7.7Summary and Conclusions

This chapter examined one of the most fundamental con-tributions to financial theory: the pricing of derivatives.The price movements of a derivative are perfectly corre-lated over short time intervals with the price movements ofthe underlying asset on which it is based. Hence, a portfo-lio of the underlying asset and a riskless security can beformed that perfectly tracks the future cash flows of the de-rivative. To prevent arbitrage, the tracking portfolio andthe derivative must have the same cost.

The no-arbitrage cost of the derivative can be com-puted in several ways. One method is direct. It forms thetracking portfolio at some terminal date and works back-ward to determine a portfolio that maintains perfect track-ing. The initial cost of the tracking portfolio is the no-arbitrage cost of the derivative. An alternative, butequivalent, approach is the risk-neutral valuation method,which assigns risk-neutral probabilities to the cash flowoutcomes of the underlying asset that are consistent withthat asset earning, on average, a risk-free return. Applyinga risk-free discount rate to the expected future cash flowsof the derivative gives its no-arbitrage present value. Bothmethods lend their unique insights to the problem of valu-ing derivatives, although the tracking portfolio approach

seems to be slightly more intuitive. Computationally, therisk-neutral approach is easier to program.

To keep this presentation relatively simple, we have re-lied on binomial models, which provide reasonable ap-proximations for the values of most derivatives. The bino-mial assumptions are usually less restrictive than theymight at first seem because the time period for the bino-mial jump can be as short as one desires. As the time periodgets shorter, the number of periods gets larger. Abinomialjump to one of two values every minute can result in 2⁶⁰(more than a billion) possible outcomes at the end of anhour. Defining the time period to be as short as one pleasesmeans that the binomial process can approximate to anydegree almost any probability distribution for the return onan investment over almost any horizon.

This chapter mainly focused on the general principlesof no-arbitrage valuation, although it applied these prin-ciples to value simple derivatives. Many other applica-tions exist. For example, the exercises that follow thischapter value risky bonds, callable bonds, and convert-ible bonds. The following chapter focuses in depth on oneof the most important applications of these principles, op-tion pricing.

Key Concepts

Result	7.1:	The no-arbitrage value of a forward	and
		contract on a share of stock (the obligation
			the current market price of a
		to buy a share of stock at a price of K, T	K
			default-free zero-coupon bond
		years in the future), assuming the stock	T
			(1r)
			fpaying K, Tyears in the future
		pays no dividends prior to T,is
			Result 7.2:The forward price for settlement in T
		K
		S	years of the purchase of a non-dividend-
		0(1 r)T
		f	paying stock with a current price of Sis
			0
		where	T
			FS(1 r)
			00f
		Scurrent price of the stock
		0

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Chapter 7

Pricing Derivatives

253

Result 7.3:In the absence of arbitrage, the forwardResult 7.5:The value of a derivative, relative to the

currency rate F(for example, Euros/dollar)value of its underlying asset, does not

0

is related to the current exchange rate (ordepend on the mean return of the

spot rate) S, by the equationunderlying asset or investor risk

0

preferences.

F1r

^0foreignResult 7.6:Valuation of a derivative based on no

S1r

0domesticarbitrage between the tracking portfolio

whereand the derivative is identical to risk-

neutral valuation of that derivative.

r the return (unannualized) on a domestic

Result 7.7:The no-arbitrage futures price is the same

or foreign risk-free security over the life

as a weighted average of the expected

of the forward agreement, as measured

futures prices at the end of the period,

in the respective country’s currency

where the weights are the risk-neutralResult 7.4:To determine the no-arbitrage value of aprobabilities, that is

derivative, find a (possibly dynamic)

F F(1 )F

portfolio of the underlying asset and a risk-^ud

free security that perfectly tracks the futureResult 7.8:The no-arbitrage futures price is the risk-

payoffs of the derivative. The value of theneutral expected future spot price at the

derivative equals the value of the trackingmaturity of the futures contract.

portfolio.

