Consider a stock with a current price of $27. Suppose that over the next 6 month
ID: 2802606 • Letter: C
Question
Consider a stock with a current price of $27. Suppose that over the next 6 months the stock price will either go up by a factor of 1.41 or down by a factor of 0.71. Consider a call option on the stock with a strike price of $25 that expires in 6 months. The risk-free rat is 6%.
A. Using the binomial model, what are the ending values of the stock price? What are the payoffs of the call option?
B. Suppose you write one call option and buy N shares of stock. How many shares must you buy to create a portfolio with a riskless payoff (i.e., a hedge portfolio)? What is the payoff of the portfolio?
C. What is the value of the call option?
D. What is the value of the put option on the stock with the same maturity and the same strike price?
Explanation / Answer
A)
Up value = 27*1.41 = 38.07
Down value = 27*0.71 = 19.17
Payoff in Up = Max(0,38.07 - 25) = 13.07
Payoff in Down = Max(0,19.17 - 25) = 0
B)
hedge ratio = (13.07 - 0 ) / (38.07 - 19.17) = 0.691534 shares per option
C)
Pu = (1 + Rf - D) / (U - D) = (1 + 0.03 - 0.71) / (1.41 - 0.71) = 0.4571
Pd = 1 - Pu = 1 - 0.4571 = 0.5429
Call = (0.4571*13.07 + 0.5429*0 ) / (1+3%) = 5.8008
D)
Put
Payoff in Up = Max(0,25 - 38.07) = 0
Payoff in Down = Max(0, 25 - 19.17) = 5.83
Call = (0.4571*0 + 0.5429*5.83 ) / (1+3%) = 3.0729