price of an asset
Carry arbitrage model for theoretical fair price
Contango, backwardation, and convenience yield
Futures prices and risk premiums
Pricing options on futures
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Ch. 9: 2
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Generic Carry Arbitrage
The Concept of Price Versus Value
Normally in an efficient market, price = value.
For a futures or forward, price is the contracted rate of
future purchase. Value is something different.
At the beginning of a contract, value = 0 for both
futures and forwards.
Notation
Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of
forward and futures contracts created at time 0 and
expiring at time T.
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Ch. 9: 3
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Generic Carry Arbitrage (continued)
The Value of a Forward Contract
Forward price at expiration:
F(T,T) = ST.
That is, the price of an expiring forward contract is
the spot price.
Value of forward contract at expiration:
VT(0,T) = ST - F(0,T).
An expiring forward contract allows you to buy the
asset, worth ST, at the forward price F(0,T). The
value to the short party is (-1) times this.
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Ch. 9: 4
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The Value of a Forward Contract (continued)
The Value of a Forward Contract Prior to Expiration
A: Go long forward contract at price F(0,T) at time 0.
B: At t go long the asset and take out a loan promising to pay
F(0,T) at T
• At time T, A and B are worth the same, ST – F(0,T).
Thus, they must both be worth the same prior to T.
• So Vt(0,T) = St – F(0,T)(1+r)-(T-t)
• See Table 9.1.
Example: Go long 45 day contract at F(0,T) = $100. Risk-free
rate = 0.10. 20 days later, the spot price is $102. The value of
the forward contract is 102 - 100(1.10)-25/365 = 2.65.
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Ch. 9: 5
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Generic Carry Arbitrage (continued)
The Value of a Futures Contract
Futures price at expiration:
fT(T) = ST.
Value during the trading day but before being marked
to market:
vt(T) = ft(T) - ft-1(T).
Value immediately after being marked to market:
vt(T) = 0.
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Ch. 9: 6
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Generic Carry Arbitrage (continued)
Forward Versus Futures Prices
Forward and futures prices will be equal
One day prior to expiration
More than one day prior to expiration if
• Interest rates are certain
• Futures prices and interest rates are uncorrelated
Futures prices will exceed forward prices if futures
prices are positively correlated with interest rates.
Default risk can also affect the difference between
futures and forward prices.
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Ch. 9: 7
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Forward and Futures Pricing When the Underlying Generates Cash
Flows
For example, dividends on a stock or index
Assume one dividend DT paid at expiration.
Buy stock, sell futures guarantees at expiration that you will
have DT + f0(T). Present value of this must equal S0, using
risk-free rate. Thus,
• f0(T) = S0(1+r)T - DT.
For multiple dividends, let DT be compound future value of
dividends. See Figure 9.1 for two dividends.
Dividends reduce the cost of carry.
If D0 represents the present value of the dividends, the model
becomes
• f0(T) = (S0 – D0)(1+r)T.
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Ch. 9: 8
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Forward and Futures Pricing When the Underlying
Generates Cash Flows (continued)
For dividends paid at a continuously compounded rate
of c,
f(0, T)  S0e (rc  c )T
Example: S0 = 50, rc = 0.08, c = 0.06, expiration in 60
days (T = 60/365 = 0.164).
f0(T) = 50e(0.08 - 0.06)(0.164) = 50.16.
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Ch. 9: 9
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Valuation of Equity Forward Contracts
When there are dividends, to determine the value of a
forward contract during its life
Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t)
where Dt,T is the value at time t of the future
dividends to time T
Or if dividends are continuous,
Vt (0, T )  St e  c (T t )  F (0, T )e  rc (T t )
Chance/Brooks
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Ch. 9: 10
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Carry Arbitrage: Currencies
Pricing Foreign Currency Forward and Futures Contracts:
Interest Rate Parity
Interest Rate Parity: the relationship between futures or
forward and spot exchange rates. Same as carry
arbitrage model in other forward and futures markets.
Proves that one cannot convert a currency to another
currency, sell a futures, earn the foreign risk-free rate,
and convert back without risk, earning a rate higher
than the domestic rate.
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Ch. 9: 11
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Carry Arbitrage: Currencies (continued)
Pricing Foreign Currency Forward and Futures Contracts: Interest
Rate Parity (continued)
S0 = spot rate in domestic currency per foreign currency. Foreign
rate is . Holding period is T. Domestic rate is r.
Take S0(1+ )-T units of domestic currency and buy (1+ )-T
units of foreign currency.
Sell forward contract to deliver one unit of foreign currency at
T at price F(0,T).
Hold foreign currency and earn rate . At T you will have one
unit of the foreign currency.
Deliver foreign currency and receive F(0,T) units of domestic
currency.
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Ch. 9: 12
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Carry Arbitrage: Currencies (continued)
Pricing Foreign Currency Forward and Futures Contracts:
Interest Rate Parity (continued)
So an investment of S0(1+ )-T units of domestic
currency grows to F (0,T) units of domestic currency
with no risk. Return should be r. Therefore
• F(0,T) = S0(1+ )-T(1 + r)T
This is called interest rate parity.
Sometimes written as
• F(0,T) = S0(1 + r)T/(1 + )T
Chance/Brooks
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Ch. 9: 13
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Carry Arbitrage: Currencies (continued)
Pricing Foreign Currency Forward and Futures Contracts: Interest
Rate Parity (continued)
Example (from a European perspective): S0 = €1.0304.
U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is
90/365 = 0.2466.
F(0,T) = €1.0304(1.0584)-0.2466(1.0359)0.2466 = €1.025
If forward rate is actually €1.03, then it is overpriced.
Buy (1.0584)-0.2466 = $0.9861 for 0.9861(€1.0304) = €1.0161.
Sell one forward contract at €1.03.
Earn 5.84% on $0.9861. This grows to $1.
At expiration, deliver $1 and receive €1.03.
Return is (1.03/1.0161)365/90 - 1 = 0.0566 (> 0.0359)
This transaction is called covered interest arbitrage.
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Ch. 9: 14
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Carry Arbitrage: Currencies (continued)
Pricing Foreign Currency Forward and Futures Contracts:
Interest Rate Parity (continued)
It is also sometimes written as
F(0,T) = S0(1 + )T(1 + r)-T
Here, the spot rate is being quoted in units of the
foreign currency.
Note that the forward discount/premium has nothing to
do with expectations of future exchange rates.
Difference between domestic and foreign rate is
analogous to difference between risk-free rate and
dividend yield on stock index futures.
Chance/Brooks
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Ch. 9: 15
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Spot Prices, Risk Premiums, and the Carry Arbitrage for
Generic Assets
First assume no uncertainty of future price. Let s be the
cost of storing an asset and i be the interest rate for the
period of time the asset is owned. Then
S0 = ST - s - iS0
If we now allow uncertainty but assume people are risk
neutral, we have
S0 = E(ST) - s - iS0
If we now allow people to be risk averse, they require a
risk premium of E(). Now
S0 = E(ST) - s - iS0 - E()
Chance/Brooks
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Ch. 9: 16
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Spot Prices, Risk Premiums, and the Carry Arbitrage for
Generic Assets (continued)
Let us define iS0 as the net interest, which is the interest
foregone minus any cash received.
Define s + iS0 as the cost of carry.
Denote cost of carry as .
Note how cost of carry is a meaningful concept only for
storable assets
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Ch. 9: 17
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The Theoretical Fair Price (Forward/Futures Pricing
Revisited)
Do the following
Buy asset in spot market, paying S0; sell futures
contract at price f0(T); store and incur costs.
At expiration, make delivery. Profit:
•  = f0(T) - S0 - 
This must be zero to avoid arbitrage; thus,
• f0(T) = S0 + 
See Figure 9.2.
Note how arbitrage and quasi-arbitrage make this hold.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 9: 18
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Pricing Models and Risk Premiums
Forward/Futures Pricing Revisited(continued)
See Figure 9.3 for an illustration of the determination of
futures prices.
Contango is f0(T) > S0. See Table 9.2.
When f0(T) < S0, convenience yield is  , an additional
return from holding asset when in short supply or a
non-pecuniary return. Market is said to be at less than
full carry and in backwardation or inverted.
See Table 9.3. Market can be both backwardation and
contango. See Table 9.4.
Chance/Brooks
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Ch. 9: 19
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Futures Prices and Risk Premia
The no risk-premium hypothesis
Market consists of only speculators.
f0(T) = E(ST). See Figure 9.4.
The risk-premium hypothesis
E(fT(T)) > f0(T).
When hedgers go short futures, they transfer risk
premium to speculators who go long futures.
E(ST) = f0(T) + E(). See Figure 9.5.
Normal contango: E(ST) < f0(T)
Normal backwardation: f0(T) < E(ST)
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Ch. 9: 20
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Put-Call-Forward/Futures Parity
Can construct synthetic futures with options.
See Table 9.5.
Put-call-forward/futures parity
Pe(S0,T,X) = Ce(S0,T,X) + (X - f0(T))(1+r)-T
Numerical example using S&P 500. On May 14, S&P 500
at 1337.80 and June futures at 1339.30. June 1340 call at
40 and put at 39. Expiration of June 18 so
T = 35/365 = 0.0959. Risk-free rate at 4.56%.
Chance/Brooks
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Ch. 9: 21
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Put-Call-Forward/Futures Parity (continued)
So Pe(S0,T,X) = 39
Ce(S0,T,X) + (X - f0(T))(1+r)-T
= 40 + (1340 - 1339.30)(1.0456)-0.0959 = 40.70.
Buy put and futures for 39, sell call and bond for 40.70
and net 1.70 profit at no risk. Transaction costs would
have to be considered.
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Ch. 9: 22
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The Intrinsic Value of an American Option on Futures
Minimum value of American call on futures
Ca(f0(T),T,X)  Max(0, f0(T) - X)
Minimum value of American put on futures
Pa(f0(T),T,X)  Max(0,X - f0(T))
Difference between option price and intrinsic value is
time value.
Chance/Brooks
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Ch. 9: 23
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The Lower Bound of a European Option on Futures
For calls, construct two portfolios.
See Table 9.6.
Portfolio A dominates Portfolio B so
Ce(f0(T),T,X)  Max[0,(f0(T) - X)(1+r)-T]
Note that lower bound can be less than intrinsic value
even for calls.
For puts, see Table 9.7.
Portfolio A dominates Portfolio B so
Pe(f0(T),T,X)  Max[0,(X - f0(T))(1+r)-T]
Chance/Brooks
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Ch. 9: 24
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Put-Call Parity of Options on Futures
Construct two portfolios, A and B.
See Table 9.8.
The portfolios produce equivalent results. Therefore they must
have equivalent current values. Thus,
Pe(f0(T),T,X) = Ce(f0(T),T,X) + (X - f0(T))(1+r)-T.
Compare to put-call parity for options on spot:
Pe(S0,T,X) = Ce(S0,T,X) - S0 + X(1+r)-T.
If options on spot and options on futures expire at same time,
their values are equal, implying
f0(T) = S0(1+r)T, which we obtained earlier (no cash flows).
Chance/Brooks
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Ch. 9: 25
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Early Exercise of Call and Put Options on Futures
Deep in-the-money call may be exercised early because
behaves almost identically to futures
exercise frees up funds tied up in option but requires
no funds to establish futures
minimum value of European futures call is less than
value if it could be exercised
See Figure 9.6.
Similar arguments hold for puts
Compare to the arguments for early exercise of call and
put options on spot.
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Ch. 9: 26
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Options on Futures Pricing Models
Black model for pricing European options on futures
C  e  rc T [f 0 (T)N(d 1 )  XN(d 2 )]
where
d1 


