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 - min_0≤t≤1 S_t)⁺, X₂ = (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).