Key Terms

at the money225	maturity date (settlement date)	216
binomial process235	mortgage passthroughs228
clearinghouse227	notional amount221
collateralized mortgage obligations (CMOs)229	numerical methods248
counterparties221	open interest219
covered interest rate parity relation234	out of the money225
discount240	principal-only securities (POs)	229
down state235	risk-neutral probabilities240
expiration date223	risk-neutral valuation method	240
forward contract216	simulation250
forward price216	spot price220
forward rate discount233	spot rate232
futures contracts216	strike price223
futures markets216	swap221
in the money225	swap maturity221
interest rate swap221	tranches229
interest-only securities (IOs)229	underlying asset215
LEAPS226	writing an option227
margin accounts220	up state235
marking to market220	zero cost instruments223

Exercises

7.1.	Using risk-neutral valuation, derive a formula for a			$1.20
	derivative that pays cash flows over the next two		$1.10
	periods. Assume the risk-free rate is 4 percent per	$1		$1.05
	period.		$.95
	The underlying asset, which pays no cash flows			$.90
	unless it is sold, has a market value that is modeled
	in the adjacent tree diagram.

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254Part IIValuing Financial Assets

	The cash flows of the derivative that correspond to	The exhibit below describes the value of the
	the above tree diagram are	firm’s assets per bond (less the amount in the
		reserve fund maintained for the intermediate
	$2.20
		coupon) and the cash payoffs of the bond. The
	$.10
		nonreserved assets of the firm are currently worth
	$0$3.05
		$100 per bond. At the Uand Dnodes, the reserve
	$.90
		fund has been depleted and the remaining assets of
	$0
		the firm per bond are worth $120 and $90,
	Find the present value of the derivative.	respectively, while they are worth $300, $110, and
		$50, respectively, in the UU, UD,and DDstates
7.2.	Aconvertible bond can be converted into a
		two periods from now.
	specified number of shares of stock at the option of
	the bondholder. Assume that a convertible bond can
		Paths forthe Value of the Firm’s Assets PerBond
	be converted to 1.5 shares of stock. Asingle share
		(Above the Node) Cash Payoffs of a Risky Bond
	of this stock has a price that follows the binomial
		(Below theNode) in a Two-Period Binomial Tree
	process
		Diagram
	Date 0Date 1
	$80
	$50
		$300
	$30
		UU
	The stock does not pay a dividend between dates 0
		$105
		$120
	and 1.
	If the bondholder never converts the bond to	U
	stock, the bond has a date 1 payoff of $100 x,	$100$110
		$5
	where xis the coupon of the bond. The conversion to
		UD
	stock may take place either at date 0 or date 1 (in the
	latter case, upon revelation of the date 1 stock price).	$90$105
	The convertible bond is issued at date 0 for
		D
	$100. What should x, the equilibrium coupon of the
		$5$50
	convertible bond per $100 at face value, be if the
	risk-free return is 15 percent per period and there	DD
	are no taxes, transaction costs, or arbitrage
		$50
	opportunities? Does the corporation save on interest
	payments if it issues a convertible bond in lieu of a
	straight bond? If so, why?
7.3.	Value a risky corporate bond, assuming that the
	risk-free interest rate is 4 percent per period,
	where a period is defined as six months. The	7.4.In many instances, whether a cash flow occurs early
	corporate bond has a face value of $100 payable	or not is a decision of the issuer or holder of the
	two periods from now and pays a 5 percent	derivative. One example of this is a callable bond,
	coupon per period; that is, interest payments of	which is a bond that the issuing firm can buy back at
	$5 at the end of both the first period and the	a prespecified call price. Valuing a callable bond is
	second period.	complicated because the early call date is not known
	The corporate bond is a derivative of the assets	in advance—it depends on the future path followed
	of the issuing firm. Assume that the assets generate	by the underlying security. In these cases, it is
	sufficient cash to pay off the promised coupon one	necessary to compare the value of the security—
	period from now. In particular, the corporation has	assuming it is held a while longer—with the value
	set aside a reserve fund of $5/1.04 per bond to pay	obtained from cash by calling the bond or
	off the promised coupon of the bond one period	prematurely exercising the call option. To solve these
	from now. Two periods from now, there are three	problems, you must work backward in the binomial
	possible states. In one of those states, the assets of	tree to make the appropriate comparisons and find the
	the firm are not worth much and the firm defaults,	nodes in the tree where intermediate cash flows occur.
	unable to generate a sufficient amount of cash.	Suppose that in the absence of a call, a callable
	Only $50 of the $105 promised payment is made on	corporate bond with a call price of $100 plus
	the bond in this state.	accrued interest has cash flows identical to those of

