Question

A stock price is currently $50. Over each of the next two 3-month periods it is expected to go up by 6% or down by 5%. The risk-free rate is 5% per annum with continuous compounding. What is the value of a six-month European call option with a strike price of $51?

Answer 1

u = 1.06 as stock price can go up by 6 % and d = 0.95 as stock price can go down by 5 %

Risk-Free Rate = 5 %, Tenure of Option = 6 months and Time Interval = 3 months each, Current Stock Price = $ 50 and Option Strike Price = $ 51

Risk-Neutral Proabability of Up Move = [e^(0.05 x 0.25) - 0.95]/[1.06 - 0.95] = 0.5689

t = 0	t = 3months	t = 6 months	Strike Price	Payoff
	Node 2	(53 x 1.06) = $ 56.18	51	(56.18 - 51) = $ 5.18
	(50 x 1.06) = $ 53	Node 4
$ 50		(53 x 0.95) = $ 50.35	51	$ 0
Node 1	(50 x 0.95) = $ 47.5	Node 5
	Node 3	(47.5 x 0.95) = $ 45.125	51	$ 0

Expected Payoff at Node 4 = (0.5689 x 5.18 + (1-0.5689) x 0) = $ 2.9469

PV at Node 2 of the Expected Payoff from Node 4 = 2.9469 / e^(0.05 x 0.25) = $ 2.91029

Expected Payoff at Node 5 = (0.5689 x 0) + (1-0.5689) x 0 = $ 0

PV at Node 3 of the Expected Payoff from Node 5 = 0 / e^(0.05 x 0.25) = $ 0

Call Price = PV of Expected Payoffs from Node 2 and Node 3 = (0.5689 x 2.91029) + (1-0.5689) x 0 / e^(0.05 x 0.25) = $ 1.6351