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^843
    Companies, 2002

you can, but not if it is European style. More generally, it often pays to exercise a put

option once it is well into the money because any additional potential gains are limited,

so American-style exercise is valuable.

What about call options? Here the answer is a little more encouraging. As long as we

stick to non-dividend-paying stocks, it will never be optimal to exercise a call option

early. Again, the reason is not complicated. A call option is worth more alive than dead,

meaning you would always be better off selling the option than exercising it. In other

words, for a call option, the exercise style is irrelevant.

Here is a challenge for the more mathematically inclined among you. We have a for-

mula for a European-style put option. What about for an American-style put? Despite a

great deal of effort, this problem has never been solved, so no formula is known. Just to

be clear, we have numerical procedures for valuing put options, but no explicit formula.

Call us if you figure one out.

MORE ON BLACK-SCHOLES

In this section, we take a closer look at the inputs into the option pricing formula and

their effects on option values. Table 24.4 summarizes the inputs and their impacts (pos-

itive or negative) on option values. In the table, a plus sign means that increasing the in-

put increases the option’s value and vice versa.

Table 24.4 also indicates that four of the five effects have common names. For fairly

obvious reasons given their names, these effects are collectively called the “greeks.” We

will discuss them in the next several sections. In some cases, the calculations can be

fairly involved, but the good news is that options calculators are widely available on the

Web. See our nearby Work the Webbox for one example.

Varying the Stock Price

The effect that the stock price has on put and call values is pretty obvious. Increasing the

stock price increases call values and decreases put values. However, the strength of the

CONCEPT QUESTIONS

24.2a What are the five factors that determine an option’s value?

24.2bWhich is worth more, an American-style put or a European-style put? Why?

818 PART EIGHT Topics in Corporate Finance

24.3

TABLE 24.4

Five Inputs Determining

the Value of an

American Option on a

Non-Dividend-Paying

Stock

Impact on Option Price from

an Increase in Input

Input Call Options Put Options Common Name

Stock price (S) 
Delta

Strike price (E) 


Time to expiration (t) Theta

Standard deviation

of return on stock ( ) Vega

Risk-free rate (R) 
Rho

Note: The effect of increasing the time to maturity is positive for an American put option, but the

impact is ambiguous for a European put.