ln(f 0 (T)/X)   2 /2 T
 T
d 2  d1   T
Chance/Brooks
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Ch. 9: 27
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Options on Futures Pricing Models (continued)
Note that with the same expiration for options on spot
as options on futures, this formula gives the same price.
Example
See Table 9.9.
Software for Black-Scholes-Merton can be used by
inserting futures price instead of spot price and risk-free
rate for dividend yield. Note why this works.
For puts
P  Xe rcT [1  N(d 2 )]  f 0 (T)e rcT [1  N(d 1 )]
Chance/Brooks
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Ch. 9: 28
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Summary
See Table 9.10 for a summary of equations.
See Figure 9.7 for linkage between forwards/futures,
underlying asset and risk-free bond.
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Ch. 9: 36
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Ch. 9: 38
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Ch. 9: 40
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Ch. 9: 41
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Ch. 9: 42
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Ch. 9: 43
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Ch. 9: 44
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Ch. 9: 46
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Chapter 11: Forward and Futures Hedging,
Spread, and Target Strategies
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The beauty of finance and speculation was that they could
be different things to different men. To some: poetry or
high drama; to others, physics, scientific and immutable; to
still others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
Chance/Brooks
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Ch. 11: 1
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Important Concepts in Chapter 11
Why firms hedge
Hedging concepts
Factors involved when constructing a hedge
Hedge ratios
Examples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures
hedges
Examples of spread and target strategies
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Ch. 11: 2
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The value of the firm may not be independent of financial
decisions because
Shareholders might be unaware of the firm’s risks.
Shareholders might not be able to identify the correct
number of futures contracts necessary to hedge.
Shareholders might have higher transaction costs of
hedging than the firm.
There may be tax advantages to a firm hedging.
Hedging reduces bankruptcy costs.
Managers may be reducing their own risk.
Hedging may send a positive signal to creditors.
Dealers hedge their market-making activities in
derivatives.
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Ch. 11: 3
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Reasons not to hedge
Hedging can give a misleading impression of the
amount of risk reduced
Hedging eliminates the opportunity to take advantage
of favorable market conditions
There is no such thing as a hedge. Any hedge is an act
of taking a position that an adverse market movement
will occur. This, itself, is a form of speculation.
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Why Hedge? (continued)
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Ch. 11: 4
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Short Hedge and Long Hedge
Short (long) hedge implies a short (long) position in
futures
Short hedges can occur because the hedger owns an
asset and plans to sell it later.
Long hedges can occur because the hedger plans to
purchase an asset later.
An anticipatory hedge is a hedge of a transaction that is
expected to occur in the future.
See Table 11.1 for hedging situations.
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Hedging Concepts
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Ch. 11: 5
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Hedging Concepts (continued)
The Basis
Basis = spot price - futures price.
Hedging and the Basis
 (short hedge) = ST - S0 (from spot market)
- (fT - f0) (from futures market)
 (long hedge) = -ST + S0 (from spot market)
+ (fT - f0) (from futures market)
If hedge is closed prior to expiration,
 (short hedge) = St - S0 - (ft - f0)
If hedge is held to expiration, St = ST = fT = ft.
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Ch. 11: 6
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The Basis (continued)
Hedging and the Basis (continued)
Example: Buy asset for $100, sell futures for $103. Hold until
expiration. Sell asset for $97, close futures at $97. Or deliver
asset and receive $103. Make $3 for sure.
Basis definition
initial basis: b0 = S0 - f0
basis at time t: bt = St - ft
basis at expiration: bT = ST - fT = 0
For a position closed at t:
 (short hedge) = St - ft - (S0 - f0) = -b0 + bt
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Hedging Concepts (continued)
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Ch. 11: 7
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The Basis (continued)
This is the change in the basis and illustrates the
principle of basis risk.
Hedging attempts to lock in the future price of an asset
today, which will be f0 + (St - ft).
A perfect hedge is practically non-existent.
Short hedges benefit from a strengthening basis.
All of this reverses for a long hedge.
See Table 11.2 for hedging profitability and the basis.
Chance/Brooks
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Ch. 11: 8
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Hedging Concepts (continued)
The Basis (continued)
Example: March 30. Spot gold $1,387.15. June
futures $1,388.60. Buy spot, sell futures. Note:
b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration,
profit should be change in basis or 1.45.
At expiration, let ST = $1,408.50. Sell gold in spot
for $1,408.50, a profit of 21.35. Buy back futures at
$1,408.50, a profit of −19.90. Net gain =1.45 or
$145 on 100 oz. of gold.
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Ch. 11: 9
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Hedging Concepts (continued)
The Basis (continued)
Example: (continued)
Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63.
Profit on spot = −9.63. Profit on futures = 9.97.
Net gain = 0.34 or $34 on 100 oz.
Note that change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34.
Behavior of the basis, see Figure 11.1.
In forward markets, the hedge is customized so there is
no basis risk.
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Ch. 11: 10
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Hedging Concepts (continued)
Some Risks of Hedging
cross hedging
spot and futures prices occasionally move opposite
quantity risk
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Ch. 11: 11
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Hedging Concepts (continued)
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Contract Choice
Which futures underlying asset?
High correlation with spot
Favorably priced
Which expiration?
The futures with maturity closest to but after the
hedge termination date subject to the suggestion not
to be in a contract in its expiration month
See Table 11.3 for example of recommended
contracts for T-bond hedge
Concept of rolling the hedge forward
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Ch. 11: 12
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Hedging Concepts (continued)
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Contract Choice (continued)
Long or short?
A critical decision! No room for mistakes.
Three methods to answer the question.
See Table 11.4.
• worst case scenario method
• current spot position method
• anticipated future spot transaction method
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Ch. 11: 13
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Hedging Concepts (continued)
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Margin Requirements and Marking to Market
low margin requirements on futures, but
cash will be required for margin calls
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Determination of the Hedge Ratio
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Hedge ratio: The number of futures contracts to hedge a
particular exposure
Naïve hedge ratio
Appropriate hedge ratio should be
Nf = −S/f
Note that this ratio must be estimated.
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Ch. 11: 15
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Determination of the Hedge Ratio (continued)
Minimum Variance Hedge Ratio
Profit from short hedge:
 = S + fNf
Variance of profit from short hedge:
  S2 + f2Nf2 + 2SfNf
The optimal (variance minimizing) hedge ratio is
Nf = −Sf/f2
This is the beta from a regression of spot price
change on futures price change.
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Ch. 11: 16
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Determination of the Hedge Ratio (continued)
Minimum Variance Hedge Ratio (continued)
Hedging effectiveness is
e* = (risk of unhedged position − risk of hedged
position)/risk of unhedged position
This is coefficient of determination from regression.
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Ch. 11: 17
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Determination of the Hedge Ratio (continued)
Price Sensitivity Hedge Ratio
This applies to hedges of interest sensitive securities.
First we introduce the concept of duration. We start
with a bond priced at B:
T

