Question

Problem 2. You are given the following data assuming a Black-Scholes model. • S = $100 • 0 = 30% • p = 0.08 • S= 0 Suppose you sell a 95-strike put with 3 months to expiration. (i) (5 points) What is the gamma of this put option? (ii) (3 points) If the underlying were to decrease by $1 over the next short while, what effect would that have on this option's delta? (iii) (2 points) If all of the data above were the same except that the maturity of the put option was 30 years, what is the approximate value of the put’s gamma?

Answer 1

The delta of a call option is given by: A = N(D,) In(S/K)+(r +o/2), and N(x)= 1 pe? dt S: underlying K: strike T: maturity o: implied volatility (at the strike) r: interest rates where d, =- OvT 24 Since gamma is the rate of change of delta with respect to the underlying: I =* , differentiating Nd) w.r.t. S gives rebehna where (x) = warne
1)Now the Gamma is calculated using the data provided and we put it in excel we get Gamma equal to 0.023.


$95 Long Put Gamma 0.0270 0.0260 0.0250 0.0240 0.0230 0.0220 0.02.10 0.0200 0.0190 0.0180 96 97 98 99 100 101 102 103 104

Call Option Put Option Theoretical Price 9.8 2.944 Delta 0.709 -0.291 Gamma 0.023 0.023 Vega 0.17 0.17 Theta -0.042 -0.021 Rho 0.151 -0.079

2) If the underlying changes to 99$ the new Delta for the option is -0.0314 which is a movement of 0.023 or Gamma FROM THE Delta at 100$ .
3)If  the maturity is extended to 30 Years the gamma tends to 0.00 .