7.2The Basics of Derivatives Pricing

Derivatives valuation has two basic components. The first component is the concept of

perfect tracking. The second is the principle of no arbitrage. Afair market price for a

derivative, obtained from a derivatives valuation model, is simply a no-arbitrage restric-

tion between the tracking portfolio and the derivative.

Perfect Tracking Portfolios

All derivatives valuation models make use of a fundamental idea: It is always possible

to develop a portfolio consisting of the underlying asset and a risk-free asset that per-

fectly tracks the future cash flows of the derivative.As a result, in the absence of arbi-

trage the derivative must have the same value as the tracking portfolio.¹⁶

Aperfect tracking portfoliois a combination of securities that perfectly replicates

the future cash flows of another investment.

The perfect tracking, used to value derivatives, stands in marked contrast to the

use of tracking portfolios (discussed in Chapters 5 and 6) to develop general models

of financial asset valuation. There our concern was with identifying a tracking portfolio

with the same systematic risk as the investment being valued. Such tracking portfolios

typically generate substantial tracking error. We did not analyze tracking error in Chap-

ters 5 and 6 because the assumptions of those models implied that the tracking error

had zero present value and thus could be ignored. This places a high degree of faith in

the validity of the assumptions of those models. The success of derivatives valuation

models rests largely on the fact that derivatives can be almost perfectly tracked, which

means that we do not need to make strong assumptions about how tracking error affects

value.

No Arbitrage and Valuation

In the absence of tracking error, arbitrage exists if it costs more to buy the tracking

portfolio than the derivative, or vice versa. Whenever the derivative is cheaper than

the tracking portfolio, arbitrage is achieved by buying the derivative and selling (or

15

See the opening vignette in Chapter 23.

¹⁶In

this text, the models developed to value derivatives assume that perfect tracking is possible. In

practice, however, transaction costs and other market frictions imply that investors can, at best, attain

almost perfect tracking of a derivative.

Grinblatt478Titman: Financial	II. Valuing Financial Assets	7. Pricing Derivatives	© The McGraw478Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 7

Pricing Derivatives

231

shorting) the tracking portfolio. On initiating the arbitrage position, the investor receives

cash because the cash spent on the derivative is less than the cash received from short-

ing the tracking portfolio. Since the future cash flows of the tracking portfolio and the

derivative are identical, buying one and shorting the other means that the future cash

flow obligations from the short sales position can be met with the future cash received

from the position in the derivative.¹⁷

Applying the Basic Principles of Derivatives Valuation to Value Forwards

This subsection considers several applications of the basic principles described above,

all of which are related to forward contracts.

Valuing a Forward Contract.Models used to value derivatives assume that arbi-

trage is impossible. Example 7.4 illustrates how to obtain the fair market value of a

forward contract using this idea and how to arbitrage a mispriced forward contract.

Example 7.4:Valuing a Forward Contract on a Share of Stock

Consider the obligation to buy a share of Microsoft stock one year from now for $100.

Assume that the stock currently sells for $97 per share and that Microsoft will pay no divi-

dends over the coming year.One-year zero-coupon bonds that pay $100 one year from now

currently sell for $92.At what price are you willing to buy or sell this obligation?

Answer:Compare the cash flows today and one year from now for two investment strate-

gies.Under strategy 1, the forward contract, the investor acquires today the obligation to buy

a share of Microsoft one year from now at a price of $100.Upon paying $100 for the share

at that time, the investor immediately sells the share for cash.Strategy 2, the tracking port-

folio, involves buying a share of Microsoft today and selling short $100 in face value of

1-year zero-coupon bonds to partly finance the purchase.The stock is sold one year from

now to finance (partly or fully) the obligation to pay the $100 owed on the zero-coupon bonds.

Denoting the stock value one year from now as the random number S˜,the cash inflows

1

and costs from these two strategies are as follows:

Cost	Cash Flow One Year
Today	from Today
	~
Strategy 1 (forward)?	S $100
	1
	~
Strategy 2 (tracking portfolio)$97–$92	S $100
	1

Since strategies 1 and 2 have identical cash flows in the future, they should have the

same cost today to prevent arbitrage.Strategy 2 costs $5 today.Strategy 1, the obligation

to buy the stock for $100 one year from now, also should cost $5.If it costs less than $5,

then going long in strategy 1 and short in strategy 2 (which, when combined, means acquir-

ing the obligation, selling short a share of Microsoft, and buying $200 face amount of the

zero-coupon bonds) has a positive cash inflow today and no cash flow consequences in the

future.This is arbitrage.If the obligation costs more than $5, then going short in strategy 1

and going long in strategy 2 (short the obligation, buy a share of stock, and short $200 face

amount of the zero-coupon bonds) has a positive cash inflow today and no future cash flows.

Thus, the investor would be willing to pay $5 as the fair value of the attractive obligation to

buy Microsoft for $100 one year from now.

¹⁷Usually,

it is impossible to perfectly track a mispricedderivative, although a forward contract is an

exception. However, any mispriced derivative that does not converge immediately to its no-arbitrage

value can be taken advantage of further to earn additional arbitrage profits, using the tracking portfolio

strategy outlined in this chapter.

