Question

Problem 2:

A stock currently sells for $50. In six months it will either rise to $60 or decline to $45. The continuous compounding risk-free interest rate is 5% per year.

1. Using the binomial approach, find the value of a European call option with an exercise price of $50.
2. Using the binomial approach, find the value of a European put option with an exercise price of $50.
3. Verify the put-call parity using the results of Questions 1 and 2.

Answer 1

(a) Expected Prices in 6-months: $ 60 and $ 45

Current Stock Price = $ 50, Call Exercise Price = $ 50

% Up Move = [(60-50) / 50] x 100 = 20 % and % Down Move = [(50-45)/50] x100 = 10 %

u = 1 + (20/100) = 1.2 and d = 1-(10/100) = 0.9

Risk-Free Rate = 5 % and Tenure = 6-months

Risk-Neutral Probability of Up Move = p = [e^{0.05 x 0.5} - 0.9] / [1.2 - 0.9] = 0.4177

After 6-months:

If price is $ 60, then call payoff = (60-50) = $ 10 and if price is $ 45, then call payoff = $ 0

Expected Payoff = 0.4177 x 10 + (1-0.4177) x 0 = $ 4.177

Call Value = PV of Expected Payoff = 4.177 / e^(0.05 x 0.5) = $ 4.07387 ~ $ 4.074

(b) After 6-months:

Put Exercise Price = $ 50

If price is $ 60, the put payoff = $ 0 and if price is $ 45, then put payoff = (50 - 45) = $ 5

Expected Payoff = 0.4177 x 0 + (1-0.4177) x 5 = $ 2.9115

Put Value = PV of Expected Payoff = 2.9115 / e^(0.05 x 0.5) = $ 2.839615 ~ $ 2.839

(c) PUT Call Parity: Call Value + PV of Exercise Price = Put Value + Stock Price

LHS Value = 4.074 + 50/e^(0.05 x 0.5) = $ 52.839

RHS Value = 2.839 + 50 = $ 52.839

As LHS = RHS, the put-call parity result holds.