5. (40pt) Write a 3-page report answering the following questions with the capture and explanation of the results. (a) (20pt) Consider an underlying asset price (which is monitored daily) with parameters: So = 100,0 = 0.4, r = 0.05, T = 0.5, h = 1/252 (daily observation). Antithetic variates (with the number of simulation trials = 10,000): - compute the price of a European-style down-and-out call option (with K = 100 and B = 110) via naive MC and antithetic variable, and report their standard errors as well as the variance reduction factor. - compute the deltas of the option via naive MC and antithetic variate (In both cases, common random number should be used.) Control variates (with the number of simulation trials = 10,000): - compute the price of a European-style Asian call option with arithmetic average and K = 110 via naive MC and control variate, and report their standard errors as well as the variance reduction factor. (Use the price with geometic average in Problem 1 as a control variate) (b) (20pt) Consider an American-style down-and-in put option with the following parameters: So = 50, T = 0.25,0 = 0.2, r = 0.1,8 = 0.05, K = 55, B = 45. Compute the price using the implicit finite difference method - when 5 = 100, m = 100, n = 100 - when 5 = 100, m = 100, n = 10,000 Compute the price using the explicit finite difference method - when S = 100, m = 100, n = 100 - when S = 100, m = 100, n= 10,000 5. (40pt) Write a 3-page report answering the following questions with the capture and explanation of the results. (a) (20pt) Consider an underlying asset price (which is monitored daily) with parameters: So = 100,0 = 0.4, r = 0.05, T = 0.5, h = 1/252 (daily observation). Antithetic variates (with the number of simulation trials = 10,000): - compute the price of a European-style down-and-out call option (with K = 100 and B = 110) via naive MC and antithetic variable, and report their standard errors as well as the variance reduction factor. - compute the deltas of the option via naive MC and antithetic variate (In both cases, common random number should be used.) Control variates (with the number of simulation trials = 10,000): - compute the price of a European-style Asian call option with arithmetic average and K = 110 via naive MC and control variate, and report their standard errors as well as the variance reduction factor. (Use the price with geometic average in Problem 1 as a control variate) (b) (20pt) Consider an American-style down-and-in put option with the following parameters: So = 50, T = 0.25,0 = 0.2, r = 0.1,8 = 0.05, K = 55, B = 45. Compute the price using the implicit finite difference method - when 5 = 100, m = 100, n = 100 - when 5 = 100, m = 100, n = 10,000 Compute the price using the explicit finite difference method - when S = 100, m = 100, n = 100 - when S = 100, m = 100, n= 10,000