Question

Today’s price of a non-dividend paying stock is $60. Use a two-step tree to value a European call option on the stock with a strike price of $60 that expires in 6 months. Each step is 3 months. The risk free rate is 5% per annum with continuous compounding. Assume that the option is written on 100 shares of stock, and that u = 1.15 and d = 0.85.

A)What is the option price today?

B) How would you hedge a position where you buy the call option today? Please label the answer for each part

Answer 1

Stock price after two time periods

Suu = 1.15*1.15*60 = 79.35

Sud = 1.15*0.85*60 = 58.65

Sdu = 0.85*1.15*60 = 58.65

Sdd = 0.85*0.85*60 = 43.35

So payoff with strike price 60 would be:

Suu = Max(79.35-60,0) = 19.35

Sud = Max(58.65-60,0) = 0

Sdu = Max(58.65-60,0) = 0

Sdd = Max(43.35-60,0) = 0

Net payoff = 0.25*19.35 = 4.8375

Option price = 4.8375*exp(-0.05*1/2) = 4.7181

Stock value at first node = Su = 60*1.15 = 69

Sd = 0.85*60 = 51

Payoff at Su = 69-60 = 9

Payoff at Sd = 0

Delta at the first node = change in call payoff/change in stock price = (9-0)/(69-51) = 0.5

So we need to short the o.5 times the stock for which the option is bought