Question

A stock price is currently $130. Over each of the next two four-month periods it is expected to go up by 15% or down by 10%. The risk-free interest rate is 8% per annum with continuous compounding. (All calculations keep four digits after the decimal point)

1. What is the value of an eight-month European call option with a strike price of $133?

2. What is the value of an eight-month American put option with a strike price of $133?

Answer 1

American option can be exercisable as and when strike price is less than stock price,

where as european option need to be exercised only at the end of period,

solution under binomial model,

Given rate of interest=8%per anum

for four months , it would be =8*4/12

0.08333

computation of probability

P=R-d/u-d

R=1.08333

d=0.90

u=1.15

therefore probability =1.0833-0.90/1.15-.090

therefore P= 73%, 1-p=27%

therefore under european option ,,

European call can be made at node D,E,F only,

therefore value of call ={73%[38.925*73%+1.55*27%]+27%[1.55*73%+0*27%]}

=$21.05284

where as under call option can be exercisable at any time before end of the period so we can exercise at node B and Node C as well

calculation as follows

{[73%*(38.925*73%+1.55*27%)/1.0833+27%(1.55*73%+0*27%)/1.0833]}

={73%*26.616+27%*1.04494}

=$18.19608

the sum was under that whenever favourable position was not there ,it means when strike price falls below future price , we dont exercise and hence therfore we put 0 in calculation whenever strike price falls below future stock price and accordingly probability allocated