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^849
    Companies, 2002

in interest rates as they are to, say, changes in volatilities. An option’s sensitivity to in-

terest rate changes is called its rho.

There are a few other greeks in addition to the ones we have discussed, but we will

end our Greek lesson here. What we now discuss is a very important use of Black-

Scholes: the calculation of implied volatilities.

Implied Standard Deviations

Thus far, we have focused on using the Black-Scholes OPM to calculate option values,

but there is another, very important, use. Of the five factors that determine an option’s

value, four can be directly observed: the stock price, the strike price, the risk-free rate,

and the life of the option. Only the standard deviation must be estimated.

The standard deviation we use in the OPM is actually a prediction of what the stan-

dard deviation of the underlying asset’s return is going to be over the life of the option.

Often, we already know the value of an option because we observe its price in the fi-

nancial markets. In such cases, we can use the value of the option, along with the four

observable inputs, to back out a value for the standard deviation. When we solve for the

standard deviation this way, the result is called the implied standard deviation(ISD,

which some people pronounce as “iz-dee”), also known as the implied volatility.

To illustrate this calculation, suppose we are given the following:

S$12

E$8

R5% per year, continuously compounded

t6 months

We also know that the call option sells for $4.59. Based on this information, how

volatile is the stock expected to be over the next three months?

824 PART EIGHT Topics in Corporate Finance

FIGURE 24.4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Put Price

Call Price

15 16 17 18 19 20

9 8 7 6 5 4 3 2 1 0

Interest Rate (%)

Option Price ($)

Input values:

S= $100

E= $100

t= 3 months

= 25%

Options Prices and Interest Rates

rho
Measures the sensitivity
of an option’s value to a
change in the risk-free
rate.

For an option-oriented
web site focusing on
volatilities, visit
http://www.ivolatility.com.

implied standard
deviation
An estimate of the future
standard deviation of the
return on an asset
obtained from the Black-
Scholes OPM.

Slide 24.27Figure 24.4