Introduction to Corporate Finance

Ross et al.: Fundamentals

of Corporate Finance, Sixth

Edition, Alternate Edition

VIII. Topics in Corporate

Finance

24. Option Valuation © The McGraw−Hill^841
    Companies, 2002

A question that sometimes comes up concerns the probabilities N(d 1 ) and N(d 2 ). Just

what are they the probabilities of? In other words, how do we interpret them? The an-

swer is that they don’t really correspond to anything in the real world. We mention this

because there is a common misconception about N(d 2 ) in particular. It is frequently

thought to be the probability that the stock price will exceed the strike price on the ex-

piration day, which is also the probability that a call option will finish in the money. Un-

fortunately, that’s not correct, at least not unless the expected return on the stock is equal

to the risk-free rate.

Tables such as Table 24.3 are the traditional means of looking up “z” values, but they

have been mostly replaced by computers. They are not as accurate because of rounding,

and they also have only a limited number of values. Our nearby Spreadsheet Strategies

box shows how to calculate Black-Scholes call option prices using a spreadsheet. Be-

cause this is so much easier and more accurate, we will do all the calculations in the rest

of this chapter using computers instead of tables.

Put Option Valuation

Our examples thus far have focused only on call options. Only a little extra work is

needed to value put options. Basically, we just pretend that a put option is a call option

and use the Black-Scholes formula to value it. We then use the put-call parity (PCP) con-

dition to solve for the put value. To see how this works, suppose we have the following:

S$40

E$40

816 PART EIGHT Topics in Corporate Finance

SPREADSHEET STRATEGIES

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Stock =65 d1 =0.4952 N(d1) =0.6898

Strike =60

Sigma =0.5 d2 =0.2452 N(d2) =0.5968

Time =0.25

Rate =0.05

XYZ stock has a price of $65 and an annual return standard deviation of 50%. The riskless

interest rate is 5%. Calculate call and put option prices with a strike of $60 and a 3-month

time to expiration.

Formula entered in E8 is =(LN(B8/B9)+(B12+0.5*B10^2)*B11)/(B10*SQRT(B11))

Formula entered in E10 is =E8–B10*SQRT(B11)

Formula entered in H8 is =NORMSDIST(E8)

Formula entered in K10 is =NORMSDIST(E10)

Formula entered in K14 is =B8*H8–B9*EXP(–B12*B11)*H10

Formula entered in K16 is =B9*EXP(–B12*B11)+K14–B8

ABCDEFGHIJK

Using a spreadsheet to calculate Black-Scholes option prices

Call = Stock x N(d1) – Strike x exp(– Rate x Time) x N(d2) = $9.47

Put = Strike x exp(– Rate x Time) + Call – Stock = $3.72