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Investment Analysis and Portfolio
Management
14
First Canadian Edition
By Reilly, Brown, Hedges, Chang
Chapter 14
Derivatives: Analysis and Valuation
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An Overview of Forward and Futures Trading
Hedging with Forwards and Futures
Valuation of Forward and Futures
Financial Futures
An Overview of Option Markets and Contracts
The Fundamental Option Valuation
Swaps
Option-Like Securities
Copyright © 2010 by Nelson Education Ltd.
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An Overview of
Forward & Futures Trading
• Forward contracts are negotiated directly between
two parties in the OTC markets
• Individually designed to meet specific needs
• Subject to default risk
• Futures contracts are bought through brokers on an
exchange
• No direct interaction between the two parties
• Exchange clearinghouse oversees delivery and settles daily
gains and losses
• Customers post initial margin account
Copyright © 2010 by Nelson Education Ltd.
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An Overview of
Forward & Futures Trading
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14-4
An Overview of
Forward & Futures Trading
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14-5
An Overview of
Forward & Futures Trading
• Futures Contract Mechanics
• Futures exchange requires each customer to post an initial
margin account in the form of cash or government
securities when the contract is originated
• The margin account is marked to market at the end of each
trading day according to that day’s price movements
• Forward contracts may not require either counterparty to
post collateral
• All outstanding contract positions are adjusted to the
settlement price set by the exchange after trading ends
Copyright © 2010 by Nelson Education Ltd.
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An Overview of
Forward & Futures Trading
• With commodity futures, it usually is the case that
delivery can take place any time during the month
at the discretion of the short position
Futures
Forwards
Design Flexibility
Standardized
Can be customized
Credit Risk
Clearinghouse risk
Counterparty risk
Liquidity Risk
Depends on trading
Negotiated risk
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Hedging with Forwards & Futures
• Create a position that will offset the price risk
of another more fundamental holding
• Short hedge: Holding a short forward position
against the long position in the commodity
• Long hedge: Supplements a short commodity
holding with a long forward position
• The basic premise behind any hedge is that
as the price of the underlying commodity
changes, so too will the price of a forward
contract based on that commodity
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14-8
Hedging with Forwards & Futures:
Defining the Basis
• Basis is spot price minus the forward price for a
contract maturing at date T:
Bt,T = St - Ft,T
where
St the Date t spot price
Ft,T the Date t forward price for a contract maturing at Date T
• Initial basis, B0,T, is always known
• Maturity basis, BT,T, is always zero. That is, forward and spot
prices converge as the contract expires
• Cover basis: Bt, T
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Hedging with Forwards & Futures:
Understanding Basis Risk
• The terminal value of the combined position is
defined as the cover basis minus the initial basis
Bt, T – B0, T = (St - Ft,T ) - (S0 – F0,T )
• Basis Risk
• Investor’s terminal value is directly related to Bt, T
• Bt, T depends on the future spot and forward prices
• If St and Ft,T are not correlated perfectly, Bt, T will change
and cause the basis risk
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Hedging with Forwards & Futures:
Understanding Basis Risk
• Hedging Exposure
•It is to the correlation between future
changes in the spot and forward
contract prices
•Perfect correlation with customized
contracts
Copyright © 2010 by Nelson Education Ltd.
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Valuation of Forwards & Futures:
The Cost to Carry Model
• The Cost of Carry
• Commissions paid for storing the commodity,
PC0,T
• Cost of financing the initial purchase, i0,T
• Cash flows received between Dates 0 and T, D0,T
F 0 ,T = S 0 + SC 0 ,T = S 0 + (PC 0 ,T + i 0 ,T - D 0 ,T
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Valuation of Forwards & Futures:
Contango & Backwardated
• Contango Market
• When F0,T > S0
• Normally with high storage costs and no dividends
• Backwardation Market
• When F0,T < S0
• Normally with no storage costs and pays dividends
• Premium for owning the commodity
• Convenience yield
• Can results from small supply at date 0 relative to what
is expected at date T (after the crop harvest)
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14-13
Financial Futures
• Interest Rate Futures
• Interest rate forwards and futures were among
the first derivatives to specify a financial security
as the underlying asset
• Forward rate agreements
