Table 4-7 shows that gammas are the same for calls and puts with the same strike, same days to expiration, and same underlying. This equality is a result of put-call parity, one part of which states that the sum of the absolute values of the call and put with the same strike price and expiration date must total 1.00. Therefore, if the absolute value of the delta of the call (or put) rises or falls, then the absolute value of the put (or call) must fall or rise an equal amount so that the sum of the two remains at 1.00.

Rows A, B, C and D in Table 4-7 and rows A, C, D and F in Table 4-8 and Figures 4-5A and 4-5C show that gammas of in-themoney and out-of-the-money options increase only slightly until about 30 days before expiration, and then they decrease to zero. Since gamma is the change in delta, a slightly changing gamma is saying that deltas change at a nearly constant rate until the last month, when they change less.

For an in-the-money call with four or five months to expiration, for example, a $1.00 stock-price rise might increase the delta from 0.75 to0.77. This is a gamma of 0.02. With only one week to expiration,however, the same $1.00 stock-price rise would raise the delta only from 0.75 to 0.76, which indicates a gamma of 0.01. Similarly, for outof-the-money options, a $1.00 stock-price rise at 90 days might cause a delta to rise from 0.33 to 0.35,

whereas the same $1.00 price rise at 10 days might cause the delta to rise only from 0.35 to 0.36.Rows E and F of Table 4-7 and rows B and E of Table 4-8 and Figure 4-5B show that gammas of at-the-money options behave differently from gammas of in-the-money and out-of-the-money options.Gammas of at-the-money options are very small and nearly constant,rising only slightly, until about one month before expiration.

Then they rise dramatically until immediately before expiration, at which point they drop to zero. You can understand why gammas respond this way by considering option mechanics at expiration. Traders exercise in-themoney options, converting them into stock positions. Therefore, their deltas at expiration are 1.00. Out-of-the-money options, however, expire worthless, so their deltas are zero.

Now consider the change in delta as an option moves from slightly out of the money to in the money immediately at expiration. The absolute value of its delta rises instantaneously from near zero to near 1.00. This is a very large gamma, nearly infinite, as a 2 cent stock price rise from 99.99 to 100.01 changes a call’s delta from 0.00 to 1.00 and a put’s delta from 1.00 to 0.00.Gammas and Volatility:Figures 4-6A and 4-6C illustrate the impact of volatility on gamma for in-the-money and out-of-the-money options.

At low levels of volatility—from roughly 10 to 20 percent—gammas rise with volatility. As volatility rises above 30 percent, however, gamma decreases as volatility rises.This happens because, as discussed earlier, as volatility rises, the absolute value of delta changes toward 0.50. With absolute deltas closer to 0.50, the change in delta is smaller, and that is a smaller gamma.Figure 4-6B shows that gammas of at-the-money options change differently as volatility changes.

Since the absolute value of deltas of atthe-money options is near 0.50 regardless of the level of volatility,rising volatility does not change the gamma. However, at very low levels of volatility, gammas of at-the-money options rise dramatically. This quick rise happens because low volatility means a low standard deviation. While a $1.00 price change in a high-volatility stock might equate to a one-half-standard-deviation move or less, the same price change in a low-volatility stock might equate to a two-standard-deviation move.

Since deltas are related to the distance to the mean in standard deviation terms, an option that is two standard deviations in the money will have a delta the absolute value of which is much greater than an option that is only one-half of a standard deviation in the money. Such an option will have a high gamma because that stock-price change would cause the absolute delta to move from 0.50 to nearly 1.00.

Similar logic applies to out-of-the-money options and their deltas.How Vega Changes Tables 4-7 and 4-8 and Figure 4-7 show that vegas, the change in option value from a one percentage point change in volatility, are biggest when options are at the money. In any column, vegas are biggest when the 100 Call and 100 Put are at the money (rows E and F in Table 4-7 and rows B and E in Table 4-8).

At-the-money options have the largest vegas because a change in volatility has the biggest absolute impact on the price of at-the-money options. In Table 4-8 (column 1), for example, the increase in volatility from 25 to 50 percent causes the 90 Call to increase in price from 11.26 (row F) to 13.94 (row C), an increase of 2.68.