Consider again the situation described in Exercise 15 of Sec. 12.2. Suppose that Wu has the Laplace distribution with parameters θ = 0 and σ = 0.1u1/2. See Eq. (10.7.5) for the p.d.f.
a. Prove that the m.g.f. of Wu is
ψ(t) = (1− t2u/100)−1, for −10u−1/2 < t < 10u−1/2.
b. Let r = 0.06 be the risk-free interest rate. Simulate a large number v of values of Wu with u = 1 and use these to estimate the price of an option to buy one share of this stock at time u = 1 in the future for the current price S0. Also compute the simulation standard error.
c. Use importance sampling to improve on the simulation in part (b). Instead of simulating Wu values directly, simulate from the conditional distribution of Wu given that Su > S0. How much smaller is the simulation standard error? |

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