• Interest rate swaps
• Basic Types
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Long-term interest rate futures
Short-term interest rate futures
Stock index futures
Currency forwards and futures
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Financial Futures
• Interest rate futures available at the
Montreal Exchange
• BAX (Three-Month Canadian Bankers’ Acceptance
Futures
• OBX (Options on Three-Month Canadian Bankers’
Acceptance Futures)
• CGB (Ten-Year Government of Canada Bond
Future)
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Stock Index Futures
• Intended to provide general hedges against stock
market movements and can be applied to a portfolio
or individual stocks
• Hedging an individual stock with an index isolates
the unsystematic portion of that security’s risk
• Can only be settled in cash, similar to the Eurodollar
(i.e., LIBOR) contract
• Stock Index Arbitrage:
• Use the stock index futures to convert a stock portfolio
into synthetic riskless positions
• Prominent in program trading
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Overview of
Options Markets & Contracts
• Option Market Conventions
• Option contracts have been traded for centuries
• Customized options traded on OTC market
• In April 1973, standardized options began trading
on the Chicago Board Option Exchange
• Contracts offered by the CBOE are standardized
in terms of the underlying common stock, the
number of shares covered, the delivery dates,
and the range of available exercise prices
• Options Clearing Corporation (OCC) acts as
guarantor of each CBOE-traded options
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Overview of
Options Markets & Contracts
• Price Quotations for Exchange-Traded Options
• Equity Options
• CBOE, AMEX, PHLX, PSE
• Typical contract for 100 shares
• Require secondary transaction if exercised
• Time premium affects pricing
• Stock Index Options
• First traded on the CBOE in 1983
• Index options can only be settled in cash
• Index puts are particularly useful in portfolio insurance
applications
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Overview of
Options Markets & Contracts
• Options on Futures Contracts
• Options on futures contracts have only been
exchange-traded since 1982
• Give the right, but not the obligation, to enter
into a futures contract on an underlying security
or commodity at a later date at a predetermined
price
• Leverage is the primary attraction of this
derivative
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Fundamentals of Option Valuation
• Risk reduction tools when used as a hedge
• Forecasting the volatility of future asset prices
• direction and magnitude
• Hedge ratio is based on the range of possible option
outcomes related to the range of possible stock
outcomes
• Risk-free hedge buys one share of stock and sells
call options to neutralize risk
• Hedge portfolio should grow at the risk-free rate
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Fundamentals of Option Valuation
• The Basic Approach
• Assume the WYZ stock price as the following
• Assume the risk-free rate is 8%
• Want the price of a call option (C0) with X =
$52.50
Price in one year
Stock price now
$65
$50
$40
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Fundamentals of Option Valuation
• Step 1: Estimate the number of call options
• Calculate the option’s payoffs for each possible
future stock price
• If stock goes to $65, option pays off $12.50
• If stock goes to $40, option pays off $0
• Determine the composition of the hedge portfolio
• It contains one share of stock and “h” call options
• If stock goes up, portfolio will pay:
$65 + (h)($12.50)
• If stock goes down, portfolio will pay:
$40 + (h)($0)
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Fundamentals of Option Valuation
• Step 1 (Continued)
• To determine the composition of the hedge
portfolio, find the number of options that equates
the payoffs
$65 + $12.50h = $40 + $0h
• Implies h = -2
• Hedge portfolio is long one share of stock and short two
call options
• Value of hedge portfolio today:
$50 - 2.00(C0)
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Fundamentals of Option Valuation
• Step 2: Determine the PV of the portfolio
• We know the hedge portfolio will pay $40 in one
year with certainty
• Thus the value of that portfolio right now is
40/(1+RFR)T
• Step 3: Compute the price of a call option
• Condition of no risk-free arbitrage
$50 - 2.00(C0) = 40/(1+RFR)T
• When T=1 and RFR=8%, solve for C0
C0=$6.48
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14-24
Fundamentals of Option Valuation
• The Binomial Option Pricing Formula
• In the jth state in any sub-period, the value
of the option can be calculated by
Cj =
where
(p )C ju + (1 - p )C jd
r
r-d
p=
u-d
and
r = one plus the risk-free rate over the sub-period
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Fundamentals of Option Valuation
• At Date 0, the binomial option pricing formula can
be expressed as follows:
N
 n
N!
N- j
j
j N- j
(u d ) S - X   r
Co =  
p (1 - p)
 j = 0 (N - j )!j!

