Question

11. An underlying asset has a spot price of So = 40, no dividend with prices moving up(u) = 1.1 or down(d) = 0.9 each of 3 periods with each period being 1 month. The continuously compounded risk free rate is 8%. Si S2 44 48.4 48 S. 40 40 40 40 40 40 40 40 Path uuu uud du duu udd dud ddu ddd Path probability 0.141 0.130 0.130 0.130 0.120 0.120 0.120 0.110 48.4 39.6 39.6 39.6 39.6 S3 53.24 43.56 43.56 43.56 35.64 35.64 35.64 29.16 32.4 32.4 Complete the binomial tree above to determine the price of an up-and-in barrier call option with a strike of 42 and a barrier of 45. a) 1.67 c) 1.97 e) 2.17 b) 1.77 d) 2.07

Answer 1

Hence, the correct answer is the first option i.e. option a) showing 1.67

Risk neutral probability of up state = p = (ert - d) / (u - d) = (e8% x 1/12 - 0.9) / (1.1 - 0.9) =  0.53344

S0	Path	Path prob.	Path prob.	S1	S2	S3	Barrier crossed i.e. if S3 > 45	Payoff = max (S3 - 42, 0) if barrier is crossed
40	uuu	p x p x p	0.1518	44	48.4	53.24	TRUE	11.24
40	uud	p x p x (1 - p)	0.1328	44	48.4	43.56	FALSE	0
40	udu	p x (1 - p) x p	0.1328	44	39.6	43.56	FALSE	0
40	duu	(1 - p) x p x p	0.1328	36	39.6	43.56	FALSE	0
40	udd	p x (1 - p) x (1 - p)	0.1161	44	39.6	35.64	FALSE	0
40	dud	(1 - p) x p x (1 - p)	0.1161	36	39.6	35.64	FALSE	0
40	ddu	(1 - p) x (1 - p) x p	0.1161	36	32.4	35.64	FALSE	0
40	ddd	(1 - p) x (1 - p) x (1 - p)	0.1016	36	32.4	29.16	FALSE	0
Total					1.0000

Barrier is crossed if S3 > Barrier = 45; TRUE means barrier has been crossed; false means barrier has not been crossed.

Payoff from the call option = max (S3 - K, 0) = max (S3 - 42, 0) if barrier is crossed and 0 otherwise.

Hence, expected payoff at the end of t = 3 months = payoff x probability = 11.24 x 0.1518 = 1.71

Hence, price = value of the call option today = Expected payoff x e-rt = 1.71 x e(-8% x 3/12) = 1.67

Hence, the correct answer is the first option i.e. option a) showing 1.67