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Strategy, Second Edition

Chapter 7

Pricing Derivatives

255

the bond in exercise 7.3. (In this case, accruedmonths. The stock currently sells for $140 a share.

interest is the $5 coupon if it is called cum-couponAssume that it pays no dividends over the coming

at the intermediate date and 0 if it is called ex-six months. Six-month zero-coupon bonds are

coupon.) What is the optimal call policy of theselling for $98 per $100 of face value.

issuing firm, assuming that the firm is trying toa.If the forward is selling for $10.75, is there an

maximize shareholder wealth? What is the value ofarbitrage opportunity? If so, describe exactly

the callable bond? Hint:Keep in mind thathow you could take advantage of it.

maximizing shareholder wealth is the same asb.Assume that three months from now: (i) the stock

minimizing the value of the bond.price has risen to $160 and (ii) three-month zero-7.5.Consider a stock that can appreciate by 50 percentcoupon bonds are selling for $99. How much has

or depreciate by 50 percent per period. Threethe fair market value of your forward contract

periods from now, a stock with an initial value ofchanged over the three months that have elapsed?

$32 per share can be worth (1) $108—three up7.7.Assume that forward contracts to purchase one

moves; (2) $36—two up moves, one down move;share of Sony and Disney for $100 and $40,

(3) $12—one up move, two down moves; or respectively, in one year are currently selling for

(4) $4—three down moves. Three periods from now,$3.00 and $2.20. Assume that neither stock pays a

a derivative is worth $78 in case (1), $4 in case (2),dividend over the coming year and that one-year

and $0 otherwise. If the risk-free rate is 10 percentzero-coupon bonds are selling for $93 per $100 of

throughout these three periods, describe a portfolioface value. The current prices of Sony and Disney

of the stock and a risk-free bond that tracks theare $90 and $35, respectively.

payoff of the derivative and requires no future casha.Are there any arbitrage opportunities? If so,

outlays. Then, fill in the question marks in thedescribe how to take advantage of them.

accompanying exhibit, which illustrates the pathsb.What is the fair market price of a forward contract

of the stock and the derivative. Hint:You need toon a portfolio composed of one-half share of Sony

work backward. Use the risk-neutral valuationand one-half share of Disney, requiring that $70

method to check your work.be paid for the portfolio in one year?

c.Is this the same as buying one-half of a forwardThree-Period Binomial Tree Diagram Underlyingcontract on each of Sony and Disney? Why orSecurity’s Price Above Node, Derivative’s Pricewhy not? (Show payoff tables.)

Below Noded.Is it generally true that a forward on a portfolio

is the same as a portfolio of forwards? Explain.

7.8.Assume the Eurodollar rate (nine-month LIBOR) is

9 percent and the Eurosterling rate (nine-month

$108

UUULIBOR for the U.K. currency) is 11 percent. What

$78is the nine-month forward $/£ exchange rate if the

$72

current spot exchange rate is $1.58/£? (Assume that

UU

nine months from now is 274 days.)

$36

$48?

UUD7.9.Assume that futures prices for Hewlett-Packard

U

$32$24$4(HP) can appreciate by 15 percent or depreciate by

?

UD10 percent and that the risk-free rate is 10 percent

$16?$12

?over the next period. Price a derivative security on

UDD

D

HPthat has payoffs of $25 in the up state and 2$5 in

?$8$0

the down state in the next period, where the actual

DD

probability of the up state occurring is 75 percent.

?

$4

7.10.Astock price follows a binomial process for two

DDD

periods. In each period, it either increases by 20

$0

percent or decreases by 20 percent. Assuming that

the stock pays no dividends, value a derivative

which at the end of the second period pays $10 for

every up move of the stock that occurred over the

previous two periods. Assume that the risk-free rate

is 6 percent per period.

