## R 92cs

Gamma defines the curvature of the derivative price curve. In Figure 1,3,2 gamma is the second derivative of the option price curve at the point under consideration, Another useful number is theta (©) Theta is defined as

Theta measures the time change in the value of a derivative security. Referring again to Figure 13 2, if time is increased, the option curve will shift to the right, Theta measures the magnitude of this shift

These parameters are sufficient to estimate the change in value of a derivative security over small time periods, and hence they can be used to define appropriate hedging strategies, In particular, using 5/, <55, and St to represent small changes in /, 5, and /, we have

Sf ^ A SS + x (SS)2 + © x <5/ as a first-order approximation to Sf 2

Example 13.3 (Call price estimation) Consider a call option with S = 43, K = 40, a = .20, r = 10, and a time to expiration of J — t = 6 months — 5 The Black-Scholes formula yields C = $5 56 We can also calculate that A = ,825, T = .14,3, and © = —6.127. (See Exercise 7.)

Now suppose that in two weeks the stock price increases to $44. We have ¿5=1 and St = 1/26; therefore the price of the call at that time is approximately

The actual value of the call at the later date according to the Black-Scholes formula is C = $6 23

-Recall that is proportional to s/St, so we must include the (<55)- term

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