Future prices of a stock are modeled with a one-period binomial tree, the period
ID: 3180209 • Letter: F
Question
Future prices of a stock are modeled with a one-period binomial tree, the period being one year.
You are given:
(i) the stocks current price is 42
(ii)the continuously risk-free interest rate is 5%
(iii) the stock pays continuous dividend at a rate of 10%
(iv) sigma = 0.3
A European call option expiring in one year on the stock has strike price 40.
Determine
(a) the number of shares purchased/sold and the amount of money borrow/lent in the replicating portfolio for the call option;
(b) the premium of the call option
Explanation / Answer
S1(H) = S0*e^(sigma8sqrt(t)) = 42*e^0.3*e^-0.1 = 42*e^0.2 = 51.30
S1(T) = 42*e^-0.3*e^-0.1 = 42*e^(-0.4) = 28.15
K=40
V1(H) = Max(51.30 - 40,0) = 11.30
V1(T) = Max(28.15-40,0) = 0
q = Risk neutral probability = (So*e^(fr*T) - S1(T))/(S1(H) - S1(T)) = (42*e^0.05 - 28.15)/(51.30-28.15) = 0.69
So, call option premium = e^(-0.05)*(0.69*11.30 + 0) = $7.42
We need to purchase/sell delta shares for each stock sold/purchased in a replicating portfolio. Hence, delta = change in option price - change in stock price = (11.30 - 0)/(51.30 - 28.15) = 0.488
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