Question

(8) Options. We come up with the following model for the stock of Shake Shack (SHAK). The stock price is 67$ today (t = 0). An analyst predicts that in a year from now (t = 1) the outcomes are S(1) = 100$ with probability P 258 with probability 1-p The annual interest rate is 2%. (a) Calculate the forward price F of the stock. (b) Calculate the price C(0) of a call option on SHAK with delivery date t = 1 with a strike price 50$. (c) Calculate the price P(0) of a put option on SHAK with delivery date t = 1 with a strike price 50$. (d) Calculate the return of the above call option. If p= 1/2, what are the expected return and the standard deviation of the return?

Answer 1

Su = 100; Sd = 25, S0 = 67; u = Su / S0 = 100 / 67 = 1.4925; d = Sd / S0 = 25 / 67 = 0.3731; r = 2%; K = 50; t = 1

(a) Forward price F = S0 x (1 + r)^t = 67 x (1 + 2%) = $ 68.34

(b) Risk neutral probability of up movement = p = (1 + r - d) / (u - d) = (1 + 2% - 0.3731) / (1.4925 - 0.3731) = 0.5779

Cu = max (Su - K, 0) = max (100 - 50, 0) = 50; Cd = max (Sd - K, 0) = max (25 - 50, 0) = 0

Hence, price C0 of the call option = [p x Cu + (1 - p) x Cd] / (1 + r)^t = [0.5779 x 50 + (1 - 0.5779) x 0] / (1 + 2%) = $ 28.33

(c) Pu = max (K - Su, 0) = max (50 - 100, 0) = 0; Pd = max (K - Sd, 0) = max (50 - 25, 0) = 25

Hence, price P0 of the put option = [p x Pu + (1 - p) x Pd] / (1 + r)^t = [0.5779 x 0 + (1 - 0.5779) x 25] / (1 + 2%) = $ 10.35

(d) Return on the call option, R = C / C0 - 1

In up state, Ru = Cu / C0 - 1 = 50 / 28.33 - 1 = 76.51%

In the down state, Rd = Cd / C0 - 1 = 0 / 28.33 - 1 = -100%

Expected return = p x Ru + (1 - p) x Rd = 1/2 x 76.51% + 1/2 x (-100%) = -11.74%

And variance of return = p x (Ru - expected return)² + (1 - p) x (Rd - expected return)² = 1/2 x (76.51% - (-11.74%))² + 1/2 x (-100% - (-11.74%))² = 0.7789

Hence, standard deviation = Variance^1/2 = 0.7789^1/2​​​​​​​ = 88.26%