Current stock price, P = $27.00
Binomial Payoffs
Strike price: X = $25.00
Current stock price: P = $27.00
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] = $0.00
P(u – d)
The Hedge Portfolio with Riskless Payoffs
Strike price: X = $25.00
Current stock price: P = $27.00
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] = $0.00
Number of shares of stock in portfolio: Ns = (Cu – Cd) / P(u-d) = 0.69153
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.
d. 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?