Question

5. Let S = $45, r = 3% (continuously compounded), d = 5%, s = 30%, T = 1.5. In this situation, the appropriate values of u and d are 1.27738 and 0.75972, respectively. Using a 2-step binomial tree, calculate the value of a $40-strike European call option.

Use: Table 1: Table of the Standard Normal Cumulative Distribution Function ?(z): https://math.ucalgary.ca/files/math/normal_cdf.pdf

a. $7.037

b. $8.305

c. $7.783

d. $8.141

e. $7.960

Answer 1

S = $ 45, K = $ 40, r = 3 %, d = 5 %, s = 30 %, T = 1.5 years and two step binomial which implies each step is t = 0.75 years long, u = 1.27738 and d = 0.75972

Let the probability of upward movement be P

Therefore, P = [EXP(r-d) x t] - d/(u-d) = [EXP(0.03 - 0.05) x 0.75] - 0.75972 / (1.27738 - 0.75972) = 0.435

Time Period	t=0	t=0.75	t=1.5	Payoff
			$ 73.4264849 (Node 4)	$ 33.4264849
		$ 57.4821 (Node 2)
	$ 45 (Node 1)		$ 43.67030101 (Node 5)	$ 3.67030101
		$ 34.1874 (Node 3)
			$ 25.97285153 (Node 6)	$ 0

Payoff at Node 4 = $ 33.4264849, Payoff at Node 5 = $ 3.67030101 and Payoff at Node 6 = $ 0

PV of Expected Payoff at Node 2 = (33.4264849 x 0.435 + 3.67030101 x 0.565) / EXP(0.03 x 0.75) = $ 16.2446

PV of Expected Payoff at Node 3 = (3.67030101 x 0.435 + 0 x 0.565) / EXP(0.03 x 0.75) = $ 1.5611

PV of Expected Payoff at Node 1 = (16.2446 x 0.435 + 1.5611 x 0.565) / EXP(0.03 x0.75) = $ 7.772 or $ 7.783 approximately.

Hence, the correct option is (c).