tailieunhanh - A Logical Approach to Actuarial Mathematics_3

Tham khảo tài liệu 'a logical approach to actuarial mathematics_3', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 6 Gamma and theta It should be apparent after reading the previous chapter that delta is an indispensable tool for understanding an option s behaviour. But because an option s delta changes continually with the 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. 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 . 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 through expiration. 54 Part 1 Options fundamentals The first situation above occasionally occurs but most options contracts expire well out-of or in-the-money. Nevertheless several points about gamma are illustrated. In any contract month gamma is the highest with the at-the-money options and it decreases as