Option-Pricing Formulas

Theoretical option values can be calculated using one of a few mathematical formulas. Two professors at the University of Chicago, Fisher Black and Myron Scholes, developed the first formula, commonly known as the Black-Scholes option-pricing model in 1973.This formula involves advanced calculus.Subsequently,other mathematicians created additional formulas that are generically known as binomial option-pricing models.

The mathematics behind these formulas are beyond the scope of this book but can be found in books by J. C. Cox and S. A. Ross, M. Rubenstein, and J. Hull. The Op-Eval Pro software that accompanies this text allows the user to choose between four formulas, the Black-Scholes model and three binomial models. For educational purposes, you should compare option values generated by the different formulas with the same inputs.

In most cases, however, you will find that the results vary little. Unless otherwise indicated, option values in exhibits in this book are calculated with a standard Black-Scholes model and presented in decimals rounded to the second place.The following examples review how changes in the inputs to the option-pricing formula affect call and put values. Each example is static; only one factor changes at a time, while all other factors remain constant.

Some dynamic examples will be presented later in this chapter.Call Values and Stock Prices Table 3-3 illustrates how call values change when the price of the underlying and time to expiration change. Table 3-3 contains 11 rows and 7 columns.The columns indicate different days prior to expiration. By looking up and down the columns and across the rows, you can see how changes in underlying price or time to expiration or both cause changes in an option’s theoretical value.

The stock price is 100, the call has 90 days to expiration, and the theoretical value of the 100 Call is 6.53. If the stock price rises by 1 to 101 (row 7, column1) and the other factors are unchanged, the theoretical value of the 100 Call increases by 0.58 to 7.11. If the stock price decreases by 1 to 99, the call value decreases by 0.54 to 5.99. In both cases, the call price changes less than the stock price.

Looking anywhere on the table, this relationship between stock price and option value always holds. Assuming that factors other than price remain constant,with any time left until expiration, an option’s theoretical value always changes less than one-for-one with a change in the stock price.Furthermore, the ratio of option-price change to stock-price change varies with the stock price and with time.

For example, when the stock price rises by 1 from 97 to 98 at 45 days, the call value rises by 0.45 from 3.04 to 3.49, or approximately 45 percent of the stock-price change. In another situation, when the stock rises from 102 to 103 at 60 days, the call value rises by 0.63 from 6.42 to 7.05, or approximately 63 percent of the stockprice change.Two conclusions can be drawn from Table 3-3.

First, as stated earlier,option prices generally change less than stock prices. Second,the amount by which option prices change depends on the time to expiration and the relationship of the stock price to the option’s strike price.Figure 3-1 shows how call values change as stock prices change.The upper line (curved) contains call values at 90 days to expiration,like column 1 in Table 3-3.

The middle line (curved) contains call values at 45 days to expiration, like column 4 in Table 3-3; and the lower line (two straight sections) contains call values at expiration, like column 7 in Table 3-3. Figure 3-1 illustrates the same two concepts from Table 3-3—that call values are directly correlated with stock prices and that the level of correlation varies depending on the relationship of the stock price to the strike price.

Call values are near zero when the stock price is significantly below the strike price. Call values rise gradually at first as the stock price moves up toward the strike price. The values then rise faster and faster as the stock price reaches and then moves above the strike price.Finally, as the stock price soars significantly above the strike price, the change in call value approaches a one-for-one relationship with change in stock price. In theory, however, call values never change exactly one for one with stock prices because, in theory, the value of the call always will contain at least a slight time premium.