Question

1. Let a stock have the price 75 at t = 0. Assume that the risk free force of interest is 6% per unit period. Find the risk neutral probabilities denoted p?, q? for one unit of time. In one unit of time the stock can achieve 95 or 63. Determine the call option value at t = 0.

Answer 1

u: up factor = 95/75 = 1.267

d: down factor = 63/75 = 0.84

r: risk-free rate = 6%

t: time = 1

exp: natural exponent

p: probability of up movement

p = (exp^(r*t)-d)/(u-d) = (exp^(6%*1)-0.84)/(1.267-0.84) = 0.5195 = 51.95%

q: probability of down movement = 1-p = 0.4805 = 48.05%

K: strike price = 75

If stock goes up

call payoff = max(S-K,0) = max(95-75,0) = 20

If stock goes down

call payoff = max(S-K,0) = max(63-75,0) = 0

Expected pay-off call option = 0.5195*20+0.4805*0 = 10.39

Value of call option = exp^(-r*t) * expected payoff = exp^(-6%*1)*10.39 = 9.785