7.3Binomial Pricing Models

In derivatives valuation models, the currentprice of the underlying asset determines

the price of the derivative today.This is a rather surprising result because in most cases

the link between the price of the derivative and the price of the corresponding under-

lying asset is usually obvious only at some future date. For example, the forward con-

tract described in Example 7.4 has a price equal to the difference between the stock

price and the agreed upon settlement price, but only at the settlement date one year

from now. Acall option has a known value at the expiration date of the call when the

stock price is known.

Tracking and Valuation: Static versus Dynamic Strategies

The pricing relation of a derivative with an underlying asset in the future translates into

a no-arbitrage pricing relation in the present because of the ability to use the underly-

ing asset to perfectly track the derivative’s future cash flows. Example 7.4 illustrates

an investment strategy in the underlying asset (a share of Microsoft stock) and a risk-

free asset (a zero-coupon bond) that tracks the payoff of the derivative at a future date

when the relation between their prices is known. In the absence of arbitrage, the track-

ing strategy must cost the same amount today as the derivative.

The tracking portfolio for a forward contract is particularly simple. As Example

7.4 illustrates, forward contracts are tracked by static investment strategies; that is, buy-

ing and holding a position in the underlying asset and a risk-free bond of a particular

maturity. The buy and hold strategy tracks the derivative, the forward contract, because

the future payoff of the forward contract is a linear function of the underlying asset’s

future payoff. However, most derivatives have future payoffs that are not linear func-

tions of the payoff of the underlying asset. Tracking such nonlinear derivatives requires

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235

a dynamic strategy: The holdings in the underlying asset and the risk-free bond need

to change frequently in order to perfectly track the derivative’s future payoffs. With a

call option on a stock, for example, tracking requires a position in the stock and a risk-

free bond, where the number of shares of stock in the tracking portfolio goes up as the

stock price goes up and decreases as the stock price goes down.

The ability to perfectly track a derivative’s payoffs with a dynamic strategy requires

that the following conditions be met:

•	The price of the underlying security must change smoothly; that is, it does notmake large jumps.
••	It must be possible to trade both the derivative and the underlying security continuously. Markets must be frictionless (see Chapter 5).20

If these conditions are violated, the results obtained will generally be approxima-

tions. One notable exception to this relates to the first two assumptions. If the price of

the underlying security follows a very specific process, known as the binomial process,

the investor can still perfectly track the derivative’s future cash flows.

With the binomial process, the underlying security’s price moves up or down over

time, but can take on only twovalues at the next point at which trading is allowed—

hence the name “binomial.” This is certainly not an accurate picture of a security’s

price process, but it is a better approximation than you first might think and it is very

convenient for illustrating how the theory of derivative pricing works. Academics and

practitioners have discovered that the dynamic tracking strategies developed from

binomial models are usually pretty good at tracking the future payoffs of the deriva-

tive. Moreover, the binomial fair market values of most derivatives approximate the

fair market values given by more complex models, often to a high degree of accuracy,

when the binomial periods are small and numerous.

Binomial Model Tracking of a Structured Bond

Exhibit 7.9 illustrates what are known as binomial trees. The tree at the top represents

the possible price paths for the S&P500, the underlying security. The leftmost point

of this tree, 1,275, represents the value of the S&P500 today. The lines connecting

1,275 with the next period represent the two possible paths that the S&P500 can take.

The two values at the end of those lines, 1,500 and 750, represent the two possible val-

ues that the S&P500 can assume one period from today. In the binomial process, one

refers to the two outcomes over the next period as the up stateand the down state.

Hence, 1,500 and 750 represent the payoffs of the S&P500 in the up state and down

state, respectively.²¹

The middle tree models the price paths for a risk-free security paying 10 percent

per period. This security is risk free because each $1.00 invested in the security pays

the same amount, $1.10, at the nodes corresponding to both the up and the down states.

²⁰Most

of the models analyzed here assume that the risk-free rate is constant. If the short-term risk-

free return can change over time, perfect tracking can be achieved with more complicated tracking

strategies. In many instances, formulas for valuing derivatives (for example, options) based on these

more complex tracking strategies can be derived, but they will differ from those presented here. These

extensions of our results are beyond the scope of this text.