Grinblatt480Titman: Financial	II. Valuing Financial Assets	7. Pricing Derivatives	© The McGraw480Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

232Part IIValuing Financial Assets

Result 7.1 generalizes Example 7.4 as follows:

Result 7.1

The no-arbitrage value of a forward contract on a share of stock (the obligation to buy ashare of stock at a price of K, Tyears in the future), assuming the stock pays no dividendsprior to T,is

K

S

⁰r)^T

(1

f

where

Scurrent price of the stock

0

and

K

the current market price of a default-free zero-coupon bond paying K,T

(1r)^T

^fyears in the future

Obtaining Forward Prices forZero-Cost Forward Contracts.The value of the for-

ward contract is zero for a contracted price, K,that satisfies

K

S 0

⁰(1r)^T

f

namely

KS(1r)^T

0f

Whenever the contracted price Kis set so that the value of forward is zero, the con-

tracted price is known as the forward price. At date 0, this special contracted price is

denoted as F.¹⁸ Assuming that no money changes hands today as part of the deal, the

0

forward price, FS(1 r)^Trepresents the price at which two parties would agree

00f

today represents fair compensation for the nondividend-paying stock Tyears in the

future. It is fair in the sense that no arbitrage takes place with this agreed-on forward

price. We summarize this important finding below.

Result 7.2

The forward price for settlement in Tyears of the purchase of a nondividend-paying stockwith a current price of Sis 0

FS(1 r)^T

00f

Currency Forward	Rates.	Currency forward rates are a variation on Result 7.2.
Specifically:

Result 7.3

In the absence of arbitrage, the forward currency rate F(for example, Euros/dollar) is 0 related to the current exchange rate (or spot rate), S, by the equation 0

F1r

^0foreign

S1r

0domestic

where

rthe return (unannualized) on a domestic or foreign risk-free security over the life

of the forward agreement, as measured in the respective country’s currency

¹⁸As

mentioned earlier, “off-market” forward prices, resulting in nonzero-cost contracts, exist.

Usually, however, forward pricerefers to the cash required in the future to buy the forward contract’s

underlying investment under the terms specified in a zero-costcontract.

Grinblatt482Titman: Financial	II. Valuing Financial Assets	7. Pricing Derivatives	© The McGraw482Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 7

Pricing Derivatives

233

To understand this result, compare the riskless future payoffs of two strategies (Aand

B) that invest one U.S. dollar today. Suppose that one U.S. dollar currently buys 1.2

Euros; that is, S1.2. If the interest rate on Euros is 10% (that is, r0.1), the

0foreign

three-part strategy Ainvolves contracting today to:

1.Convert US$1 to 1.2 Euros.

2.Invest the Euros at 10% and receive 1.32 Euros (10% more than 1.2) one year

from now.

3.Convert the 1.32 Euros back to 1.32/Fdollars one year from now (at the

0

Euros per U.S. dollar forward rate Fagreed to today).

0

Let’s now compare strategy Ato strategy B, which consists of

1.Taking the same dollar and investing it in the United States, earning 5 percent

(that is, r 0.05), which produces US$1.05 one year from now.

domestic

To prevent arbitrage, strategy Ahas to generate the same payoff next year as the

US$1.05 generated by strategy B. This can only happen if:

(1.2) (1.1)

1.05

F

0

or equivalently

F10.1

0,

1.210.05

which is the equation obtained if applying Result 7.3 directly.

By subtracting 1 from each side of the equation in Result 7.3 and simplifying, we

obtain

Fr r

⁰foreigndomestic

1

S1r

0domestic

Note that the right-hand side quotient of this equation is approximately the same as its

numerator, r r. Thus, Result 7.3 implies that the forward rate discount¹⁹

foreigndomestic

relative to the spot rate

F ^SF

0^{0 0} 1

SS

00

which represents the percentage discount at which one buys foreign currency in the

forward market relative to the spot market, is approximately the same as the interest

rate differential, r r. For example, if the one-year interest rate in Switzer-

foreigndomestic

land is 10 percent and the one-year interest rate in the United States is 8 percent, then

a U.S. company can purchase Swiss francs forward for approximately 2 percent less

than the spot rate, which is reflected by Fbeing about 2 percent larger than S.

0 0

¹⁹It

is a discount instead of a premium because exchange rates are measured as units of foreign

currency that can be purchased per unit of domestic currency. Depending on the currency, it is sometimes

customary to express the exhange rate as the number of units of domestic currency that can be purchased

per unit of foreign currency. In this case, rand rare reversed in Result 7.3 and in the

foreigndomestic

approximation of Result 7.3 expressed in this sentence. Moreover, we would refer to FS 1 as the

00

forward rate premium.

Grinblatt484Titman: Financial	II. Valuing Financial Assets	7. Pricing Derivatives	© The McGraw484Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

234Part IIValuing Financial Assets

Result 7.3 is sometimes known as the covered interest rate parity relation.This

relation describes how forward currency rates are determined by spot currency rates

and interest rates in the two countries. Example 7.5 illustrates how to implement the

covered interest rate parity relation.

Example 7.5:The Relation Between Forward Currency Rates and Interest

Rates

Assume that six-month LIBOR on Canadian funds is 4 percent and the US$ Eurodollar rate

(six-month LIBOR on U.S.funds) is 10 percent and that both rates are default free.What is

the six-month forward Can$/US$ exchange rate if the current spot rate is Can$1.25/US$?

Assume that six months from now is 182 days.

Answer:As noted in Chapter 2, LIBOR is a zero-coupon rate based on an actual/360

day count.Hence:

Canada

United States

	182	182
Six-month interest rate (unannualized):	2.02% 4%	5.06% 10%
	360	360

Can$ 1.211.0202

By Result 7.3, the forward rate is 1.25.

US$1.0506