CPt
B
t
t 1 (1  y B )
where CPt is the cash payment at time t and yB is the
yield, or discount rate.
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Ch. 11: 18
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Determination of the Hedge Ratio (continued)
Price Sensitivity Hedge Ratio (continuation)
An approximation to the change in price for a yield change is
B  B
DUR B (y)
1  yB
with DURB being the bond’s duration, which is a weightedaverage of the times to each cash payment date on the bond, and 
represents the change in the bond price or yield.
Duration has many weaknesses but is widely used as a measure of
the sensitivity of a bond’s price to its yield.
Modified duration (MD) measures the bond percentage price
change for a given change in yield.
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Ch. 11: 19
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Determination of the Hedge Ratio (continued)
Price Sensitivity Hedge Ratio (continuation)
The hedge ratio is as follows:
N*f
 MD B  B 
 
 f 
 MD 

f 

Where MDB  −(/B) /yB and
MDf  −(f/f) /yf
Note the concepts of implied yield and implied duration
of a futures. Also, technically, the hedge ratio will
change continuously like an option’s delta and, like
delta, it will not capture the risk of large moves.
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Determination of the Hedge Ratio (continued)
Price Sensitivity Hedge Ratio (continued)
Alternatively,
Nf = −(Yield beta)PVBPB/PVBPf
• where Yield beta is the beta from a regression of
spot bond yield on futures yield and
• PVBPB, PVBPf is the present value of a basis
point change in the bond and futures prices.
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Determination of the Hedge Ratio (continued)
Stock Index Futures Hedging
Appropriate hedge ratio is
Nf = −(S/f)(S/f)
where S is the beta from the CAPM and f is the
beta of the futures, often assumed to be 1.
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Hedging Strategies
Long Hedge With Foreign Currency Futures
American firm planning to buy foreign inventory and
will pay in foreign currency.
See Table 11.5.
Short Hedge With Foreign Currency Forwards
British subsidiary of American firm will convert
pounds to dollars.
See Table 11.6.
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Hedging Strategies (continued)
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Intermediate and Long-Term Interest Rate Hedges
First let us look at the CBOT T-note and bond contracts
T-bonds: must be a T-bond with at least 15 years to
maturity or first call date
T-note: three contracts (2-, 5-, and 10-year)
A bond of any coupon can be delivered but the
standard is a 6% coupon. Adjustments, explained in
Chapter 10, are made to reflect other coupons.
Price is quoted in units and 32nds, relative to $100
par, e.g., 93 14/32 is $93.4375.
Contract size is $100,000 face value so price is
$93,437.50
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Hedging Strategies (continued)
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Intermediate and Long-Term Interest Rate Hedges
(continued)
Hedging a Long Position in a Government Bond
See Table 11.7 for example.
Anticipatory Hedge of a Future Purchase of a Treasury
Note
See Table 11.8 for example.
Hedging a Corporate Bond Issue
See Table 11.9 for example.
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Hedging Strategies (continued)
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Stock Market Hedges
First look at the contracts
We primarily shall use the S&P 500 futures. Its
price is determined by multiplying the quoted price
by $250, e.g., if the futures is at 1300, the price is
1300($250) = $325,000
Stock Portfolio Hedge
See Table 11.10 for example.
Anticipatory Hedge of a Takeover
See Table 11.11 for example.
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Spread Strategies
Intramarket Spreads
Based on changes in the difference in carry costs
See Figure 11.2 for illustration.
Treasury Bond Futures Spreads
See Figure 11.3 and Figure 11.4 for illustration the
relationship between changes in spreads and interest
rates.
See Table 11.12 for calculation of Tbond futures spread
profits.
See Figure 11.5 for illustration of stock index spreads
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Intermarket Spread Strategies
Intermarket spread strategies involve two futures contracts
on different underlying instruments
Intermarket spread strategies tend to be more risky than
intramarket spreads because there is both the change in
spreads and the change in underlying instruments
NOB denotes notes over bonds
Intermarket spread strategies could also involve various
equity markets
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Target Strategies: Bonds
Target Duration with Bond Futures
Number of futures needed to change modified duration
N*f
 MD T - MD B  B 

 




