Question

Use a two-step binomial model to evaluate a call option on a stock with the following price projections. The current stock price is $80 and the strike price on the options is $82. The option expires in 6 months so each step is 3 months. The risk- free rate is 5%. What is the value of the call option? Note: to be eligible for partial credit, please show your work as much as possible and be sure to clearly indicate values the values obtained through the binomial pricing process. $882 $80

Answer 1

S0 =	Stock price today	=	80
r=	risk free interest rate	=	1.25%
u=	up factor	=	1.05
d=	Down factor	=	0.95
X =	Exercise price	=	82
	We first compute the possible values of the stock at each node in the binomial tree:
t=1
S+ =	= 80*1.05	=	84
S- =	= 80*0.95	=	76
t = 2 = T
S++ =	= 80*1.05*1.05	=	88.2
S+ - =	= 80*1.05*0.95	=	79.8
S- - =	= 80*0.95*0.95	=	72.2
	Intrinsic value of the call option at expiration
c++ =	= Max(0, S++ - X)
	= Max(0, 88.2 - 82)	=	6.2
c+ - =	= Max(0, S+ - - X)
	= Max(0, 79.8 - 82)	=	0
c- - =	= Max(0, S- - - X)
	= Max(0, 72.2 - 82)	=	0
∏=	Risk neutral probability	=	(1+r-d)/(u-d)
∏=	Risk neutral probability	=	(1+0.0125-0.95)/(1.05-0.95)
		=	0.6250
	1- ∏=	=	0.3750
	Compute the value of call option at each node for t=1
c+ =	Call price t=1	=	[∏c++ + (1-∏)c+ - ]/ (1+r)
c+=	[0.625*6.2 + 0.375*0] /[1+0.0125 ]	=	3.83
c- =	Call price t=1	=	[∏c+ - + (1-∏)c- - ]/ (1+r)
	[0.625*0 + 0.375*0] /[1+0.0125 ]	=	-
	Finally, value of call option
c =	Call price t=0	=	[∏c+ + (1-∏)c - ]/ (1+r)
c =	Call price today
	[0.625*3.83 + 0.375*0] /[1+0.0125 ]	=	2.36

up factor = 84/80 = 1.05

down factor = 76/80 = 0.95

Answer is $2.36 for call option.

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