[10 points] We have discussed the estimation of the discounted payoff of an Asian option. An alternative model is the European option whose price at time T is given by the expectation a = E{e'1T[S(T) K,0]+}. In words, the value of a European option at time T is solely determined from the nal price M) = 8(0) exp [(61 - n" + zx], where X is a random variable from N (0, 1). In this univariate case, xl can be evaluated numerically, but we will attempt to use MC with antithetic sampling to estimate it. Let X = '1(U) and X* = (D'1(1 U), where U ~ (\"(0, 1) and (I)( -) is the c.d.f. of the standard normal distribution. Also dene h(X) = e-flT [5m K, 0]*. (a) Validate the effectiveness of the method by showing that Var{% [h(X) + h(X*)]} g %Var[h(X)]. (b) (C) Perform independent MC experiments to estimate A using an interest rate 51 = 0.1, volatilities 2 E {0.2, 0.3,0.4,0.5}, strike price K : 100, initial priceto-strike ratios S(0)/K e {0.9,1,1.1}, and time horizons T = {05,1}. In each experiment, use 71 = 104 antithetic pairs, so that replication 1' collects the sample Z\") = 5110(0)) +h(X,("))]. Report the estimates of A, ~ 1 n . a = Z\"), n n; their standard errors, their relative errors, the respective estimated variance reduction ratios compared to the standard MC estimates based on 211 samples, and the respective eiciency ratios. Comment on the variance reduction as a function of the volatility 62, the initial price S (0), and the length of the time horizon T