Question

Suppose that firm D's shares are currently selling for $50. After six months it is estimated that the share price will either rise to $54.25 or fall to $46.50. If the share price rises to $54.25 in six months, six months from that date (1 year from today) the price is estimated to either rise to $58.86 or fall to $50.45. If the share price falls to $46.50 in six months, six months from that date (1 year from today) the price is estimated to either rise to $50.45 or fall to $43.25. The six month risk free rate is 2.5%.

Based on the tree diagram of stock prices, what is the risk neutral probability of a share price increase over either six month subperiod?

Based on the two stage binomial model, what should be the value today of a call option with exercise price $55 that expires in one year?

Answer 1

S0 =	Stock price today	=	50
r=	risk free interest rate	=	2.50%
u=	up factor	=	1.085
d=	Down factor	=	0.93
X =	Exercise price	=	55
	We first compute the possible values of the stock at each node in the binomial tree:
t=1
S+ =	= 50*1.085	=	54.25
S- =	= 50*0.93	=	46.5
t = 2 = T
S++ =	= 50*1.085*1.085	=	58.86125
S+ - =	= 50*1.085*0.93	=	50.4525
S- - =	= 50*0.93*0.93	=	43.245
	Intrinsic value of the call option at expiration
c++ =	= Max(0, S++ - X)
	= Max(0, 58.86125 - 55)	=	3.86125
c+ - =	= Max(0, S+ - - X)
	= Max(0, 50.4525 - 55)	=	0
c- - =	= Max(0, S- - - X)
	= Max(0, 43.245 - 55)	=	0
∏=	Risk neutral probability	=	(1+r-d)/(u-d)
∏=	Risk neutral probability	=	(1+0.025-0.93)/(1.085-0.93)
		=	0.6129
	1- ∏=	=	0.3871
	Compute the value of call option at each node for t=1
c+ =	Call price t=1	=	[∏c++ + (1-∏)c+ - ]/ (1+r)
c+=	[0.6129*3.86125 + 0.3871*0] /[1+0.025 ]	=	2.31
c- =	Call price t=1	=	[∏c+ - + (1-∏)c- - ]/ (1+r)
	[0.6129*0 + 0.3871*0] /[1+0.025 ]	=	-
	Finally, value of call option
c =	Call price t=0	=	[∏c+ + (1-∏)c - ]/ (1+r)
c =	Call price today
	[0.6129*2.31 + 0.3871*0] /[1+0.025 ]	=	1.38

Probability of increase is 0.6129

Call option price is 1.38