²¹For

reasons that will become more clear in Chapter 8, we assume that the future payoffs of the

S&P500 include dividend payments.

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236Part IIValuing Financial Assets

EXHIBIT7.9One-Period Binomial Trees

Underlying Asset: S&P 500

$1,500 (Up State)

$1,275

$750 (Down State)

Risk-Free Security with 10% Return

$1.10 (Up State)

$1

$1.10 (Down State)

Derivative: Price of a Structured Bond

$331.75 (Up State)

= $100.00 + 6.75% ($100.00) + $225.00

?

$106.75 (Down State)

= $100.00 + 6.75% ($100.00)

The bottom tree models the price paths for the derivative, which in this case is a

structured bond. The upper node corresponds to the payoff of the structured bond when

the up state occurs (which leads to an S&P500 increase of 225 points to 1500); the

lower node represents the down state, which corresponds to an S&Pvalue of 750. The

structured bond pays $106.75, $100.00 in principal plus 6.75 percent interest at the

maturity of the bond one period from now. In addition, if the S&P500 goes up in value,

there is an additional payment of $1 for every 1 point increase in the S&P500. Since

this occurs only in the up state, the payoff of the structured bond at the up state node

is $331.75, while the payoff at the down state node is $106.75.

The structured bond pictured in Exhibit 7.9 is a simplified characterization of a

number of these bonds. Investors, such as pension fund managers, who are prohib-

ited from participating directly in the equity markets have attempted to circumvent

this restriction by investing in corporate debt instruments such as structured bonds

that have payoffs tied to the appreciation of a stock index. Of course, a corporation

that issues such a bond needs compensation for the future payments it makes to the

bond’s investors. Such payments reflect interest and principal, as well as a payoff

that enables the investor to enjoy upside participation in the stock market. Generally,

the corporation cannot observe a fair market price for derivatives, like structured

bonds, because they are not actively traded. The question mark at the leftmost node

of the bottom tree in Exhibit 7.9 reflects that a fair value is yet to be determined.

The next section computes this fair value using the principle of no arbitrage. It

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Chapter 7

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237

involves identifying a portfolio of financial instruments that are actively traded,

unlike the structured bond. This portfolio has the property that its future cash flows

are identical to those of the structured bond.

Using Tracking Portfolios to Value Derivatives

Since each node has only two future values attached to it, binomial processes allow

perfect tracking of the value of the derivative with a tracking portfolio consisting of

the underlying security and a risk-free bond. After identifying the tracking portfolio’s

current value from the known prices of its component securities, the derivative can be

valued. This is the easy part because, according to the principle of no arbitrage, the

value of the derivative is the same as that of its tracking portfolio.

Identifying the Tracking Portfolio.Finding the perfect tracking portfolio is the

major task in valuing a derivative. With binomial processes, the tracking portfolio is

identified by solving two equations in two unknowns, where each equation corresponds

to one of the two future nodes to which one can move.

The equation corresponding to the up nodeis

SB(1 r) V	(7.1)
ufu

where

number of units (shares) of the underlying asset

Bnumber of dollars in the risk-free security

rrisk-free rate

f

Svalue of the underlying asset at the up node

u

Vvalue of the derivative at the up node

u

The left-hand side of equation (7.1), SB(1 r), is the value of the tracking

uf

portfolio at the up node. The right-hand side of the equation, V, is the value of the

u

derivative at the up node. Equation (7.1) thus expresses that, at the up node, the track-

ing portfolio, with units of the underlying asset and Bunits of the risk-free security,

should have the same up-node value as the derivative; that is, it perfectly tracks the

derivative at the up node.

