## 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 standard 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 financial 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:

R = 5% per year, continuously compounded t = 6 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?

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

VIII. Topics in Corporate Finance

24. Option Valuation

### CHAPTER 24 Option Valuation

If we plug all this information into the Black-Scholes formula, we would be left with one unknown, the standard deviation (t). However, it's not possible to directly solve for t, so trial and error must be used. In other words, we just start plugging in values for t until we find one that produces the call price of \$4.59.

For a stock option, .50 is a good place to start. If you plug this in, you will see that the calculated call value is \$4.38, which is too low. Recall that option values increase as we increase t, so we might try .60. Now the option value is \$4.52, so we're getting close, but we're still low. At .65, the calculated value is \$4.61, which is just a little too high. After a little more work, we discover that the implied volatility is .64, or 64 percent.

The options calculator at www.numa.com makes it easy to calculate ISDs..

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