MD f
 f 

Goal is to move the modified duration from its current
value to a new target value
See Table 11.13 for illustration.
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Target Strategies: Equities
Alpha Capture
Number of futures to hedge systematic risk
N *f
S
 S  
f 
Goal is to move the eliminate systematic risk
See Table 11.14 for illustration.
Target Beta (see Table 11.15 for illustration.)
N*f
Chance/Brooks
S
 T  S  
f 
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Target Strategies: Equities (continued)
Tactical Asset Allocation
Strategic asset allocation – long run target weights for
each asset class
Tactical asset allocation – short run deviations in
weights for each asset class
See Table 11.16 for illustration.
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Summary
Table 11.17 recaps the types of hedge situations, the nature
of the risk and how to hedge the risk
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Appendix 11: Taxation of Hedging
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Hedges used by businesses to protect inventory and in
standard business transactions are taxed as ordinary
income.
Transactions must be shown to be legitimate hedges and
not just speculation outside of the norm of ordinary
business activities. This is called the business motive test.
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Chapter 12: Swaps
Let us not forget there were plenty of financial disasters
before quants showed up on Wall Street, and the
subsequent disasters (including the current one) had plenty
of help from the non-quants.
Aaron Brown
Risk Professional, April 2010, p. 18
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Ch. 12: 1
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Important Concepts in Chapter 12
The concept of a swap
Different types of swaps, based on underlying currency,
interest rate, or equity
Pricing and valuation of swaps
Strategies using swaps
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Ch. 12: 2
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Definition of a swap
Four types of swaps
Currency
Interest rate
Equity
Commodity (not covered in this book)
Characteristics of swaps
No cash up front
Notional amount
Settlement date, settlement period
Credit risk
Dealer market
See Figure 12.1 for growth in world-wide notional amount
See Figure 12.2 for growth in world-wide gross market value
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Ch. 12: 3
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Interest Rate Swaps
The Structure of a Typical Interest Rate Swap
Example: On December 15 XYZ enters into $50
million notional amount swap with ABSwaps.
Payments will be on 15th of March, June,
September, December for one year, based on
LIBOR. XYZ will pay 7.5% fixed and ABSwaps
will pay LIBOR. Interest based on exact day count
and 360 days (30 per month). In general the cash
flow to the fixed payer will be
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Ch. 12: 4
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Interest Rate Swaps (continued)
The Structure of a Typical Interest Rate Swap
(continued)
The payments in this swap are
 Days 
($50,000,0 00)(LIBOR - 0.075) 

 360 
Payments are netted.
See Figure 12.3 for payment pattern
See Table 12.1 for sample of payments after-thefact.
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Interest Rate Swaps (continued)
The Pricing and Valuation of Interest Rate Swaps
How is the fixed rate determined?
A digression on floating-rate securities. The price
of a LIBOR zero coupon bond for maturity of ti days
is
1
B0 (t i ) 
1  L 0 (t i )(t i /360)
• Starting at the maturity date and working back,
we see that the price is par on each coupon date.
See Figure 12.4.
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Ch. 12: 6
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Interest Rate Swaps (continued)
The Pricing and Valuation of Interest Rate Swaps
(continued)
By adding the notional amounts at the end, we can
separate the cash flow streams of an interest rate
swap into those of a fixed-rate bond and a floatingrate bond.
See Figure 12.5.
The value of a fixed-rate bond (q = days/360):
n
VFXRB 
 RqB (t )  B (t )
0
i
0
n
i 1
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Ch. 12: 7
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Interest Rate Swaps (continued)
The Pricing and Valuation of Interest Rate Swaps
(continued)
The value of a floating-rate bond
VFLRB  1 (at time 0 or a payment date)
At time t, between 0 and 1,
VFLRB 
1  L 0 (t 1 )q
(between payment dates 0 and 1)
1  L t (t 1 )(t 1  t)/360
The value of the swap (pay fixed, receive floating)
is, therefore,
VS  VFLRB  VFXRB
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Ch. 12: 8
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Interest Rate Swaps (continued)
The Pricing and Valuation of Interest Rate Swaps (continued)
To price the swap at the start, set this value to zero and solve
for R




 1  1  B 0 (t n ) 
R    n

q

 
B
(t
)
0 i 

 i 1

See Table 12.2 for an example.
Note how dealers quote as a spread over Treasury rate.
To value a swap during its life, simply find the difference
between the present values of the two streams of payments.
See Table 12.3. Market value reflects the economic value, is
necessary for accounting, and gives an indication of the credit
risk.

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Ch. 12: 9
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Interest Rate Swaps (continued)
The Pricing and Valuation of Interest Rate Swaps
(continued)
A basis swap is equivalent to the difference between
two plain vanilla swaps based on different rates:
• A swap to pay T-bill, receive fixed, plus
• A swap to pay fixed, receive LIBOR, equals
• A swap to pay T-bill, receive LIBOR, plus pay
the difference between the LIBOR and T-bill
fixed rates
• See Tables 12.4 and Table 12.5 for examples of
pricing and valuation of a basis swap.
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Interest Rate Swaps (continued)
Interest Rate Swap Strategies
See Figure 12.6 for example of converting floatingrate loan into fixed-rate loan
Other types of swaps
• Index amortizing swaps
• Diff swaps
• Constant maturity swaps
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Currency Swaps
Example: Reston Technology enters into currency
swap with GSI. Reston will pay euros at 4.35% based
on NP of €10 million semiannually for two years. GSI
will pay dollars at 6.1% based on NP of $9.804 million
semiannually for two years. Notional amounts will be
exchanged.
See Figure 12.7.
Note the relationship between interest rate and currency
swaps in Figure 12.8.
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Currency Swaps (continued)
Pricing and Valuation of Currency Swaps
Let dollar notional amount be NP$. Then euro notional amount
is NP€ = 1/S0 for every dollar notional amount. Here euro
notional amount will be €10 million. With S0 = $0.9804, NP$
= $9,804,000.
For fixed payments, we use the fixed rate on plain vanilla
swaps in that currency, R$ or R€.
No pricing is required for the floating side of a currency swap.
See Table 12.6.
During the life of the swap, we value it by finding the
difference in the present values of the two streams of
payments, adjusting for the notional amounts, and converting
to a common currency. Assume new exchange rate is $0.9790
three months later.
See Table 12.7 for calculations of values of streams of
payments per unit notional amount.
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Currency Swaps (continued)
Pricing and Valuation of Currency Swaps (continued)
Dollars fixed for NA of $9.804 million
= $9,804,000(1.01132335) = $9,915,014
Dollars floating for NA of $9.804 million
= $9,804,000(1.013115) = $9,932,579
Euros fixed for NA of €10 million
= €10,000,000(1.00883078) = €10,088,308
Euros floating for NA of €10 million
= €10,000,000(1.0091157) = €10,091,157
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Currency Swaps (continued)
Pricing and Valuation of Currency Swaps (continued)
Value of swap to pay € fixed, receive $ fixed
• $9,915,014 - €10,088,308($0.9790/€) = $38,560
Value of swap to pay € fixed, receive $ floating
• $9,932,579 - €10,088,308($0.9790/€) = $56,125
Value of swap to pay € floating, receive $ fixed
• $9,915,014 - €10,091,157($0.9790/€) = $35,771
Value of swap to pay € floating, receive $ floating
• $9,932,579 - €10,091,157($0.9790/€) = $53,336
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Currency Swaps (continued)
Currency Swap Strategies
A typical case is a firm borrowing in one currency
and wanting to borrow in another. See Figure 12.9
for Reston-GSI example. Reston could get a better
rate due to its familiarity to GSI and also due to
credit risk.
Also a currency swap be used to convert a stream of
foreign cash flows. This type of swap would
probably have no exchange of notional amounts.
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Equity Swaps
Characteristics
One party pays the return on an equity, the other
pays fixed, floating, or the return on another equity
Rate of return is paid, so payment can be negative
Payment is not determined until end of period
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Ch. 12: 17
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Equity Swaps (continued)
The Structure of a Typical Equity Swap
Cash flow to party paying stock and receiving fixed
Example: IVM enters into a swap with FNS to pay
S&P 500 Total Return and receive a fixed rate of
3.45%. The index starts at 2710.55. Payments
every 90 days for one year. Net payment will be


 90 
($25,000,000) .0345
  Return on stock index over settlement period 
 360 


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Ch. 12: 18
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Equity Swaps (continued)
The Structure of a Typical Equity Swap (continued)
The fixed payment will be
• $25,000,000(.0345)(90/360) = $215,625
See Table 12.8 for example of payments. The first
equity payment is
 2764.90 
$25,000,000
 1  $501,282
2710.55