Typically, everything is a known number in an expression like equation (7.1),

except for and B. In Exhibit 7.9, for example, where the underlying asset is the S&P

500 and the derivative is the structured bond, the known corresponding values are

r.1

f

S$1,500

u

and

V$331.75

u

Thus, to identify the tracking portfolio for the derivative in Exhibit 7.9 the first equa-

tion that needs to be solved is:

$1,500 B(1.1) $331.75

(7.1a)

Equation (7.1) represents one linear equation with two unknowns, and B. Pair

this equation with the corresponding equation for the down node

SB(1 r) V	(7.2)
dfd

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238Part IIValuing Financial Assets

For the numbers in Exhibit 7.9, this is
$750 B(1.1) $106.75	(7.2a)

Solving equations (7.1) and (7.2) simultaneously yields a unique solution for and B,

which is typical when solving two linear equations for two unknown variables.

Example 7.6 illustrates this technique for the numbers given in Exhibit 7.9 and pro-

vides some tips on methods for a quick solution.²²

Example 7.6:Finding the Tracking Portfolio

A unit of the S&P 500 sells for $1,275 today.One period from now, it can take on one of

two values, each associated with a good or bad state.If the good state occurs, it is worth

$1,500.If the bad state occurs, it is worth $750.If the risk-free interest rate is 10 percent

per period, find the tracking portfolio for a structured bond, which has a value of $331.75

next period if the S&P 500 unit sells for $1,500, and $106.75 if the S&P 500 sells for $750.

Answer:A quick way to find the tracking portfolio of the S&P 500 and the risk-free asset

is to look at the differences between the up and down node values of both the derivative

and the underlying asset.(This is equivalent to subtracting equation (7.2a) from (7.1a) and

solving for .) For the derivative, the structured bond, the difference is $225 $331.75

$106.75, but for the S&P 500, it is $750 $1,500 $750.

Since the tracking portfolio has to have the same $225 difference in outcomes as the

derivative, and since the amount of the risk-free security held will not affect this difference,

the tracking portfolio has to hold 0.3 units of the S&P 500.If it held more than 0.3 units, for

example, the difference in the tracking portfolio’s future values would exceed $225.Given

that is 0.3 units of the S&P 500 and that Bdollars of the risk-free asset is held in addi-

tion to 0.3 units, the investor perfectly tracks the derivative’s future value only if he or she

selects the correct value of B.To determine this value of B, study the two columns on the

right-hand side of the table below, which outlines the values of the tracking portfolio and the

derivative in the up and down states next period:

	Next Period Value
Today’s Value	Up StateDown State

Tracking portfolio	0.3($1,275) B	0.3($1,500) 1.1B	0.3($750) 1.1B
Derivative	?	$331.75	$106.75

Comparing the two up-state values in the middle column implies that the amount of the risk-

free security needed to perfectly track the derivative next period solves the equation

$450 1.1B$331.75

Thus, B $118.25/1.1 or $107.50.(The minus sign implies that $107.50 is borrowed

at the risk-free rate.) Having already chosen to be 0.3, this value of Balso makes the

down-state values of the tracking portfolio and the derivative the same.

Finding the Current Value of the Tracking Portfolio.The fair market value of the

derivative equals the amount it costs to buy the tracking portfolio. Buying shares of

²²Example

7.6 tracks the structured bond over the period with a static portfolio in the S&P500 and a

riskless security. With multiple periods, readjust these weights as each new period begins to maintain

perfect tracking of the option’s value. This readjustment is discussed in more detail shortly.

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Chapter 7

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239

the underlying security today at a cost of Sper share, and Bdollars of the risk-free

asset cost SBin total. Hence, the derivative has a no-arbitrage price of

V S B	(7.3)
This calculation is demonstrated in Example 7.7.

Example 7.7:Valuing a Derivative Once the Tracking Portfolio Is Known

What is the no-arbitrage value of the derivative in Example 7.6? See Exhibit 7.9 for the num-

bers to use in this example.

Answer:The tracking portfolio, which requires buying 0.3 units of the S&P 500 (for $1,275

per unit) and borrowing $107.50 at the risk-free rate costs

$275 0.3($1,275) $107.50

To prevent arbitrage, the derivative should also cost the same amount.

Result 7.4 summarizes the results of this section.

Result 7.4

To determine the no-arbitrage value of a derivative, find a (possibly dynamic) portfolio ofthe underlying asset and a risk-free security that perfectly tracks the future payoffs of thederivative. The value of the derivative equals the value of the tracking portfolio.

