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^845
    Companies, 2002

Call option delta N(d 1 )

Put option delta N(d 1 ) 
 1

The “N(d 1 )” that we need to calculate these deltas is the same one we used to calculate

option values, so we already know how to do it. Remember that N(d 1 ) is a probability,

so its value ranges somewhere between zero and one.

For a small change in the stock price, the change in an option’s price is approxi-

mately equal to its delta multiplied by the change in the stock price:

Change in option value 
Delta Change in stock value

To illustrate this, suppose we are given the following:

S$120

E$100

R8% per year, continuously compounded

80% per year

t6 months

Using the Black-Scholes formula, the value of a call option is $37.72. The delta (N(d 1 ))

is .75, which tells us that if the stock price changes by, say, $1, the option’s value will

change in the same direction by $.75.

We can check this directly by changing the stock price to $121 and recalculating the

option value. If we do this, the new value of the call is $38.47, an increase of $.75, so

the approximation is pretty accurate (it is off in the third decimal point).

If we price a put option using these same inputs, the value is $13.94. The delta is

.75
1, or 
.25. If we increase the stock price to $121, the new put value is 13.70,

a change of 
.24, so, again, the approximation is fairly accurate as long as we stick to

relatively small changes.

820 PART EIGHT Topics in Corporate Finance

FIGURE 24.1

80 90 100 110 120

25

20

15

10

5

0

Stock Price ($)

Put Price Call Price

Input values:

E= $100

t= 3 months

R= 5%

= 25%

Option Price ($)

Put and Call Option Prices