Investment Analysis and Portfolio Management First Canadian Edition

Investment Analysis and Portfolio Investment Analysis and Portfolio ManagementManagement

First Canadian EditionFirst Canadian EditionBy Reilly, Brown, Hedges, ChangBy Reilly, Brown, Hedges, Chang1414

Copyright © 2010 by Nelson Education Ltd. 14-2

•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

Chapter 14Derivatives: Analysis and Valuation


• 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

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



• 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


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


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


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


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


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

TTTTT DiPCSSCSF ,0,0,00,00,0


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)


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 • Long-term interest rate futures• Short-term interest rate futures• Stock index futures• Currency forwards and futures


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)


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


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



• 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



• 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


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



• 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

Stock price now

Price in one year

$50

$65

$40



• 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)



• 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)



• 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 is40/(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


• The Binomial Option Pricing Formula• In the jth state in any sub-period, the value

of the option can be calculated by


r

CpCpC

jdjuj

1

where

andr = one plus the risk-free rate over the sub-period

du

drp



• At Date 0, the binomial option pricing formula can be expressed as follows:

njNjjNjN

0jo rXSdupp

jjN

NC

1

!!

!

m is the smallest integer number of up moves guaranteeing that the option will be in the money at expiration



• The hedge ratio for any state j becomes

jujd

jj

CC

Sduh


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


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)



where:C0 = market value of call option

S = current market price of underlying stockX = exercise price of call optione-(RFR)T = discount function for continuously compounded variablesN(d1) = cumulative density function of d1 defined as

N(d2) = cumulative density function of d2 defined as

2121 5.0Sln [T]TRFRXd

21

12 [T]dd



• Properties of the Model• Option’s value is a function of five variables

• 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)





• Estimating Volatility• The standard deviation of returns to the underlying

asset can be estimated in two ways1.Traditional mean and standard deviation of a series

of price relatives2.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



• 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 in-the-money contracts

• Any violation of the assumptions upon which the Black-Scholes model is based could lead to a misevaluation of the option contract



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


Swaps


Swaps


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


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 mid-2007

• Types: • Total Return Swap• Credit Default Swap (CDS)• Collateralized Debt Obligations (CDOs)


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.



• Warrants Valuation• Expiration Date Value, WT

0,max X

NN

XNVW

W

WTT

where:N = the current number of outstanding sharesNW = the shares created if the warrants are

exercisedVT = the value of the firm before the warrants are

exercisedX = the exercise price



• Warrants Valuation

TW

T CNN

W

1

1or

where:CT = the expiration date value of a regular call option with otherwise identical terms as the warrant



• 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



• 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



• 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)


Option Like Securities

Investment Analysis and Portfolio Management First Canadian Edition

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