Risk-Neutral Valuation of Derivatives: The Wall Street Approach

One of the most interesting things about Examples 7.6 and 7.7 is that there was no

mention of the probabilities of the up and down states occurring. The probability of the

up move determines the mean return of the underlying security, yet the value of

the derivativein relation to the value of the underlying assetdoes not depend on this

probability.

In addition, it was not necessary to know how risk averse the investor was.²³If

the typical investor is risk neutral, slightly risk averse, highly risk averse, or even risk

loving, one obtains the same value for the derivativein relation to the value of the

underlying assetregardless of investor attitudes toward risk. Why is this? Well, when-

ever arbitrage considerations dictate pricing, risk preferences should not affect the rela-

tion between the value of the derivative and the value of the underlying asset. Whether

risk neutral, slightly risk averse, or highly risk averse, you would still love to obtain

something for nothing with 100 percent certainty. The valuation approach this chapter

develops is simply a way to determine the unique pricing relation that rules this out.

Result 7.5 summarizes this important finding.

Result 7.5

The value of a derivative, relative to the value of its underlying asset, does not depend onthe mean return of the underlying asset or investor risk preferences.

Why Mean Returns and Risk Aversion Do Not Affect the Valuation of Deriva-

tives.The reason that information about probabilities or risk aversion does not enter

into the valuation equation is that such information is already captured by the price of

the underlying asset on which we base our valuation of the derivative. For example,

²³Given

two investments with the same expected return but different risk, a risk-averse individual

prefers the investment with less risk. To make a risk-averse individual indifferent between the two

investments, the riskier investment would have to carry a higher expected return.

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assume the underlying asset is a stock. Holding risk aversion constant, the more likely

the future stock price is up and the less likely the stock price is down, the greater the

current stock price will be. Similarly, holding the distribution of future stock price out-

comes constant, if the typical investor is more risk averse, the current stock price will

be lower. However, the wording of Result 7.5 is about the value of the derivative given

the current stock price.Thus, Result 7.5 is a statement that, once the stock price is

known,risk aversion and mean return information are superfluous, not that they are

irrelevant.

An Overview of How the Risk-Neutral Valuation Method Is Implemented.Result

7.5 states that the no-arbitrage price of the derivative in relation to the underlying secu-

rity is the same, regardless of risk preferences. This serves as the basis for a trick known

as the risk-neutral valuation method, which is especially useful in valuing the more

complicated derivatives encountered on Wall Street.

The risk-neutral valuation methodis a three-step procedure for valuing

derivatives.

1.Identify risk-neutral probabilitiesthat are consistent with investors being risk

neutral, given the current value of the underlying asset and its possible future

values. Risk-neutral probabilitiesare a set of weights applied to the future

values of the underlying asset along each path. The expectedfuture asset value

generated by these probabilities, when discounted at the risk-free rate, equals the

current value of the underlying asset.

2.Multiply each risk-neutral probability by the corresponding future value for

the derivative and sum the products together.

3.Discountthe sum of the products in step 2 (the probability-weighted average

of the derivative’s possible future values) at the risk-free rate. This simply

means that we divide the result in step 2 by the sum of 1 plus the risk-free

rate.

The major benefit of the risk-neutral valuation method is that it requires fewer steps

to value derivatives than the tracking portfolio method. However, it is not really a

different method, but a shortcut for going through the tracking portfolio method’s val-

uation steps, developed in the last section. As such, it is easier to program into a com-

puter and has become a fairly standard tool on Wall Street. Moreover, it has its own

useful insights that aid in the understanding of derivatives.

The risk-neutral valuation method obtains derivative prices in the risk preference

scenario that is easiest to analyze—that of risk-neutral preferences. While this sce-

nario may not be the most realistic, pretending that everyone is risk neutral is a per-

fectly valid way to derive the correct no-arbitrage value that applies in all risk pref-

erence scenarios.

The next two subsections walk through the three-step procedure in great detail.

Step 1: Obtaining Risk-Neutral Probabilities.Example 7.6 has a unique probability

of the good state occurring, which implies that the underlying asset, the S&P500, is

expected to appreciate at the risk-free rate. Example 7.8 solves for that probability.

