Annuities and Other Retirement Products: Designing the Payout Phase (Directions in Development)_3

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Annuities and Other Retirement Products: Designing the Payout Phase (Directions in Development)_3 6 Gamma and theta Gamma quantifies the It should be apparent after reading the previ- rate of change of the ous chapter that delta is an indispensable tool for delta with respect to a understanding an option’s behaviour. But because an option’s delta changes continually with the change in the underlying underlying, we need to be able to assess its own rate of change. Gamma quantifies the rate of change of the delta with respect to a change in the underlying. To understand gamma is to understand how quickly or slowly a delta can change. Suppose XYZ is trading at a price of 100, and there are just two hours until the front-month options contract expires. The typical daily range of XYZ is two points, so we expect it to be between 99 and 101 at the time of expiration. Now suppose that XYZ starts to move erratically, and for the next two hours it trades between 99 and 101. During this time, what is the delta of the expiring 100 call? If XYZ settles below 100, the 100 call will expire worthless, with a delta of zero. If XYZ settles above 100, the call will close at parity, with a delta of 1.00. During these last two hours it would have been pointless to calculate the delta because it is changing so rapidly. This rapid and most extreme change of delta, however, is an example of the highest possible gamma that an option can have. If we consider the out-of-the-money options in the same contract month, such as the 105 calls and the 95 puts, we can be almost certain that they will expire worthless. Their deltas are zero and will not change. They have no gamma. Likewise in-the-money, parity options such as the 90 calls and the 110 puts have no gamma because their deltas will remain at 1.00 through expiration. O ptions fundamentals 54 P art 1  The first situation above occasionally occurs, but most options con- tracts expire well out-of or in-the-money. Nevertheless, several points about gamma are illustrated. In any contract month, gamma is the high- est with the at-the-money options, and it decreases as the strike prices become more distant from the money, whether they are in-the-money or out-of-the-money. As a contract month approaches expiration, the gammas of both the at- the-money options, and the options near-the-money, increase. The effect of time decay, however, causes the gammas of the far out-of-the-money and far in-the-money options to approach zero. Generally speaking, how- ever, time decay least affects the gammas of options in the 0.10 and 0.90 delta ranges. This all becomes complicated, of course, by the fact that deltas change with time. You should simply remember that as time passes, the nearer an option is to the underlying, the more its gamma increases. Table 6.1 is a typical example of a set of options with deltas and gammas in one contract month: 90 days until expiration; implied volatility at 30 per cent; interest rate at 3 per cent; options multiplier at $50, so multiply call and put values times $50. Table 6.1 December Corn at $3.80 Strike Call value Call delta Put value Put delta Gamma × $50 × $50 31/4 320 63.00 0.90 0.10 0.003 340 47.00 0.80 7 0.20 0.005 337/8 360 0.67 14 0.33 0.007 380 22.00 0.53 22 0.47 0.008 400 15.00 0.40 35 0.60 0.007 85/8 481/2 0.27 0.73 0.006 420 51/2 651/4 440 0.19 0.81 0.005 The gamma-delta calculation is a matter of simple addition or subtraction. Here, the December 400 call with a 0.40 delta has a gamma of 0.007. This means that if the December futures contract moves up one point, from 380 to 381, the delta of the call will increase to 0.407, rounded to 0.41. G amma and theta 6 55  If the futures contract moves down one point, the delta of the same call will d ...