Question

A stock price is $10 now. In 1 month it can go to $11 or $9. The annual interest rate is 5% with continuous compounding. Construct a synthetic long call strategy with strike price 9.5 What is the cost of this synthetic trading strategy?

Answer 1

value of call option if price goes to $11 = max (spot - strike, 0) = $11-$10 = $1

value of call option if prices goes to $9 = max (spot - strike, 0) = $0

Annual interest rate = 5%,

let Probability of stock going up be p

Therefore probability of stock going down = 1-p

For calculating risk neutral probabilites,

Expected returns from stock = returns from risk free rate

[1 x p + -1 x (1-p)] / 10 = e^rt-1

[1 x p + -1 x (1-p)] / 10 = e^{0.05 x 1/12}-1

[2p-1] / 10 = 0.004175

therefore p = 0.5209

p = 0.5209

1-p = 1-0.5209 = 0.4791

therefore future value of call option = $1 x 0.5209 + $0 x 0.4791 = $0.52

Therefore, present value of option = $0.52 / e^{0.05 x 1/12} = $0.52 / 1.004175 = $0.517 = $0.52

cost of this synthetic trading strategy = value of option = $0.52