Question

A stock price is currently priced at £100. Over each of the next two three-month periods it is expected to down by 7% or go up by 8%. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a six-month European call option with a strike price of £95?

If possible, please provide a detailed step by step as I would like to fully comprehend it rather than just copying it. Thank you :)

Answer 1

Converting continuous interest rate to normal rate for 3 months.

Three month rate = e^(5%*3/12)-1 = 1.2578%

S0 =	Stock price today	=	100
r=	risk free interest rate	=	1.2578%
u=	up factor	=	1.08
d=	Down factor	=	0.93
X =	Exercise price	=	95
	We first compute the possible values of the stock at each node in the binomial tree:
t=1
S+ =	= 100*1.08	=	108
S- =	= 100*0.93	=	93
t = 2 = T
S++ =	= 100*1.08*1.08	=	116.64
S+ - =	= 100*1.08*0.93	=	100.44
S- - =	= 100*0.93*0.93	=	86.49
	Intrinsic value of the call option at expiration
c++ =	= Max(0, S++ - X)
	= Max(0, 116.64 - 95)	=	21.64
c+ - =	= Max(0, S+ - - X)
	= Max(0, 100.44 - 95)	=	5.44
c- - =	= Max(0, S- - - X)
	= Max(0, 86.49 - 95)	=	0
∏=	Risk neutral probability	=	(1+r-d)/(u-d)
∏=	Risk neutral probability	=	(1+0.0125784515406344-0.93)/(1.08-0.93)
		=	0.5505
	1- ∏=	=	0.4495
	Compute the value of call option at each node for t=1
c+ =	Call price t=1	=	[∏c++ + (1-∏)c+ - ]/ (1+r)
c+=	[0.5505*21.64 + 0.4495*5.44000000000001] /[1+0.0125784515406344 ]	=	14.18
c- =	Call price t=1	=	[∏c+ - + (1-∏)c- - ]/ (1+r)
	[0.5505*5.44000000000001 + 0.4495*0] /[1+0.0125784515406344 ]	=	2.96
	Finally, value of call option
c =	Call price t=0	=	[∏c+ + (1-∏)c - ]/ (1+r)
c =	Call price today
	[0.5505*14.18 + 0.4495*2.96] /[1+0.0125784515406344 ]	=	9.02

Call option fair price is 9.02