Question

Consider a two-period binomial model with u=1.2 and d=0.85. What is the current equilibrium price of a call with a strike price of $20.00, if shares of stock are currently trading at $20, and the one-period risk free rate is 4%?

Answer 1

S0 =	Stock price today	=	20
r=	risk free interest rate	=	4%
u=	up factor	=	1.2
d=	Down factor	=	0.85
X =	Exercise price	=	20
	We first compute the possible values of the stock at each node in the binomial tree:
t=1
S+ =	= 20*1.2	=	24
S- =	= 20*0.85	=	17
t = 2 = T
S++ =	= 20*1.2*1.2	=	28.8
S+ - =	= 20*1.2*0.85	=	20.4
S- - =	= 20*0.85*0.85	=	14.45
	Intrinsic value of the call option at expiration
c++ =	= Max(0, S++ - X)
	= Max(0, 28.8 - 20)	=	8.8
c+ - =	= Max(0, S+ - - X)
	= Max(0, 20.4 - 20)	=	0.4
c- - =	= Max(0, S- - - X)
	= Max(0, 14.45 - 20)	=	0
∏=	Risk neutral probability	=	(1+r-d)/(u-d)
∏=	Risk neutral probability	=	(1+0.04-0.85)/(1.2-0.85)
		=	0.5429
	1- ∏=	=	0.4571
	Compute the value of call option at each node for t=1
c+ =	Call price t=1	=	[∏c++ + (1-∏)c+ - ]/ (1+r)
c+=	[0.5429*8.8 + 0.4571*0.399999999999999] /[1+0.04 ]	=	4.77
c- =	Call price t=1	=	[∏c+ - + (1-∏)c- - ]/ (1+r)
	[0.5429*0.399999999999999 + 0.4571*0] /[1+0.04 ]	=	0.21
	Finally, value of call option
c =	Call price t=0	=	[∏c+ + (1-∏)c - ]/ (1+r)
c =	Call price today
	[0.5429*4.77 + 0.4571*0.21] /[1+0.04 ]	=	2.58

Call price is 2.58