Question

Consider a call option with strike price $100 on a stock. There are two periods until the option expires. The stock can either move up by u = 1.25 or down by a factor d = 0.8 in each period. Assume the interest rate for each period is given by r̂ = e^r with r = 5% and that the current stock price is $95. Find the value of the call option.

(Please write quick explanation for each step of the process, thank you so much!)

Answer 1

Risk neutral probability of up state = p = (e^r - d) / (u - d) = (e^0.05 - 0.8) / (1.25 - 0.8) = 0.55838

Node	S(T)	Payoff	Probability	Probability	Probability x payoff
		max(S - K, 0) = max (S - 100,0)	Formula	Value
Suu	148.44	48.44	p2	0.311788464	15.10
Sud	95.00	-	2p(1 - p)	0.493183501	-
Sdd	60.80	-	(1-p)2	0.195028035	-
Total				1	15.10

Hence, value of the call option today = 15.10 x e^{-r x t} = 15.10 x e^{-0.05 x 2} = $  13.67