Question

The Call option on the stock has a $13 exercise price and one-year maturity. The volatility of the stock is 10%. The probability of an up or down movement is an equal 50%. The risk-free interest rate is 6% per annum The current stock price is $13. Stock movement is 2 times a year. Value the premium of the option based on Binomial Model.

Answer 1

Strike Price = $ 13, RIks-Free Rate = 6 % per annum, Option Maturity = 1 year, Number of price change per annum = 2

Volatility = s = 10 %, Probability of up move = 50 %

Magnitude of Up Move = u = EXP[s x (t)^(1/2)] where t is the length of each time period (time between two consecutive price movements)

u = EXP[0.1 x (0.5)^(1/2)] ~ 1.0733 and d = 1/u = 1/1.0733 ~ 0.9317

Risk-Neutral Probability of Up Move = [EXP(0.06 x 0.5) - 0.9317] / [1.0733 - 0.9317] ~ 0.6974

t= 0	t= 0.5	t= 1	Payoff (Strike Price $ 13
	Node 1	13.953 x 1.0733 ~ $ 14.976 (node 3)	(14.976 - 13) ~ 1.976
Node 0	13 x 1.0733 ~ $ 13.953
$ 13	Node 2	12.112 x 1.0733 ~ $ 12.999 (node 4)	0
	13 x 0.9317 ~ $ 12.112
		12.112 x 0.9317 ~ $ 11.285 (node 5)	0

Present Value of Payoff at Node 1 = [1.976 x 0.6974 + 0 x (1-0.6974)] / EXP(0.06 x 0.5) ~ $ 1.3028

Present Value of Payoff at Node 2 = [0 x 0.6974 + 0 x (1-0.6974)] / EXP(0.06 x 0.5) = $ 0

Call Option Premium = PV of of Payoff at Node 0 = [ 1.3028 x 0.6974 + 0 x (1-0.6974)] / EXP(0.06 x 0.5) ~ $ 0.8817