7.6.Consider a forward contract on IBM requiring7.11.Astock has a 30% per year standard deviation of its

purchase of a share of IBM stock for $150 in sixlogged returns. If you are modeling the stock price

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Strategy, Second Edition

256Part IIValuing Financial Assets

	to value a derivative maturing in six months with	ABC stock is currently selling at $1 per share.
	eight binomial periods, what should uand dbe?	Each period the price either rises 10% or falls by
7.12.	Find the risk neutral probabilities and zero cost date	5% (i.e., after one period, the stock price of ABC
	0 forward prices (for settlement at date 1) for the	can either rise to $1.10 or fall to $0.95). The
	stock in exercise 7.2. As in that exercise, assume a	probability of a rise is 0.5. The risk-free interest
	risk-free rate of 15% per period.	rate is 5% per period.
		a.Determine the price at which you expect the
7.13.	AEuropean “Tootsie Square” is a financial
		Tootsie Square on ABC to trade.
	contract, which, at maturity, pays off the square
		b.Suppose that you wanted to form a portfolio to
	of the price of the underlying stock on which it is
		track the payoff on the Tootsie Square over the
	written. For instance, if the price of the
		first period. How many shares of ABC stock
	underlying stock is $3 at maturity, the Tootsie
		should you hold in this portfolio?
	Square contract pays off $9. Consider a two
	periodTootsie Square written on ABC stock.

References and Additional Readings

Arrow, Kenneth J. “The Role of Securities in the Optimal

Allocation of Risk-Bearing.” Review of Economic

Studies31 (1964), pp. 91–96.

Balducci, Vince; Kumar Doraiswani; Cal Johnson; and

Janet Showers. “Currency Swaps: Corporate

Applications and Pricing Methodology,” pamphlet.

New York: Salomon Brothers, Inc., Bond Portfolio

Analysis Group, 1990.

Black, Fischer, and Myron Scholes. “The Pricing of

Options and Corporate Liabilities.” Journal of

Political Economy81 (May–June 1973), pp.

637–59.

Breeden, Douglas T., and Robert H. Litzenberger. “Prices

of State-Contingent Claims Implicit in Option

Prices.” Journal of Business51 (1978), pp. 621–52.Cox, J., and S. Ross. “The Valuation of Options for

Alternative Stochastic Processes.” Journal of

Financial Economics3 (Jan.–Mar. 1976),

pp.145–66.

Cox, John C.; Stephen A. Ross; and Mark Rubinstein.

“Option Pricing: ASimplified Approach.” Journal of

Financial Economics7 (Sept. 1979), pp. 229–63.

Cox, John C., and Mark Rubinstein. Options Markets.

Engelwood Cliffs, NJ: Prentice-Hall, 1985.

Grinblatt, Mark. “An Analytic Solution for Interest Rate

Swap Spreads.” Working Paper, University of

California, Los Angeles, 1995.

Grinblatt, Mark, and Narasimhan Jegadeesh. “The

Relative Pricing of Eurodollar Futures and

Forwards.” Journal of Finance51, no. 4 (Sept.

1996), pp. 1499–1522.

Grinblatt, Mark, and Francis Longstaff. “Financial

Innovation and the Role of Derivative Securities: An

Empirical Analysis of the Treasury Strips Program.”

Journal of Finance.

Harrison, Michael J., and David M. Kreps. “Martingales

and Arbitrage in Multiperiod Securities Markets.”

Journal of Economic Theory20 (1979),

pp.381–408.

Hull, John C. Options, Futures, and Other Derivatives,3d

ed. Upper Saddle River, N.J.: Prentice-Hall, 1997.

Ingersoll, Jonathan. Theory of Financial Decision

Making.Totowa, NJ: Rowman & Littlefield, 1987.

Jarrow, Robert, and Stuart Turnbull. Derivative Securities.

Cincinnati, Ohio: South-Western Publishing, 1996.

Lewis, Michael. Liar’s Poker: Rising through the

Wreckage on Wall Street. New York: W. W. Norton,

1989.

Merton, Robert C. “Theory of Rational Option Pricing.”

Bell Journal of Economics and Management Science

4 (Spring 1973), pp. 141–83.

———. Continuous Time Finance.Oxford, England:

Basil Blackwell, 1990.

Rendelman, Richard J., Jr., and Brit J. Bartter. “Two-State

Option Pricing.” Journal of Finance34 (Dec. 1979),

pp. 1093–1110.

Ross, Stephen A. “ASimple Approach to the Valuation of

Risky Streams.” Journal of Business51 (July 1978)

pp. 453–75.