Example 7.8:Attaching Probabilities to Up and Down Nodes

For Example 7.6, solve for the probability of the up state occurring that is consistent with

expected appreciation of the underlying asset at the 10 percent risk-free rate.See Exhibit

7.9 for the numbers to use in this example.

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Answer:With a 10 percent risk-free rate per period in Example 7.6, the expected value

of the S&P 500 in the next period if investors are risk neutral is 110 percent of the current

value, $1275.Hence, the risk-neutral probability that makes the expected future value of

the underlying asset 110 percent of today’s value solves

	$1,275(1.1) $1,500$750(1 )
Thus, .87.

Note from Example 7.8 that .87 is the risk-neutral probability, not the actual

probability, of the good state occurring, which remains unspecified. The risk-neutral

probability is simply a number consistent with $1,275 as the current value of the under-

lying asset and with the assumption that investors are risk neutral, an assumption that

may not be true.

The ability to form risk-free investments by having a long position in the tracking

portfolio and a short position in the derivative, or vice versa, is what makes it possi-

ble to ignore the true probabilities of the up and down state occurring. This is a sub-

tle point. In essence, we are pretending to be in a world that we are not in—a world

where all assets are expected to appreciate at the risk-free rate. To do this, throw away

the true probabilities of up and down moves and replace them with up and down prob-

abilities that make future values along the binomial tree consistent with risk-neutral

preferences.

Step 2: Probability-Weight the Derivative’s Future Values and Sum the Weighted

Value.Once having computed the risk-neutral probabilities for the underlying asset,

apply these same probabilities to the future outcomes of the value of the derivative to

obtain its risk-neutral expected future value. This is its expected futurevalue, assum-

ing that everyone is risk neutral, which is not the same as the derivative’s true expected

value.

Step 3: Discount at the Risk-Free Rate.Discounting the risk-neutral expected value

at the risk-free rate gives the no-arbitrage present value of the derivative. By dis-

counting, we mean that we divide a number by the sum of 1 plus an interest rate (or

1 plus a rate of return). In this particular case, the number we discount is the deriva-

tive’s expected future value, assuming everyone is risk neutral. The discount rate used

here is the risk-free rate, meaning that we simply divide by the sum of 1 and the risk-

free rate. It generally shrinks that number we are dividing by, reflecting the fact that

money today has earning power.

We will have much more to say about why we discount and how we discount in

the next part of the text, particularly in Chapter 9. We have already been discounting

in this part of the text without specifically referring to it. For example, the no-arbitrage

value of the forward contract (see Result 7.1) requires that we discount the price we

pay, K,at the risk-free rate—in this case, over multiple periods. Without elaborating

on this in too much detail, suffice it to say that discounting is a procedure for taking

a future value and turning it into a present value, the latter being a number that repre-

sents a fair current value for a payment or receipt of cash in the future. Hence, the dis-

counted value of Kin Result 7.1 turns a future payment of Kinto its present value,

which we will denote by PV(K).Similarly, the discounting of the risk-neutral expected

future value of a derivative is a way of turning that expected future value into a pres-

ent value.

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242Part IIValuing Financial Assets

Example 7.9 demonstrates how to value derivatives using the risk-neutral valua-

tion method.

Example 7.9:Using Risk-Neutral Probabilities to Value Derivatives

Apply the risk-neutral probabilities of .87 and .13 from Example 7.8 to the cash flows of the

derivative of Example 7.6 and discount the resulting risk-neutral expected value at the risk-

free rate.See Exhibit 7.9 for the numbers to use in the example.

Answer:.87($331.75) .13($106.75) yields a risk-neutral expected future value of

$302.50, which has a discounted value of $302.50/1.1 $275.

Relating Risk-Neutral Valuation to the Tracking Portfolio Method.It is indeed

remarkable, but not coincidental, as our earlier arguments suggested, that Examples 7.9

and 7.7 arrive at the same answer. Indeed, this will always be the case, as the follow-

ing result states.

