Question

Consider a one-step binomial tree on stock with a current price of $200 that can go either up to $230 or down to $170 in 2 years. The stock does not pay dividend. Continuously compounding interest rate is 5%. Use the tree to compute the value of a 2-year $210-strike European call option on the stock. Answer in four decimal place.

Answer 1

1)            Current share price = $ 200

After 2 years, there are 2 possibilities

1. Share price going up to Rs 230
% Up Movement = (Share Price after up movement – Share Price) / Share Price
% Up Movement = (230 – 200) / 200
% Up Movement = 30/200
% Up Movement = 15%

2. Share price going down to Rs 170
% Down Movement = (Share Price - Share Price after down movement) / Share Price
% Down Movement = (200 – 170) / 200
% Down Movement = 30/200
% Down Movement = 15%

Therefore u = 1+0.15 = 1.15
                   d = 1-0.15 = 0.85

Calculation of probability of upside movement :
Probability of upmove is given by the formula : [ert-d]/[u-d]

Probability of upmove = [e(.05*2) – 0.85] / [1.15 - 0.85]
Probability of upmove = [e(.10) – 0.85] / [0.30]
Probability of upmove = [1.1052 – 0.85]/0.30
Probability of upmove = 0.2552/0.30
Probability of upmove = 85.07%

Probability of downmove = 1 – 0.8507 = 14.93%

Exercise Price of Option = $210
Payoff from call option = max[Share Price – Exercise Price,0]
Payoff from call option when share price is 230 = 230 - 210 = $20
Payoff from call option when share price is 170 = 0 since call option lapses

Expected payoff = Probability of up move * 20 + Probability of down move * 0
                                = (85.07% * 20) + (14.93% *0)
                                = 17.014

PV of expected payoff at time 0 = 17.014 /ert
                            = 17.014 / e(.05*2)
                            = 17.014 / e.10
                            = 17.014/1.1052
                            = 15.3945

Price of a 2 year $210 strike European call option on the stock is $15.3945