So the first net payment is IVM pays $285,657.
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Ch. 12: 19
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Equity Swaps (continued)
The Structure of a Typical Equity Swap (continued)
If IVM had received floating, the payoff formula
would be
If the swap were structured so that IVM pays the
return on one stock index and receives the return on
another, the payoff formula would be
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Ch. 12: 20
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Equity Swaps (continued)
Pricing and Valuation of Equity Swaps
For a swap to pay fixed and receive equity, we replicate as
follows:
• Invest $1 in stock
• Issue $1 face value loan with interest at rate R. Pay
interest on each swap settlement date and repay amount at
swap termination date. Interest based on q = days/360.
• Example: Assume payments on days 180 and 360.
– On day 180, stock worth S180/S0. Sell stock and
withdraw S180/S0 - 1
– Owe interest of Rq
– Overall cash flow is S180/S0 – 1 – Rq, which is
equivalent to the first swap payment. $1 is left over.
Reinvest in the stock.
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Ch. 12: 21
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Equity Swaps (continued)
Pricing and Valuation of Equity Swaps (continued)
On day 360, stock is worth S360/S180.
Liquidate stock. Pay back loan of $1 and interest of
Rq.
Overall cash flow is S360/S180 – 1 – Rq, which is
equivalent to the second swap payment.
The value of the position is the value of the swap.
In general for n payments, the value at the start is
n
1  B0 (t n )  Rq  B0 (t i )
i 1
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Ch. 12: 22
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Equity Swaps (continued)
Pricing and Valuation of Equity Swaps (continued)
Setting the value to zero and solving for R gives




 1  1  B (t )
R    n 0 n 
 q  B (t ) 
 0 i 
 i 1

which is the same as the fixed rate on an interest rate
swap. See Table 12.9 for pricing the IVM swap.
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Ch. 12: 23
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Equity Swaps (continued)
Pricing and Valuation of Equity Swaps (continued)
To value the swap at time t during its life, consider the party
paying fixed and receiving equity.
To replicate the first payment, at time t
• Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at
rate R maturing at next payment date.
• At the next payment date (assume day 90), shares are
worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1
(equivalent to the equity payment on the swap), plus $1
left over, which is reinvested in the stock. Pay the loan
interest, Rq (which is equivalent to the fixed payment on
the swap).
• Do this for each payment on the swap.
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Ch. 12: 24
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Equity Swaps (continued)
Pricing and Valuation of Equity Swaps (continued)
The cost to do this strategy at time t is
n
 St 
   Bt (t n )  Rq  Bt (t i )
i 1
 S0 
This is the value of the swap. See Table 12.10 for an example
of the IVM swap.
To value the equity swap receiving floating and paying equity,
note the equivalence to
• A swap to pay equity and receive fixed, plus
• A swap to pay fixed and receive floating.
So we can use what we already know.
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Equity Swaps (continued)
Pricing and Valuation of Equity Swaps (continued)
Using the new discount factors, the value of the fixed
payments (plus hypothetical notional amount) is
• 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677)
+ 1(0.9677) = 1.00159884
The value of the floating payments (plus hypothetical notional
amount) is
• (1 + 0.03(90/360))(0.9971) = 1.00457825
The plain vanilla swap value is, thus,
• 1.00457825 – 1.00159884 = 0.00297941
For a $25 million notional amount,
• $25,000,000(0.00297941) = $74,485
So the value of the equity swap is (using -$227,964, the value
of the equity swap to pay fixed)
• -$227,964 + $74,485 = -$153,479
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Equity Swaps (continued)
Pricing and Valuation of Equity Swaps (continued)
For swaps to pay one equity and receive another,
replicate by selling short one stock and buy the
other. Each period withdraw the cash return,
reinvesting $1. Cover short position by buying it
back, and then sell short $1. So each period start
with $1 long one stock and $1 short the other.
For the IVM swap, suppose we pay the S&P and
receive NASDAQ, which starts at 2710.55 and goes
to 2739.60. The value of the swap is
 1915.71   2739.60 
  0.03312974


 1835.24   2710.55 
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Equity Swaps (continued)
Pricing and Valuation of Equity Swaps (continued)
For $25 million notional amount, the value is
• $25,000,000(0.03312974) = $828,244
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Equity Swaps (continued)
Equity Swap Strategies
Used to synthetically buy or sell stock
See Figure 12.10 for example.
Some risks
• default
• tracking error
• cash flow shortages
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Some Final Words About Swaps
Similarities to forwards and futures
Offsetting swaps
Go back to dealer
Offset with another counterparty
Forward contract or option on the swap
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Summary
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Ch. 12: 42
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Ch. 12: 51
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Chapter 13: Interest Rate Forwards
and Options
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As with a second-hand car, you never really know what an
OTC option is worth until you actually sell it or buy it.
Placing a value on it in the interim is, in some ways, only a
more sophisticated version of pinning the tail on the
donkey. .
Richard Thomson
Apocalypse Roulette, 1998, p. 149
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Ch. 13: 1
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Important Concepts in Chapter 13
The notion of a derivative on an interest rate
Pricing, valuation, and use of forward rate agreements
(FRAs), interest rate options, swaptions, and forward
swaps
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Ch. 13: 2
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A derivative on an interest rate:
The payoff of a derivative on a bond is based on the
price of the bond relative to a fixed price.
The payoff of a derivative on an interest rate is based
on the interest rate relative to a fixed interest rate.
In some cases these can be shown to be the same,
particularly in the case of a discount instrument. In
most other cases, however, a derivative on an interest
rate is a different instrument than a different on a bond.
See Figure 13.1 for notional amount of FRAs and interest
rate options over time.
See Figure 13.2 for gross market value of FRAs and
interest rate options over time.
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Ch. 13: 3
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Forward Rate Agreements
Definition
A forward contract in which the underlying is an
interest rate
An FRA can work better than a forward or futures on a
bond, because its payoff is tied directly to the source of
risk, the interest rate.
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Ch. 13: 4
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Forward Rate Agreements (continued)
The Structure and Use of a Typical FRA
Underlying is usually LIBOR
Payoff is made at expiration (contrast with swaps)
and discounted. For FRA on m-day LIBOR, the
payoff is
Example: Long an FRA on 90-day LIBOR expiring
in 30 days. Notional amount of $20 million.
Agreed upon rate is 5 percent. Payoff will be
 (LIBOR - 0.05)(90/3 60) 

($20,000,0 00) 
 1  LIBOR(90/360) 
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Ch. 13: 5
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Forward Rate Agreements (continued)
Some possible payoffs. If LIBOR at expiration is 4
percent,
 (0.04 - 0.05)(90/3 60) 
  $49,505
($20,000,0 00) 
 1  0.04(90/360) 
So the long has to pay $49,505. If LIBOR at
expiration is 6 percent, the payoff is
 (0.06 - 0.05)(90/3 60) 
  $49,261
($20,000,0 00) 
 1  0.06(90/360) 
Note the terminology of FRAs: A  B means FRA
expires in A months and underlying matures in B
months.
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Ch. 13: 6
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Forward Rate Agreements (continued)
The Pricing and Valuation of FRAs
Let F be the rate the parties agree on, h be the
expiration day, and the underlying be an m-day rate.
L0(h) is spot rate on day 0 for h days, L0(h+m) is
spot rate on day 0 for h + m days. Assume notional
amount of $1.
To find the fixed rate, we must replicate an FRA:
• Short a Eurodollar maturing in h+m days that
pays 1 + F(m/360). This is a loan that can be
paid off early or transferred to another party
• Long a Eurodollar maturing in h days that pays
$1
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Ch. 13: 7
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Forward Rate Agreements (continued)
The Pricing and Valuation of FRAs (continued)
On day h,
• Loan we owe has a market value of
1  F(m/360)