Result 7.6

Valuation of a derivative based on no arbitrage between the tracking portfolio and the deriv-ative is identical to risk-neutral valuation of that derivative.24

It is worth repeating that one cannot take the true expected future value of a

derivative, discount it at the risk-free rate, and hope to obtain its true present value.

The true expected future value is based on the true probabilities, not the risk-neutral

probabilities.

AGeneral Formula forRisk-Neutral Probabilities.One determines risk-neutral

probabilities from the returns of the underlying asset at each of the binomial outcomes,

and not by the likelihood of each binomial outcome. The risk-neutral probabilities, ,

are those probabilities that make the “expected” return of an asset equal the risk-free

rate. That is, must solve

u (1 )d 1r

f

where

rrisk-free rate

f

u1per period rate of return of the underlying asset at the up node

d1per period rate of return of the underlying asset at the down node

When rearranged, this says

1r d
f
	(7.4)
u d

Risk-Neutral Probabilities and Zero Cost Forward and Futures Prices.One infers

risk-neutral probabilities from the terms and market values of traded financial instru-

ments. Because futures contracts are one class of popularly traded financial instruments

with known terms (the futures price) and known market values, it is often useful to

infer risk-neutral probabilities from them.

²⁴Another

related method of valuing derivatives is the state price valuation method,which was

derived from research in mathematical economics in the 1950s by Gerard Debreu, a 1986 Nobel Prize

winner, and Kenneth Arrow, a 1974 Nobel Prize winner. Since a state price is the risk-neutral probability

times the risk-free rate, the state price valuation technique yields the same answers for derivatives as the

two other methods discussed earlier.

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Chapter 7

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For example, corporations often enter into derivative contracts that involve options

on real assets. Indeed, in the next section, we will consider a hypothetical case involv-

ing the option to buy jet airplanes. To value this option correctly, using data about the

underlying asset, the jet airplane, is a heroic task. For reasons beyond the scope of this

text, the future values postulated for an asset like a jet airplane must be adjusted in a

complex way to reflect maintenance costs on the plane, revenue from carrying pas-

sengers or renting the plane out, obsolescence, and so forth. In addition, the no-

arbitrage-based valuation relationship derived is hard to envision if one is required to

“sell short a jet airplane” to take advantage of an arbitrage.

In these instances, corporations often use forward and futures prices as inputs for

their derivative valuation models.²⁵Such forwards and futures, while derivatives them-

selves, can be used to value derivatives for which market prices are harder to come by,

like jet airplane options, without any of the complications alluded to above.

To use the prices of zero-cost forwards or futures to obtain risk-neutral probabilities,

it is necessary to slightly modify the risk-neutral valuation formulas developed above. We

begin with futures and later argue that forwards should satisfy the same formula.

Recall that futures prices are set so that the contract has zero fair market value.

The amount earned on a futures contract over a period is the change in the futures

price. This profit is marked to market in that the cash from this profit is deposited in—

or, in case of a loss, taken from—one’s margin account at a futures brokerage firm.

Thus, a current futures price of F, which can appreciate in the next period to Fin the

u

up state or depreciate to Fin the down state, corresponds to margin cash inflow F F

du

if the up state occurs, and a negative number, F F(a cash outflow of F F) if the

dd

down state occurs. (We omit time subscripts here to simplify notation.) In a risk-neutral

setting with as the risk-neutral probability of the up state, the expected cash received

at the end of the period is

(F F) (1 )(F F)

ud

The investor in futures spends no money to receive this expected amount of cash. Thus,

in a risk-neutral world, the zero cost of entering into the contract should equal the dis-

counted²⁶expected cash received at the end of the period; that is

(F F)(1 )(F F)

0^ud

(1r)

f

Rearranging this equation implies:

Result 7.7

The no-arbitrage futures price is the same as a weighted average of the expected futuresprices at the end of the period, where the weights are the risk-neutral probabilities; that is

F F(1 )F	(7.5)
ud

If the end of the period is the maturity date of the futures contract, then FS

uu

and FS, where Sand S, respectively, are the spot prices underlying the futures

ddud

contract. Substituting Sand Sinto the last equation implies that for this special case

ud

F S(1 )S

ud

We can generalize this as follows:

25

Indeed, forward contracts to buy airplanes are fairly common.

