It was crucial for our no arbitrage computations that there were only two possible values of the stock. Suppose that a stock is now at 100, but in 1 month may be at 130, 110 or 80 in outcomes that we call 1, 2 and 3.

(a) Find all the (nonnegative)

probabilities p1; p2 and p3 D 1  p1  p2 that make the stock price a martingale.

(b) Find the maximum and minimum values, v1 and v0, of the expected value of the call option .S1  105/C among the martingale probabilities.

(c) Show that we can start with v1 in cash, buy x1 shares of stock and we have v1 C x1.S1  S0/  .S1  105/C in all three outcomes with equality for 1 and 3.

(d) If we start with v0 in cash, buy x0 shares of stock and we have v0 C x0.S1  S0/  .S1  105/C in all three outcomes with equality for 2 and 3.

(e) Use

(c) and

(d) to argue that the only prices for the option consistent with absence of arbitrage are those in OEv0; v1.