1  L h (m)(m/360)
• Pay if off early. Collect $1 on the ED we hold.
So total cash flow is
1  F(m/360)
1
1  L h (m)(m/360)
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Ch. 13: 8
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Forward Rate Agreements (continued)
The Pricing and Valuation of FRAs (continued)
• This can be rearranged to get
(L h (m)  F)(m/360)
1  L h (m)(m/360)
This is the payoff of an FRA so this strategy is
equivalent to an FRA. With no initial cash flow, we
set this to zero and solve for F:
 1  L0 (h  m)((h  m)/360)  360 
F  
 1

1  L0 (h/360)

 m 
This is just the forward rate in the LIBOR term
structure. See Table 13.1 for an example.
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Ch. 13: 9
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Forward Rate Agreements (continued)
The Pricing and Valuation of FRAs (continued)
Now we determine the market value of the FRA
during its life, day g. If we value the two replicating
transactions, we get the value of the FRA. The ED
we hold pays $1 in h – g days. For the ED loan we
took out, we will pay 1 + F(m/360) in h + m – g
days. Thus, the value is

 

1
1  F(m/360)




VFRA 

 1  L (h  g)((h  g)/360)   1  L (h  m  g)((h  m  g)/360) 
g
g

 

See Table 13.2 for example.
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Ch. 13: 10
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Forward Rate Agreements (continued)
Applications of FRAs
FRA users are typically borrowers or lenders with a
single future date on which they are exposed to
interest rate risk.
See Table 13.3 and Figure 13.3 for an example.
Note that a series of FRAs is similar to a swap;
however, in a swap all payments are at the same
rate. Each FRA in a series would be priced at
different rates (unless the term structure is flat).
You could, however, set the fixed rate at a different
rate (called an off-market FRA). Then a swap
would be a series of off-market FRAs.
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Ch. 13: 11
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Interest Rate Options
Definition: an option in which the underlying is an
interest rate; it provides the right to make a fixed
interest payment and receive a floating interest payment
or the right to make a floating interest payment and
receive a fixed interest payment.
The fixed rate is called the exercise rate.
Most are European-style.
Chance/Brooks
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Ch. 13: 12
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Interest Rate Options (continued)
The Structure and Use of a Typical Interest Rate Option
With an exercise rate of X, the payoff of an interest
rate call is
The payoff of an interest rate put is
The payoff occurs m days after expiration.
Example: notional amount of $20 million,
expiration in 30 days, underlying of 90-day LIBOR,
exercise rate of 5 percent.
Chance/Brooks
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Ch. 13: 13
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Interest Rate Options (continued)
The Structure and Use of a Typical Interest Rate Option
(continued)
If LIBOR is 1 percent at expiration, payoff of a call is
($20,000,0 00) Max(0,0.01  0.05)(90/360)   $0
The payoff of a put is
($20,000,0 00) Max(0,0.05  0.01)(90/360)   $200,000
If LIBOR is 9 percent at expiration, payoff of a call is
($20,000,0 00) Max(0,0.09  0.05)(90/360)   $200,000
The payoff of a put is
($20,000,0 00) Max(0,0.05  0.09)(90/360)   $0
Chance/Brooks
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Ch. 13: 14
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Interest Rate Options (continued)
Pricing and Valuation of Interest Rate Options
A difficult task; binomial models are preferred, but the
Black model is sometimes used with the forward rate
as the underlying.
When the result is obtained from the Black model, you
must discount at the forward rate over m days to reflect
the deferred payoff.
Then to convert to the premium, multiply by (notional
amount)(days/360).
See Table 13.4 for illustration.
Chance/Brooks
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Ch. 13: 15
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Interest Rate Options (continued)
Interest Rate Option Strategies
See Table 13.5 and Figure 13.4 for an example of the
use of an interest rate call by a borrower to hedge an
anticipated loan.
See Table 13.6 and Figure 13.5 for an example of the
use of an interest rate put by a lender to hedge an
anticipated loan.
Chance/Brooks
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Ch. 13: 16
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Interest Rate Options (continued)
Interest Rate Caps, Floors, and Collars
A combination of interest rate calls used by a borrower
to hedge a floating-rate loan is called an interest rate
cap. The component calls are referred to as caplets.
A combination of interest rate puts used by a lender to
hedge a floating-rate loan is called an interest rate
floor. The component puts are referred to as floorlets.
A combination of a long cap and short floor at
different exercise prices is called an interest rate collar.
Chance/Brooks
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Ch. 13: 17
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Interest Rate Options (continued)
Interest Rate Caps, Floors, and Collars (continued)
Interest Rate Cap
• Each component caplet pays off independently of
the others.
• See Table 13.7 for an example of a borrower using
an interest rate cap.
• To price caps, price each component caplet
individually and add up the prices of the caplets.
Chance/Brooks
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Ch. 13: 18
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Interest Rate Options (continued)
Interest Rate Caps, Floors, and Collars (continued)
Interest Rate Floor
• Each component floorlet pays off independently of
the others
• See Table 13.8 for an example of a lender using an
interest rate floor.
• To price floors, price each component floorlet
individually and add up the prices of the floorlets.
Chance/Brooks
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Ch. 13: 19
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Interest Rate Options (continued)
Interest Rate Caps, Floors, and Collars (continued)
Interest Rate Collars
• A borrower using a long cap can combine it with a
short floor so that the floor premium offsets the cap
premium. If the floor premium precisely equals the
cap premium, there is no cash cost up front. This is
called a zero-cost collar.
• The exercise rate on the floor is set so that the
premium on the floor offsets the premium on the cap.
• By selling the floor, however, the borrower gives up
gains from falling interest rates below the floor
exercise rate.
• See Table 13.9 for example.
Chance/Brooks
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Ch. 13: 20
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Interest Rate Options (continued)
Interest Rate Options, FRAs, and Swaps
Recall that a swap is like a series of off-market FRAs.
Now compare a swap to interest rate options. On a
settlement date, the payoff of a long call is
•0
if LIBOR  X
• LIBOR – X
if LIBOR > X
The payoff of a short put is
• – (X – LIBOR) if LIBOR  X
•0
if LIBOR > X
These combine to equal LIBOR – X. If X is set at R,
which is the swap fixed rate, the long cap and short floor
replicate the swap.
Chance/Brooks
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Ch. 13: 21
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Interest Rate Swaptions and Forward Swaps
Definition of a swaption: an option to enter into a swap at a
fixed rate.
Payer swaption: an option to enter into a swap as a fixedrate payer
Receiver swaption: an option to enter into a swap as a
fixed-rate receiver
Chance/Brooks
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Ch. 13: 22
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The Structure of a Typical Interest Rate Swaption
Example: MPK considers the need to engage in a $10
million three-year swap in two years. Worried about
rising rates, it buys a payer swaption at an exercise rate of
11.5 percent. Swap payments will be annual.
• At expiration, the following rates occur (Eurodollar
zero coupon bond prices in parentheses):
– 360 day rate: 0.12 (0.8929)
– 720 day rate: 0.1328 (0.7901)
– 1080 day rate: 0.1451 (0.6967)
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Interest Rate Swaptions and Forward Swaps
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Ch. 13: 23
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The Structure of a Typical Interest Rate Swaption
(continued)
• The rate on 3-year swaps is, therefore,
1  0.6967