Rubinstein, Mark. “Presidential Address: Implied

Binomial Trees.” Journal of Finance49 (July 1994),

pp. 771–818.

Schwartz, Eduardo. “The Valuation of Warrants:

Implementing a New Approach.” Journal of

Financial Economics4 (Jan. 1977), pp. 79–93.Shleifer, Andrei and Robert Vishny. “The Limits of

Arbitrage.” Journal of Finance 52 (1997),

pp.35–55.

Grinblatt530Titman: Financial	II. Valuing Financial Assets	8. Options	© The McGraw530Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

CHAPTER

8

Options

Learning Objectives

After reading this chapter, you should be able to:

1.Explain the basic put-call parity formula, how it relates to the value of a forward

contract, and the types of options to which put-call parity applies.

2.Relate put-call parity to a boundary condition for the minimum call price and

know the implications of this boundary condition for pricing American call

options and determining when they should be exercised.

3.Gather the information needed to price a European option with (a) the binomial

model and (b) the Black-Scholes Model, and then implement these models to

price an option.

4.Understand the Black Scholes formula in the following form:

SN(d) PV(K)N(d T).

011

	and interpret N(d).
5.	1 Provide examples that illustrate why an American call on a dividend-paying stock oran American put (irrespective of dividend policy) might be exercised prematurely.
6.	Understand the effect of volatility on option prices and premature exercise.

In 1980, Chrysler, then on the verge of bankruptcy, received loan guarantees from the

U.S. government. As partial payment for the guarantees, Chrysler gave the U.S.

government options to buy Chrysler stock for $13 a share until 1990. At the time these

options (technically known as “warrants”) were issued, Chrysler stock was trading at

about $7.50 per share. By mid-1983, however, Chrysler stock had risen to well over

$30 a share and in July of that year, Chrysler repaid the guaranteed loans in full.

Lee Iacocca, Chrysler’s CEO at the time, argued that the government had not

shelled out a dime to provide the loan guarantees and asked that the warrants be

returned to Chrysler at little or no cost. However, many congressmen felt that the

government had taken a risk by providing the guarantees and if the warrants were to

be returned to Chrysler, they should be returned at their full market value. Ultimately,

the government decided to hold an auction for the warrants. Determined to recover

its warrants, Chrysler participated in the auction, which occurred when Chrysler

257

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Strategy, Second Edition

258Part IIValuing Financial Assets

stock was trading at about $28 a share. Chrysler’s bid, about $22 per warrant, was

the winning bid in the auction.

One of the most important applications of the theory of derivatives valuation is the

pricing of options. Options, introduced earlier in the text, are ubiquitous in finan-

cial markets and, as seen in this chapter and in much of the remainder of this text, are

often found implicitly in numerous corporate financial decisions and corporate securities.

Options have long been important to portfolio managers. In recent years, they have

become increasingly important to corporate treasurers. There are several reasons for

this. For one, a corporate treasurer needs to be educated about options for proper

oversight of the pension funds of the corporation. In addition, the treasurer needs to

understand that options can be useful in corporate hedging¹and are implicit in many

securities the corporation issues, such as convertible bonds, mortgages, and callable

bonds.²

Options also are an important form of executive compensation and, in some

cases, a form of financing for the firm.

An understanding of option valuation also is critical for any sophisticated financial

decision maker. Consider the opening vignette, for example. As compensation for a

government loan guarantee—which by itself is an option—Chrysler issued stock

options that gave the government the right to buy Chrysler stock at $13 a share. Both

Chrysler and the government felt that this agreement (along with some additional

compensation that Chrysler provided) was a fair deal at the time. However, as Chrysler

recovered in 1983, the value of the loan guarantee declined to zero and the value of

the stock options rose. With no active market in which to observe traded prices for the

Chrysler options, it was difficult to know exactly how much the options were worth.

This chapter applies the no-arbitrage tracking portfolio approach to derive the “fair

market” values of options, like those issued by Chrysler, using both the binomial

approach and a continuous-time approach known as the Black-Scholes Model. It then

discusses several practical considerations and limitations of the binomial and Black-

Scholes option valuation models. Among these is the problem of estimating volatility.

The chapter generalizes the Black-Scholes Model to a variety of underlying securities,

including bonds and commodities. It concludes with a discussion of the empirical

evidence on the Black-Scholes Model.