BLACK-SCHOLES OPTION PRICING MODEL Several models have been developed to determine the theoretical value

BLACK-SCHOLES OPTION PRICING MODEL Several models have been developed to determine the theoretical value

of an option. The most popular one was developed by Fischer Black and Myron Scholes in 1973 for valuing European call options on common stock. Recall that a European option is one that cannot be exercised prior to the expiration date.

Basically, the idea behind the arbitrage argument in deriving the option pricing model is that if the payoff from owning a call option can

be replicated by (1) purchasing the stock underlying the call option and (2) borrowing funds, then the price of the option will be (at most) the cost of creating the payoff replicating strategy.

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By imposing certain assumptions (to be discussed later) and using arbitrage arguments, the Black-Scholes option pricing model computes the fair (or theoretical) price of a European call option on a non-dividend- paying stock with the following formula:

C = SN(d

– rt

1 ) − Xe N(d 2 )

where ln SK +

--------------------------------------------------------- ( ⁄ ) ( r + 0.5 s d )t 1 = st

d 2 = d 1 − st ln = natural logarithm

C = call option price S = price of the underlying asset K = strike price r

= short-term risk-free rate

e = 2.718 (natural antilog of 1) t

= time remaining to the expiration date (measured as a fraction of

a year) s

= standard deviation of the value of the underlying asset N(.) = the cumulative probability density 3

Notice that five of the factors that we indicated in Chapter 8 that influence the price of an option are included in the formula. Anticipated cash dividends are not included because the model is for a non-dividend- paying stock. In the Black-Scholes option pricing model, the direction of the influence of each of these factors is the same as stated in Chapter 8. Four of the factors—strike price, price of underlying asset, time to expi- ration, and risk-free rate—are easily observed. The standard deviation of the price of the underlying asset must be estimated.

The option price derived from the Black-Scholes option pricing model is “fair” in the sense that if any other price existed, it would be possible to earn riskless arbitrage profits by taking an offsetting position in the underlying asset. That is, if the price of the call option in the mar- ket is higher than that derived from the Black-Scholes option pricing model, an investor could sell the call option and buy a certain quantity of

3 The value for N(.) is obtained from a normal distribution function that is tabulated in most statistics textbooks or from spreadsheets that have this built-in function.

APPENDIX

the underlying asset. If the reverse is true, that is, the market price of the call option is less than the “fair” price derived from the model, the investor could buy the call option and sell short a certain amount of the underly- ing asset. This process of hedging by taking a position in the underlying asset allows the investor to lock in the riskless arbitrage profit.

To illustrate the Black-Scholes option pricing formula, assume the following values:

Strike price

Time remaining to expiration = 183 days Stock price

Expected price volatility = standard deviation = 25% Risk-free rate

In terms of the values in the formula: S = $47

K = $45 t = 0.5 (183 days/365, rounded) s = 0.25 r = 0.10

Substituting these values into the Black-Scholes option pricing model, we get:

ln $47 $45 ( 2 ⁄ ) + [ 0.10 + ( 0.5 0.25 ) ]0.5

d 1 = ---------------------------------------------------------------------------------------------------- = 0.6172

d 2 = 0.6172 − 0.25 0.5 = 0.4404 From a normal distribution table:

N(0.6172) = 0.7315 and N(0.4404) = 0.6702

Then:

C = $47(0.7315) − $45(e –(0.10)(0.5) )(0.6702) = $5.69 Let’s look at what happens to the theoretical option price if the

expected price volatility is 40% rather than 25%. Then:

ln $47 $45 ( ⁄ ) + [ 0.10 +

d 1 = ---------------------------------------------------------------------------------------------------- = 0.4719

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d 2 = 0.4719 − 0.40 0.5 = 0.1891 From a normal distribution table:

N(0.4719) = 0.6815 and N(0.1891) = 0.5750

Then:

C = $47(0.6815) − $45(e –(0.10)(0.5) )(0.5750) = $7.42 Notice that the higher the assumed expected price volatility of the

underlying asset, the higher the price of a call option. Exhibit App.1 shows the option value as calculated from the Black- Scholes option pricing model for different assumptions concerning (1) the standard deviation, (2) the risk-free rate, and (3) the time remaining to expiration. Notice that the option price varies directly with all three variables. That is, (1) the lower (higher) the volatility, the lower (higher) the option price; (2) the lower (higher) the risk-free rate, the lower (higher) the option price; and, (3) the shorter (longer) the time remain- ing to expiration, the lower (higher) the option price. All of this agrees with what we stated in Chapter 8 about the effect of a change in one of the factors on the price of a call option.

Value of a Put Option How do we determine the value of a put option? There is a relationship

among the price of the underlying asset, the call option price, and the put option price. This relationship, called the put-call parity relationship, is given below for European options:

Put option price = Call option price + Present value of strike price

– Price of the underlying asset or, using the notation we used previously,

P – = C + Xe rt – S

If there are cash distributions on the underlying asset (e.g., dividends), these would be added to the right-hand side of this equation. The rela- tionship is approximately true for American options.

If this relationship does not hold, arbitrage opportunities exist. That is, portfolios consisting of long and short positions in the underlying asset and related options that provide an extra return with (practical) certainty will exist.

APPENDIX

EXHIBIT APP.1 Comparison of Black-Scholes Call Option Prices Varying One

Factor at a Time

Base case: Strike price = $45 Current stock price = $47 Time remaining to expiration = 183 days Risk-free rate = 10% Expected price volatility = standard deviation = 25%

Panel A: Holding All Factors Constant Except Expected Price Volatility

Panel B: Holding All Factors Constant Except for the Risk-Free Rate

Panel C: Holding All Factors Constant Except for the Time Remaining to Expiration

APPENDIX

EXHIBIT APP.2 Relation Between Call and Put Option Features and the Value of an

Option Relation to Call Relation to Put

Factor Description Option Value Option Value

S Value of the underlying asset, S Direct relation Inverse relation X Exercise price, X Inverse relation Direct relation

r Time value of money, r Direct relation Inverse relation s

Volatility of the value of the underly- Direct relation Direct relation ing asset, s

t Time to maturity, t Direct relation Direct relation

If we can calculate the fair value of a call option, the fair value of a put with the same strike price and expiration on the same stock can be calculated from the put-call parity relationship.