 360 
R 

  0.1275
 0.8929  0.7901  0.6967  360 
• So MPK could enter into a swap at 12.75 percent in
the market or exercise the swaption and enter into a
swap at 11.5 percent. Obviously it would exercise
the swaption. What is the swaption worth?
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Interest Rate Swaptions and Forward Swaps
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Ch. 13: 24
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The Structure of a Typical Interest Rate Swaption
(continued)
• Exercise would create a stream of 11.5 percent fixed
payments and LIBOR floating receipts. MPK could
then enter into the opposite swap in the market to
receive 12.75 fixed and pay LIBOR floating. The
LIBORs offset leaving a three-year annuity of
12.75 – 11.5 = 1.25 percent, or $125,000 on $10
million notional amount. The value of this stream of
payments is
$125,000(0.8929 + 0.7901 + 0.6967) = $297,463
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Interest Rate Swaptions and Forward Swaps
(continued)
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Ch. 13: 25
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The Structure of a Typical Interest Rate Swaption
(continued)
In general, the value of a payer swaption at expiration is
The value of a receiver swaption at expiration is
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Interest Rate Swaptions and Forward Swaps
(continued)
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Ch. 13: 26
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The Equivalence of Swaptions and Options on Bonds
Using the above example, substituting the formula for the
swap rate in the market, R, into the formula for the
payoff of a swaption gives
• Max(0,1 – 0.6967 – 0.115(0.8929+0.7901+0.6967))
This is the formula for the payoff of a put option on a
bond with 11.5 percent coupon where the option has an
exercise price of par. So payer swaptions are equivalent
to puts on bonds. Similarly, receiver swaptions are
equivalent to calls on bonds.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Interest Rate Swaptions and Forward Swaps
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Ch. 13: 27
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Swaption and Callable Bonds
One application of swaptions relates to callable bonds
Recall callable bond issuer has sold (issued) bonds and
purchased a call option
A receiver swaption is comparable to the embedded
call option of a bond
See Figure 13.6
Chance/Brooks
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Ch. 13: 28
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Pricing Swaptions
We do not cover this advanced topic here, but note that
based on the previous result, we would price swaptions
using models for pricing options on bonds.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Interest Rate Swaptions and Forward Swaps
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Ch. 13: 29
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Forward Swaps
Definition: a forward contract to enter into a swap; a
forward swap commits the parties to entering into a swap
at a later date at a rate agreed on today.
Example: The MPK situation previously described. Let
MPK commit to a three-year pay-fixed, receive-floating
swap in two years. To find the fixed rate at the time the
forward swap is agreed to, we need the term structure of
rates for one through five years (Eurodollar zero coupon
bond prices shown in parentheses).
Chance/Brooks
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Interest Rate Swaptions and Forward Swaps
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Ch. 13: 30
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Forward Swaps (continued)
• 360 days:
0.09 (0.9174)
• 720 days:
0.1006 (0.8325)
• 1080 days:
0.1103 (0.7514)
• 1440 days:
0.12 (0.6757)
• 1800 days:
0.1295 (0.6070)
We need the forward rates two years ahead for periods of
one, two, and three years.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Interest Rate Swaptions and Forward Swaps
(continued)
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Ch. 13: 31
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Forward Swaps (continued)
 1  0.1103(1080 /360)  360 
One year  
 1
  0.1080
 360 
 1  0.1006(720/ 360)
 1  0.12(1440/3 60)
 360 
 1
Two years  
  0.1161
 1  0.1006(720/ 360)  720 
 1  0.1295(1800 /360)  360 
 1
Three years  
  0.1238
 1  0.1006(720/ 360)
 1080 
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Interest Rate Swaptions and Forward Swaps
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Ch. 13: 32
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Forward Swaps (continued)
The Eurodollar zero coupon (forward) bond prices
1
 0.9025
B0 (720,1080 ) 
1  0.1080 (360 / 360 )
1
 0.8116
B0 (720,1440 ) 
1  0.1161(720 / 360 )
1
 0.7292
B0 (720,1800 ) 
1  0.1238 (1080 / 360 )
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Forward Swaps (continued)
The rate on the forward swap would be
1 - 0.7292
 0.1108
0.9025  0.8116  0.7292
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An Introduction to Derivatives and Risk Management, 9th ed.
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Ch. 13: 34
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Applications of Swaptions and Forward Swaps
Anticipation of the need for a swap in the future
Swaption can be used
• To exit a swap
• As a substitute for an option on a bond
• Creating synthetic callable or puttable debt
Remember that forward swaps commit the parties to a
swap but require no cash payment up front. Options give
one party the choice of entering into a swap but require
payment of a premium up front.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
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Summary
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Ch. 13: 36
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Ch. 13: 37
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Ch. 13: 38
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Ch. 13: 39
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Ch. 13: 40
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Ch. 13: 41
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Ch. 13: 42
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Ch. 13: 43
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An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 13: 44
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Ch. 13: 45
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An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 13: 46
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Ch. 13: 47
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Ch. 13: 48
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Ch. 13: 51
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Chapter 15: Financial Risk Management:
Techniques and Applications
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Risk managers need to be perceived like good goalkeepers,
always in the game and occasional at the heart of it, like in
a penalty shoot-out. .
Anonymous
"Confessions of a Risk Manager,” The Economist, 8/7/08
Chance/Brooks
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Ch. 15: 1
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Important Concepts in Chapter 15
The concept and practice of risk management
The benefits of risk management
The difference between market and credit risk
How market risk is managed using delta, gamma, vega,
and Value-at-Risk
How credit risk is managed, including credit derivatives
Risks other than market and credit risk
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Ch. 15: 2
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Definition of risk management: the practice of defining the
risk level a firm desires, identifying the risk level it
currently has, and using derivatives or other financial
instruments to adjust the actual risk level to the desired
risk level.
Chance/Brooks
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Ch. 15: 3
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Why Practice Risk Management?
The Impetus for Risk Management
Firms practice risk management for several reasons:
Interest rates, exchange rates and stock prices are
more volatile today than in the past.
Significant losses incurred by firms that did not
practice risk management
Improvements in information technology
Favorable regulatory environment
Sometimes we call this activity financial risk
management.
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Ch. 15: 4
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The Benefits of Risk Management
What are the benefits of risk management, in light of
the Modigliani-Miller principle that corporate financial
decisions provide no value because shareholders can
execute these transactions themselves?
Firms can practice risk management more
effectively.
There may tax advantages from the progressive tax
system.
Risk management reduces bankruptcy costs.
Managers are trying to reduce their own risk.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 15: 5
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Why Practice Risk Management? (continued)
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The Benefits of Risk Management (continued)
By protecting a firm’s cash flow, it increases the
likelihood that the firm will generate enough cash to
allow it to engage in profitable investments.
Some firms use risk management as an excuse to
speculate.
Some firms believe that there are arbitrage
opportunities in the financial markets.
Note: The desire to lower risk is not a sufficient reason
to practice risk management.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 15: 6
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Managing Market Risk
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Market risk: the uncertainty associated with interest rates,
foreign exchange rates, stock prices, or commodity prices.
Example: A dealer with the following positions:
A four-year interest rate swap with $10 million
notional principal in which it pays a fixed rate and
receives a floating rate.
A 3-year interest rate call with $8 million notional
principal. The dealer is short and the exercise rate is
12%.
