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Question 1 - Consider a two-period CRR binomial setting where interest rate is r = 0 and the intial stock price is S_{0} = 100. The stock price can go up by a factor of u = 1.1 or down by a factor of d = 0.9. Consider the American option where the payoffs at times 0, 1 and 2 are defined as
X_{0} = 16, X_{1} = (105 - min_{0≤t≤1} S_{t})^{+}, X_{2} = (max_{0≤t≤2} S_{t} - 95)^{+}.
Find the optimal exercise time and the price of this option at t = 0.
Note: Be careful in finding the payoffs. This is a path-dependent option.
Question 2 - Assume W is a standard Brownian motion and
P(t, T)| = e^{((T-t)W_t + 1/6(T-t)^3+½(T-t)^2+(T-t))}.
Find the short rate r(t).