227 years with strike 33 assuming an interest rate of

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Unformatted text preview: ues of the stock. Suppose that a stock is now at 100, but in one month my 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 = 1 p1 p2 that make the stock proce a martingale. (b) Find the maximum and minimum values, v1 and v0 , of the expected value of the the call option (S1 105)+ among the martingale probabilities. (c) Show that we can start with v1 in cash, buy x1 shares of stock and we have v1 + x1 (S1 S0 ) (S1 105)+ 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 + x0 (S1 S0 ) (S1 105)+ 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 [v0 , v1 ]. 6.4. The Cornell hockey team is playing a game against Harvard that it will either win, lose, or draw. A gambler o↵ers you the following three payo↵s, each for a $1 bet Bet 1 Bet 2 Bet 3 win 0 2 .5 lose 1 2 1.5 draw 1.5 0 0 (a) Assume you are...
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