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Exercise 1 (A2-step American option). Consider an American option maturing at time t = 2 on a stock with price S, which evolves according to the 2-step binomial model with constant interest rate r = 5% given in Figure 1 (left). The American option is defined by the payoff function (w) given in Figure 1 (right). Stock American option S.(HX) - 22 (W) = 15 SCH) - 15 (O) (N) - 4 (1) 5-10 SNT) - 12 S.(TW) - 9 9.(0) 0 (NT) - 2 (TH) - 4 S.(T) - 6 9.(T) = 13 S.(TT) - 4 P ( (IT) - 14 Interest rate 5% Interest rate 5% FIGURE 1. A 2-step American option with payoff function gaw) as given in the right (1) Compute the value V of this American option using the recursive formula in Proposition 5.1 in Lecture note 2 (ii) Write down all possible stopping time for this model (there are 5 of them). (ii) Compute the value Vo(IT) of this American option under each stopping time . Verify that the maximum value among all r agrees the result of (i). 5. Pricing American options In this section, we consider pricing general path-dependent American options under the N-step binomial model. 5.1. Definition and examples of American options. Denote the stock price by (Sposisy and as- sume constant interest rate r. We will be considering a general American option on this stock such that the payoff at time k if exercised is a function on both the stock price Sx and the sample path we (H.TK that the market takes up to time k. Namely, we fix a nonnegative function g so that (w) = payoff of the American option at time k assuming time evolution. (247) Proposition 5.1. Consider the path-dependent American option as above. For each sample path w of length