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 one 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.