8.6Black-Scholes Valuation

Up to this point, we have valued derivatives using discrete modelsof stock prices,

which consider only a finite number of future outcomes for the stock price and only a

finite number of points in time. We now turn our attention to continuous-time models.

These models allow for an infinite number of stock price outcomes and they can

describe the distribution of stock and option prices at any point in time.

Grinblatt574Titman: Financial	II. Valuing Financial Assets	8. Options	© The McGraw574Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 8

Options

279

Black-Scholes Formula

Chapter 7 noted that if time is divided into large numbers of short periods, a binomial

process for stock prices can approximate the continuous time lognormal process for

these prices. Here, particular values are selected for uand dwhich are related to the

standard deviation of the stock return, which also is known as the stock’s volatility.

The limiting case where the time periods are infinitesimally small is one where the

binomial formula developed in the last section converges to an equation known as the

Black-Scholes formula:

Result	8.8	(The Black-Scholes formula.) If a stock that pays no dividends before the expiration of an
		option has a return that is lognormally distributed, can be continuously traded in friction-
		less markets, and has a constant variance, then, for a constant risk-free rate,16
		the value of
		a European call option on that stock with a strike price of KandTyears to expiration is
		given by

cSN(d) PV(K)N(d T)	(8.3)
0011

where

d 1 InS 0/PV(K) T

T2

The Greek letter is the annualized standard deviation of the natural logarithm of the stock

return, ln( ) represents the natural logarithm, and N(z) is the probability that a normally

distributed variable with a mean of zero and variance of 1 is less than z.

The Black-Scholes formula provides no-arbitrage prices for European call options

and American call options on underlying securities with no cash dividends until expi-

ration (because such options should have the same values as European call options with

the same features). The Black-Scholes formula also is easily extended to price Euro-

pean puts (see exercise 8.2). Finally, the formula reasonably approximates the values

of more complex options. For example, to price a call option on a stock that pays div-

idends, one can calculate the Black-Scholes values of options expiring at each of the

ex-dividend dates and those expiring at the true expiration date. The largest of these

option values is a quick and often accurate approximation of the value obtained from

more sophisticated option pricing models. This largest value is known as the pseudo-

American valueof the call option.

Example 8.8 illustrates how to use a normal distribution table to compute Black-

Scholes values.

Example 8.8:Computing Black-Scholes Warrant Values forChrysler

Use the normal distribution table (Table A.5 in Appendix A) at the end of the book (or a

spreadsheet function like NORMSDIST in Microsoft Excel) to calculate the fair market value

of the Chrysler warrants described at the beginning of this chapter.Assume that at the time

¹⁶The

formula also holds if the variance and risk-free rate can change in a predictable way. In the

former case, use the average volatility (the square root of the variance) over the life of the option. In the

latter case, use the yield associated with the zero-coupon bond maturing at the expiration date of the

option. If the risk-free rate or the volatility changes in an unpredictable way, and is not perfectly

correlated with the stock price, no risk-free hedge between the stock and the option exists. Additional

securities (such as long- and short-term bonds or other options on the same stock) may be needed to

generate this hedge and the Black-Scholes formula would have to be modified accordingly. However, for

most short-term options, the modification to the option’s value due to unpredictable changes in interest

rates is a negligible one.

Grinblatt577Titman: Financial	II. Valuing Financial Assets	8. Options	© The McGraw577Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

280Part IIValuing Financial Assets

Chrysler is making a bid of $22 for the warrants (1) Chrysler stock is worth $28 a share, (2)

Chrysler’s stock return volatility is 30 percent per year (which is 0.3 in decimal form), (3)

the risk-free rate is 10 percent per year (annually compounded), (4) the warrants are Amer-

ican options, but no dividends are expected to be paid over the life of the option, and there

is no dilution effect, so it is possible to value the warrants as European options, (5) the

options expire in exactly seven years, and (6) the strike price is $13.

Answer:The present value of the strike price is 13/1.1⁷$6.67.The volatility times the

square root of time to maturity is the product of 0.3 and the square root of 7, or 0.79.The

Black-Scholes equation says that the value of the call is therefore

c$28N(d) $6.67N(d .79)

011

where

ln($28/$6.67)1

d (.79)2.21

¹.792

The normal distribution values for d(2.21) and d .79 (1.42) in Table A.5 in Appendix

11

A indicates that the probability of a standard normal variable being less than 2.21 is .9864,

and the probability of it being less than 1.42 is .9222.Thus, the no-arbitrage call value is

$28(.9864) $6.67(.9222)$21.47

The estimate of $21.47 in this example is less than Chrysler’s bid of $22.00, which

would indicate, if the assumptions used are correct, that Chrysler got a slightly poor

deal by winning the auction. While the example ignored the possibility of dividends,

the existence of dividends would make the value of this option even lower.

Dividends and the Black-Scholes Model

It is especially important with continuous-time modeling to model the path of the stock

price when stripped of its dividend rights if dividends are paid before the expiration

date of the option. For example, the Black-Scholes Model assumes that the logarithm

of the return on the underlying stock is normally distributed. This means that in any

finite amount of time, the stock price may be close to zero, albeit with low probabil-

ity. Subtracting a finite dividend from this low value would result in a negative ex-

dividend stock price. Hence, unless one is willing to describe the dividend for each of

an infinite number of stock price outcomes in this type of continuous price model, it

is important to model the process for the stock stripped of its right to the dividend.

This was illustrated with the binomial model earlier, but it is more imperative here.¹⁷