Gammas of Option Values Are Positive

Plus signs () are always associated with gammas of both calls and puts because change in delta is positively correlated with change in the price of the underlying. Table 4-2 illustrates this: An increase in stock price causes an increase in the deltas of both the 100 Call and the 100 Put. The delta of the 100 Call, for example, increases from 0.55 to 0.58, an amount exactly equal to the gamma of 0.03. Table 4-3, on the other hand, shows that a decrease in the stock price causes a decrease in the deltas of both options.

The delta of the 100 Call decreases from 0.55 to 0.51, and the delta of the 100 Put decreases from 0.46 to 0.50. These are examples of the change in delta not being exactly equal to the change in the gamma.This is a positive correlation: Stock price up, delta up (Table 4-2), and stock price down, delta down (Table 4-3). Note that the gammas of calls and puts with the same underlying, same strike, and same expiration are nearly equal.

In Tables 4-2 and 4-3, they appear equal because they are calculated to only two decimal places. Gammas of same-strike calls and puts are nearly equal mowing to put-call parity, a corollary of which is that the sum of the absolute values of the call delta and put delta equals 1.00 (or very nearly 1.00), assuming that the call and put have the same underlying, same strike, and same expiration.

This means that if the absolute value of the delta of the call increases, then the absolute value of the put must decrease by an equal amount. Otherwise, the sum of the  absolute values of the deltas would no longer equal 1.00. Since the deltas of the call and put change by nearly the same amount, their gammas also must be nearly equal because gamma is the change in delta.

Vega:Volatility will be discussed in depth in Chapter 7, but vega will be defined here. Vega is the change in option value that results from a one percentage point change in the volatility assumption, assuming that other factors remain constant. Mathematically, vega is the first derivative of option price with respect to change in volatility. Since first derivatives are theoretically instantaneous rates of change, and since vega estimates the impact of a one percentage point change, there frequently will be rounding errors.

Vega answers this question: If volatility changes by one percentage point, how much do I make or lose? Vegas of Option Values Are Positive:Vegas of both call values and put values are always positive because changes in option value are positively correlated with changes in volatility; that is, volatility up, option value up, and volatility down, option value down.

Another result of the put-call parity concept is that vegas of calls and puts with the same underlying, strike, and expiration are equal.According to put-call parity, there is a quantifiable relationship between the price of the underlying instrument and the prices of calls and puts with the same strike and same expiration. In order for the
put-call parity relationship to be maintained when the call value increases, the same-strike put must rise by an identical amount with a given change in volatility.

Thus vegas of calls and puts with the same underlying, strike, and expiration must be equal.Vega Is Not Greek:Readers familiar with the Greek alphabet may note that vega is not a Greek letter. The derivation of this term’s use remains murky, but one belief postulates that option traders wanted a short word beginning with a v (for volatility) that sounds like delta, gamma, and theta.Exactly who coined the term and when it was first used are not known.

Some mathematicians and traders use another Greek letter such as kappa or lambda instead of vega. Why no uniform terminology exists to represent a concept as important as volatility is one of the many quirks of the options business.Theta:Theta is an estimate of the change in option value given a one-unit change in time to expiration, assuming that other factors remain constant.

Theta answers this question: If time passes, how much do I make or lose? Table 4-5 illustrates what happens to call and put values when days to expiration are reduced from 60 to 53. The call value decreases from 5.19 to 4.86, a change equal to the theta of 0.33.The put value decreases from 4.59 to 4.33, a change also equal to its theta of 0.26.

Although the amount of change in option values exactly equals the thetas in this example, slight differences may occur owing to rounding, especially when the calculation goes out to several decimal points.The definition of theta raises an important question: What is one unit of time? Mathematically, theta is the first derivative of option value with respect to change in time to expiration. This means, theoretically,that one unit of time is instantaneous.