 # 6. Consider a CRR model with T = 2, S0 = \$100, S1 = \$200 or S1 = \$5...

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6. Consider a CRR model with T = 2, S0 = \$100, S1 = \$200 or S1 = \$50 (the same model as in Exercise 2). Now consider an American put option with strike price K = \$120. Assume that the risk free interest rate is r = 0.1. (a) Use a binary tree to compute the arbitrage free initial price of the American put option. (b) Determine an explicit superhedging strategy 0* for this option. (c) Suppose that you can buy the American put option at time zero for \$1 less than its arbitrage free price. Explicitly describe a strategy that yields an arbitrage opportunity for a buyer of the American put option. (d) Try to automate the arbitrage free pricing of an American put option in a computer program where T, S0, u, d and K are variables. 4. Let T : S {0, 1, 2, , T be a random variable on a finite sample space S2, and let {Fit= = 0, 1, 2, , I} be the filtration generated by the stock prices in a binomial model, that is, Ft = 0 (So, S1 , St) for t = 0, 1, , T. (The result actually holds for any filtration.) (a) Show that if T is an (Fi}-stopping time, then for every t = 0,1, , T, {T t} E Ft. (b) Show that if every t = 0,1, T, {T t} E Ft, then T is an {Fi}-stopping time (that is, the converse of part a is true).

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