[
]
m is the smallest integer number of up moves guaranteeing
that the option will be in the money at expiration
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Fundamentals of Option Valuation
• The hedge ratio for any state j becomes
hj =
(u - d )S j
( C jd
- C ju
)
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14-27
The Black-Scholes Valuation Model
• Binomial model is discrete method for valuing
options because it allows security price
changes to occur in distinct upward or
downward movements
• Prices can change continuously throughout
time
• Advantage of Black-Scholes approach is
relatively simple, closed-form equation
capable of valuing options accurately under a
wide array of circumstances
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14-28
The Black-Scholes Valuation Model
Assuming the continuously compounded risk-free
rate and the stock’s variance (i.e., 2) remain
constant until the expiration date T, Black and
Scholes used the riskless hedge intuition to
derive the following formula for valuing a call
option on a non-dividend-paying stock:
C0 = SN(d1) – X(e-(RFR)T)N(d2)
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The Black-Scholes Valuation Model
where:
C0 = market value of call option
S = current market price of underlying stock
X = exercise price of call option
e-(RFR)T = discount function for continuously compounded
variables
N(d1) = cumulative density function of d1 defined as
](
[
12
d 1 = (ln (S X )+(RFR + 0.5s 2 )T )  s [T]
)
N(d2) = cumulative density function of d2 defined as
12
=
s
d 2 d1
[T]
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The Black-Scholes Valuation Model
• Properties of the Model
• Option’s value is a function of five variables
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•
•
•
•
Current security price
Exercise price
Time to expiration
Risk-free rate
Security price volatility
• Functionally, the Black-Scholes model holds that
C = f (S, X, T, RFR, σ)
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The Black-Scholes Valuation Model
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The Black-Scholes Valuation Model
• Estimating Volatility
• The standard deviation of returns to the
underlying asset can be estimated in two ways
1. Traditional mean and standard deviation of a
series of price relatives
2. Estimate implied volatility from Black-Scholes
formula
• If we know the current price of the option (call it C*)
and the four other variables, we can calculate the level
of σ
• No simple closed-form solution exists for performing this
calculation; it must be done by trial and error
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The Black-Scholes Valuation Model
• Problems With Black-Scholes Valuation
• Stock prices do not change continuously
• Arbitrageable differences between option values and prices
(due to brokerage fees, bid-ask spreads, and inflexible position
sizes)
• Risk-free rate and volatility levels do not remain constant until
the expiration date
• Empirical studies showed that the Black-Scholes model
overvalued out-of-the-money call options and undervalued inthe-money contracts
• Any violation of the assumptions upon which the Black-Scholes
model is based could lead to a misevaluation of the option
contract
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Swaps
• Swaps
• They are agreements to exchange a series of cash flows on
periodic settlement dates over a certain time period (e.g.,
quarterly payments over two years). The length of a swap
is termed the tenor of the swap that ends on termination
date.
• Forward-Based Interest Rate Contracts
• Forward Rate Agreement (FRA)
• Interest Rate Swaps
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Swaps
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Swaps
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Extensions of Swaps
• Equity Index-Linked Swaps
• Equivalent to portfolios of forward contracts
calling for the exchange of cash flows based on
two different investment rates:
• A variable-debt rate (e.g., three-month LIBOR)
• Return to an equity index (e.g., Standard & Poor’s 500)
• Payment is based on:
• Total return, or
• Percentage index change for settlement period plus a
fixed spread adjustment
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Extensions of Swaps
• Credit-Related Swaps
• These swaps are designed to help investors
manage their credit risk exposures
• One of the newest swap contracting extensions
introduced in the late 1990s
• Credit-related swaps have grown in popularity,
exceeding $45 trillion in notional value by mid2007
• Types:
• Total Return Swap
• Credit Default Swap (CDS)
• Collateralized Debt Obligations (CDOs)
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Option-Like Securities
• Warrants
• Equity call option issued directly by company
whose stock serves as the underlying asset
• Key feature that distinguishes it from an ordinary
call option is that, if exercised, the company will
create new shares of stock to give to the warrant
holder
• Thus, exercise of a warrant will increase total
number of outstanding shares, which reduces the
value of each individual share. Because of this
dilutive effect, the warrant is not as valuable as
an otherwise comparable option contract.
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Option-Like Securities
• Warrants Valuation
• Expiration Date Value, WT
V + N X
W
- X ,0 
W T = max  T
 N + NW

where:
N = the current number of outstanding shares
NW = the shares created if the warrants are exercised
VT = the value of the firm before the warrants are
exercised
X = the exercise price
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Option-Like Securities
• Warrants Valuation
or


1
WT = 
CT
1 + (N W N )
where:
CT = the expiration date value of a regular call option
with otherwise identical terms as the warrant
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Option-Like Securities
• Convertible Bonds
• A convertible security gives its owner the right,
but not the obligation, to convert the existing
investment into another form
• Typically, the original security is either a bond or
a share of preferred stock, which can be
exchanged into common stock according to a
predetermined formula
• A hybrid security
• Bond or preferred stock holding
• A call option that allows for the conversion
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Option-Like Securities
• Convertible Bonds
• Conversion ratio
• number of shares of common stock for which a
convertible security may be exchanged
• Conversion parity price
• price at which common stock can be obtained
by surrendering the convertible instrument at
par value
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Option-Like Securities
• Callable Bond
• Provides the issuer with an option to call the
bond under certain conditions and pay it off with
funds from a new issue sold at a lower yield
• Bond with an embedded option (Chapter 12)
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Option Like Securities
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