Question

A stock initially selling at $100 could increase by 10% with probability of 60% or decrease by 10% with probability of 40% every six months. Suppose that the continuous interest rate is 5% per six-month. A call option on the stock specifies an exercise price of $100 and a time to expiration of one year, what is its price using binomial pricing method? A) 6.78 B) 4.75 C)6.84 D. None of the above

Answer 1

Discrete interest rate = e^5% -1 = 5.13%

S0 =	Stock price today	=	100
r=	risk free interest rate	=	5.127%
u=	up factor	=	1.1
d=	Down factor	=	0.9
X =	Exercise price	=	100
	We first compute the possible values of the stock at each node in the binomial tree:
t=1
S+ =	= 100*1.1	=	110
S- =	= 100*0.9	=	90
t = 2 = T
S++ =	= 100*1.1*1.1	=	121
S+ - =	= 100*1.1*0.9	=	99
S- - =	= 100*0.9*0.9	=	81
	Intrinsic value of the call option at expiration
c++ =	= Max(0, S++ - X)
	= Max(0, 121 - 100)	=	21
c+ - =	= Max(0, S+ - - X)
	= Max(0, 99 - 100)	=	0
c- - =	= Max(0, S- - - X)
	= Max(0, 81 - 100)	=	0
∏=		=
∏=	Probability of up	=	0.6000
	1- ∏=	=	0.4000
	Compute the value of call option at each node for t=1
c+ =	Call price t=1	=	[∏c++ + (1-∏)c+ - ]/ (1+r)
c+=	[0.6*21 + 0.4*0] /[1+0.0512710963760241 ]	=	11.99
c- =	Call price t=1	=	[∏c+ - + (1-∏)c- - ]/ (1+r)
	[0.6*0 + 0.4*0] /[1+0.0512710963760241 ]	=	-
	Finally, value of call option
c =	Call price t=0	=	[∏c+ + (1-∏)c - ]/ (1+r)
c =	Call price today
	[0.6*11.99 + 0.4*0] /[1+0.0512710963760241 ]	=	6.84

Answer is 6.84

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