Problem 3: We consider a model with three assets: one bond and two stocks. There are only times 0 and 1. The bond price is equal to 1 at both times (no interest). The stock prices are both equal to 1 at time 0. There are three possibilities for the stock prices at time 1: either the first stock price stays at 1 and the second stock price increases to 1.5 (case A), or the second stock price stays at 1 and the first increases to 1.5 (case B), or the two stock prices both decrease to 0.5 (case C). We consider a call option on the first stock, with maturity 1 and strike price 1.

(1) Compute the payoff of this option, in each of the three cases A, B, C.

(2) We consider a portfolio with ? bonds, ? first stocks and ? second stocks. Compute the value of the portfolio at time 0, and the value at time 1 in each of the three cases

A, B, C.

(3) For which values of ?,?,? do we replicate the call option above? (hint: we have to

solve a system of three linear equations with three unknowns).

(4) Compute the initial value of the replicating portfolio of the call option.

(5) We buy the call option at time 0. Which price should be pay?