Consider a stock with a current price of P = $27. Suppose that over the next 6 m
ID: 2769992 • Letter: C
Question
Consider a stock with a current price of P = $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 which expires in 6 months. The risk-free rate is 6%.
1. Using the binomial model, what are the ending values of the stock price? What are the payoffs of the call option?
2. Suppose you write 1 call option and buy Ns shares of stock. How many shares must you buy to create a portfolio with a riskless payoff (which is called a hedge portfolio)? What is the payoff of the portfolio?
d. 3. What is the present value of the hedge portfolio’s riskless payoff? What is the value of the call option?
d. 4. What is a replicating portfolio? What is arbitrage?
Explanation / Answer
1
Current stock price, P = $27
Risk-free rate, rRF = 6%
Strike price, X =$25
Up factor for stock price, u =1.41
Down factor for stock price, d =0.71
Years to expiration, t =0.50
Binomial Payoffs
Stirke Price =$25
Current stock price =27
Up factor for stock price: u = 1.41
Down factor for stock price: d =0.71
Up option payoff: Cu = MAX[0,P(u)-X] = $13.07
Down option payoff: Cd =MAX[0,P(d)-X] =0
$13.07
Option payoff: Cd =MAX[0,P(d)-X] =
Binomial Payoffs
Stirke Price =$25
Current stock price =27
Up factor for stock price: u = 1.41
Down factor for stock price: d =0.71
Up option payoff: Cu = MAX[0,P(u)-X] = $13.07
Down option payoff: Cd =MAX[0,P(d)-X] =0
Current stock price P = $27------ Ending "up" stock price = P (u) = $38.07 Option payoff: Cu = MAX[0,P(u)-X] =$13.07
Ending "down" stock price = P (d) =Option payoff: Cd =MAX[0,P(d)-X] =
(2.) Suppose you write 1 call option and buy Ns shares of stock. How many shares must you buy to create a portfolio with a riskless payoff (which is called a hedge portfolio)? What is the payoff of the portfolio? We can form a portfolio by writing 1 call option and purchasing Ns shares of stock. We want to choose Ns such that the payoff of the portfolio if the stock price goes up is the same as if the stock price goes down. This is a hedge portfolio because it has a riskless payoff. Ns = Cu - Cd = 0.69153 P(u - d)