Variation Of Gamma For At The Money Option
Previously, in most examples concerning 6 it was said that a $1 change in stock price would cause the option price to change by 6. Although in these examples this is a useful interpretation of 6, it is not totally correct. This is because 6 changes even if the stock price changes by a small amount and also with the passage of time. Thus, while the stock price changes by $1, 6 takes many different values. This means that it is not correct to calculate the option price change, caused by a $1 change in stock price, using only one 6-value. The right way to deal with 6 is, if the stock price changes by a small1 amount, then the option price changes by 6 times this amount. Since 6 changes if the stock price changes, it would be nice to have a unit that measures delta's sensitivity to stock price movements. This unit is called 'gamma' and is indicated by the Greek letter 7. Mathematically, 7 is the derivative of 6 with respect to the stock price. If gamma is small, stock price movements only cause small changes in delta. However, if gamma is large, delta is highly sensitive to stock price changes. So, an investor who owns an option with a large gamma has to adjust the number of stocks in
his portfolio frequently to keep this portfolio deltaneutral. Both call and put options have a positive gamma and, as will be shown shortly, these gammas appear to be the same for both European call and put options. The fact that gamma is positive for both the call and the put option is logical. After all, the delta of a call option is an increasing function in the stock price, and increases from 0 to 1 as the stock price increases from 0 to infinity; the delta of a put option increases from —1 to 0 as the stock price increases from 0 to infinity. Before the formula of gamma is given for the European call and put option, note that gamma is the second derivative of the option price with respect to the stock price. Indeed, delta is the first derivative of the option price with respect to the stock price, and because gamma is the derivative of delta with respect to the stock price it is clear that this is true. For a European call and put option, gamma is given by:
dS d20t N0(di) > 0 7put'European" d St = a (St)2" StaPffit > 0 (6:2)
where d1 is defined as in equation (2.3) and:
The above formulae again show that the gamma of a European call option is equal to the gamma of a European put option. Respectively, Figures 6.1 and 6.2 indicate the way in which gamma varies with the stock price and the time to maturity.
- Figure 6.1 Variation of gamma with stock price.
Figure 6.2 Variation of gamma with time to maturity.
Figure 6.2 Variation of gamma with time to maturity.
It is worth noting that gamma becomes very big if an at-the-money option is close to expiring. This is caused by the fact that small stock price changes heavily affect the probability that this option will expire in the money. Since, for options close to expiration, \6\ is approximately equal to this probability, small stock price changes heavily affect 6. This explains why at-the-money options close to maturity have big gammas.
Lastly, from a practical point of a view it is good to remember that the gamma of any option is largest when it is close to 'at the money'. This has the direct implication that prices of options close to 'at the money' are very sensitive to stock price movements and, therefore, need to be re-hedged frequently. Also, when the stock is close to the strike, options with a short time to maturity have higher gammas than options with a long time to maturity. So, the prices of at-the-money options with a short time to maturity are more sensitive to changes in the underlying stock and, therefore, need to be re-hedged more frequently than at-the-money options with a long time to maturity.
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