Find the optimal exercise time and the price
Course:- Other Subject
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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 S0 = 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

X0 = 16, X1 = (105 - min0≤t≤1 St)+, X2 = (max0≤t≤2 St - 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).

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