²⁶The

discount rate does not matter here. With a zero-cost contract, expected future profit has to

be zero.

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Result 7.8

The no-arbitrage futures price is the risk-neutral expected future spot price at the maturityof the futures contract.

Result 7.8 is a general result that holds in both a multiperiod and single-period setting.

For example, a futures contract in January 2002, which is two years from maturity, has

a January 2002 futures price equal to the risk-neutral expected spot price in January

2004.

Example 7.10 illustrates how futures prices relate to risk-neutral probabilities.

Example 7.10:Using Risk-Neutral Probabilities to Obtain Futures Prices

Apply the risk-neutral probabilities of .87 and .13 from Example 7.8 to derive the futures

price of the S&P 500.Exhibit 7.10, which modifies a part of Exhibit 7.9, should aid in this

calculation.

Answer:Using Result 7.8, the S&P futures price is $1,402.50 .87($1,500) .13($750).

Note that, consistent with Result 7.2, this is the same as $1,275(1.1), which is the future

value of the spot price from investing it at the 10 percent risk-free rate of interest.

It is also possible to rearrange equation (7.5) to identify the risk-neutral proba-

bilities, and 1 from futures prices. This yields

F F
d
	(7.6)
F F
ud

EXHIBIT7.10Futures Prices

Underlying Security: S&P 500

$1,500 (Up State)

$1,275

$750 (Down State)

Risk-Free Security with 10% Return

$1.10 (Up State)

$1	$1.10 (Down State)

Derivative: Futures Cash

$1,500 – F (Up State)

?	$750 – F (Down State)

Grinblatt506Titman: Financial	II. Valuing Financial Assets	7. Pricing Derivatives	© The McGraw506Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 7

Pricing Derivatives

245

Example 7.11 provides a numerical illustration of this.

Example 7.11:Using Futures Prices to Determine Risk-Neutral Probabilities

If futures prices for Boeing 777 airplanes can appreciate by 10 percent (up state) or depre-

ciate by 10 percent (down state), compute the risk-neutral probabilities for the up and down

states.

Answer:In the up state, F1.1F;in the down state, F.9F.Thus, applying equation

ud

F .9F

,

(7.6), the risk-neutral probability for the up state, .5making 1 .5

1.1F .9F

as well.

Earlier in this chapter, we mentioned that futures and forward prices are essentially

the same, with the notable exception of long-term interest rate contracts. Hence, the

evolution of forward prices for zero-cost contracts could be used just as easily to com-

pute risk-neutral probabilities. This would be important to consider in Example 7.11

because forward prices for airplanes exist, but futures prices do not.

In applying equations (7.5) and (7.6) to forwards rather than futures, it is impor-

tant to distinguish the evolution of forward prices on new zero-cost contracts from the

evolution of the value of a forward contract. For example, in January 2003 American

Airlines may enter into a forward contract with Boeing to buy Boeing 777s one year

hence (that is, January 2004), at a prespecified price. Most likely, this prespecified price

is set so that the contract has zero up-front cost to American. However, as prices of

777s increase or decrease over the next month (February 2003), the value of the for-

ward contract becomes positive or negative because the old contract has the same for-

ward price.Equations (7.5) and (7.6) would not apply to the value of this contract.

Instead, these equations compare the forward price for 777s at the beginning of the

month with subsequent forward prices for new contracts to buy 777s at the maturity of

the original contract,that is, new contracts in subsequent months that mature in Janu-

ary 2004. Such new contracts would be zero-cost contracts.

It is also possible to use the binomial evolution of the value,not the forward price

of the old American Airlines forward contract to determine the risk-neutral probabili-

ties. However, because this contract sometimes has positive and sometimes negative

value, one needs to use equation (7.4) for this computation.

It is generally impossible to observe a binomial path for the value of a single for-

ward contract when that contract is not actively traded among investors (and it usually

isn’t). In contrast, parameters that describe the path of forward prices for new zero-cost

contracts are usually easier to observe and estimate.