See Table 15.1 for current term structure and forward
rates. We obtain the call price as $73,745 and the swap
rate is 11.85%.
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Managing Market Risk (continued)
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Delta Hedging
We estimate the delta by repricing the swap and option
with a one basis point move in all spot rates and
average the price change.
See Table 15.2 for estimated swap and option deltas.
• We are long the swap so we have a delta of
$2,130.5, round to $2,131.
• We are short the option so we have a delta of
-$244.
• Our overall delta is $1,887.
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Ch. 15: 8
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Managing Market Risk (continued)
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Delta Hedging (continued)
We need a Eurodollar futures position that gains $1,887
if rates move down and loses that amount if rates move
up. Thus, we require a long position of
$1,887/$25 = 75.48 contracts. Round to 75. Overall
delta:
$2,131 (from swap)
-$244 (from option)
75(-$25) (from futures)
= $12 (overall)
Chance/Brooks
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Ch. 15: 9
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Managing Market Risk (continued)
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Gamma Hedging
Here we deal with the risk of large price moves, which
are not fully captured by the delta.
See Table 15.3 for the estimation of swap and option
gammas. Swap gamma is -$12,500, and option gamma
is $5,000. Being short the option, the total gamma is $17,500.
Eurodollar futures have zero gamma so we must add
another option position to offset the gamma. We
assume the availability of a one-year call with delta of
$43 and gamma of $2,500.
Chance/Brooks
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Ch. 15: 10
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Managing Market Risk (continued)
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Gamma Hedging (continued)
We use x1 Eurodollar futures and x2 of the one-year
calls. The swap and option have a delta of $1,887 and
gamma of -$17,500. We solve the following equations:
$1,887 + x1(-$25) + x2($43) = $0 (zero delta)
-$17,500 + x1($0) + x2($2,500) = $0 (zero gamma)
Solving these gives x1 = 87.52 (go long 88
Eurodollar futures) and x2 = 7 (go long 7 times
$1,000,000 notional principal of one-year option)
Chance/Brooks
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Ch. 15: 11
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Managing Market Risk (continued)
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Vega Hedging
Swaps, futures, and FRAs do not have vegas.
We estimate the vegas of the options
On our 3-year option, if volatility increases
(decreases) by .01, option will increase (decrease)
by $42 (-$42). Average is $42. We are short this
option, so vega = -$42.
One-year option has estimated vega of $3.50.
Overall portfolio has vega of
($3.50)(7 million) - $42 = -$17.50.
Chance/Brooks
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Ch. 15: 12
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Vega Hedging (continued)
We add a Eurodollar futures option, which has delta of
-$12.75, gamma of -$500, and vega of $2.50 per
$1MM.
Solve the following equations
$1,887 + x1(-$25) + x2($43) + x3(-$12.75) = 0
(delta)
-$17,500 + x1($0) + x2($2,500) + x3(-$500) = 0
(gamma)
-$42 + x1($0) + x2($3.50) + x3($2.50) = 0 (vega)
The coefficients are the multiples of $1,000,000
notional principal we need.
Solutions are x1 = 86.61, x2 = 8.09375, x3 = 5.46875.
Chance/Brooks
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Ch. 15: 13
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Managing Market Risk (continued)
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Vega Hedging (continued)
Any type of hedge (delta, delta-gamma, or deltagamma-vega) must be periodically adjusted.
Virtually impossible to have a perfect hedge.
Chance/Brooks
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Ch. 15: 14
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Value-at-Risk (VAR)
A dollar measure of the minimum loss that would be
expected over a given time with a given probability.
Example:
VAR of $1 million for one day at 0.05 means that
the firm could expect to lose at least $1 million over
a one day period 5% of the time.
Widely used by dealers and increasingly by end users.
See Table 15.4 for example of discrete probability
distribution of change in value.
VAR at 5% is $3 million loss.
See Figure 15.1 for continuous distribution.
Chance/Brooks
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Ch. 15: 15
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Value-at-Risk (VAR) (continued)
VAR calculations require use of formulas for expected
return and standard deviation of a portfolio:
E(R p )  w1E(R 1 )  w 2 E(R 2 )
σ p  w1 σ12  w 2 σ 22  2w1w 2σ1σ 2ρ
2
2
where
E(R1), E(R2) = expected returns of assets 1 and 2
1, 2 = standard deviations of assets 1 and 2
 = correlation between assets 1 and 2
w1, w2 = % of one’s wealth invested in asset 1 or 2
Chance/Brooks
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Ch. 15: 16
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Managing Market Risk (continued)
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Value-at-Risk (VAR) (continued)
Three methods of estimating VAR
Analytical method: Uses knowledge of the parameters
(expected return and standard deviation) of the probability
distribution and assumes a normal distribution.
• Example: $20 million of S&P 500 with expected return of
0.12 and volatility of 0.15 and $12 million of Nikkei 300
with expected return of 0.105 and volatility of 0.18.
Correlation is 0.55. Using the above formulas, the overall
portfolio expected return is 0.1144 and volatility is 0.1425.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 15: 17
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Value-at-Risk (VAR) (continued)
For a weekly VAR, convert these to weekly figures.
Expected return = 0.1144/52 = 0.0022
Volatility = 0.1425/52 = 0.0198.
With a normal distribution, we have
VAR = 0.0022 - 1.65(0.0198) = -0.0305
So the VAR is $32,000,000(0.0305) = $976,000.
Chance/Brooks
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Ch. 15: 18
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Value-at-Risk (VAR) (continued)
Example using options: 200 short 12-month calls on S&P 500,
which has volatility of 0.15 and price of $14.21. Total value of
$1,421,000.
Based on monthly data, expected return is 0.0095 and volatility
is 0.0412.
Upside 5 % is 0.0095 + 1.65(0.0412) = 0.0775, which is
720(1.0775) = 775.80.
Option would be worth 775.80 - 720 = 55.80 so loss is
55.80 - 14.21= 41.59 per option.
Total loss = 200(500)(41.59) = $4.159 million. This is the
VAR.
Chance/Brooks
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Ch. 15: 19
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Value-at-Risk (VAR) (continued)
One assumption often made is that the expected return
is zero. This is not likely to be true.
Sometimes rather than use the precise option price from
a model, a delta is used to estimate the price. This
makes the analytical method be sometimes called the
delta-normal method.
Volatility and correlation information is necessary. See
the web site www.riskmetrics.com, where data of this
sort are provided free.
Chance/Brooks
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Ch. 15: 20
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Value-at-Risk (VAR) (continued)
 Historical method: Uses historical information on the
user’s portfolio to obtain the distribution.
Example: See Figure 15.2. For portfolio of $15
million, VAR at 5% is approximately a loss of 10%
or $15,000,000(0.10) = -$1,500,000.
Historical method is subject to limitation that the
past holdings of the portfolio may not have the same
distributional properties as the future holdings. It
also is limited by the results of the chosen time
period, which might not be representative of the
future.
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Ch. 15: 21
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Value-at-Risk (VAR) (continued)
 Monte Carlo Simulation method: Uses Monte Carlo
method, as described in Appendix 14, to generate
random outcomes on the portfolio’s components.
Chance/Brooks
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Ch. 15: 22
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Managing Market Risk (continued)
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Value-at-Risk (VAR) (continued)
A Comprehensive Calculation of VAR
We do an example of a portfolio of $25 million in
the S&P 500. We want a 5% 1-day VAR using each
method. We collect a sample of daily returns on the
S&P 500 for the past year and obtain the following
parameter estimates: Average daily return =
0.0457% and daily standard deviation = 1.3327%.
These result in annual figures of
0.0457 (253)  0.1156
1.3327 253  0.2120
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 15: 23
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Value-at-Risk (VAR) (continued)
A Comprehensive Calculation of VAR (continued)
Analytical method: We have 0.0457% − (1.65)1.3327%
= −2.1533%. So the VAR is
• 0.021533($25,000,000) = $538,325
• The 0.21 standard deviation is historically a bit high. Reestimating with a standard deviation of 0.15 gives us a
daily standard deviation of 0.9430. Then we obtain
0.0474% − 1.65(0.9430) = −1.5086% and a VAR of
• 0.015086($25,000,000) = $377,150
• Are our data normally distributed? Observe Figure 15.3.
Chance/Brooks
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Ch. 15: 24
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Value-at-Risk (VAR) (continued)
A Comprehensive Calculation of VAR (continued)
Historical method: Here we rank the returns from
worst to best. For 253 returns we obtain the 5%
worst by observing the 0.05(253) = 12.65 worst
return. We shall make it the 13th worst. This would
be −2.0969%. Thus, the VAR is
• 0.020969($25,000,000) = $524,225
Chance/Brooks
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Ch. 15: 25
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Value-at-Risk (VAR) (continued)
A Comprehensive Calculation of VAR (con