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A stock price is currently priced at £100. Over each of the next two three-month periods...

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 :)

Solutions

Expert Solution

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


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