$(15$ points$)$ A stock is governed by a geometric Brownian motion with an initial price of

$$60,$ an interest rate of $1\%,$ and a volatility of $40\%.$ You monitor the stock price each

week for twenty$-$six weeks $($ one half of a year$).$ Compute the price of a down$-$and$-$in put

option with a strike price of $$60$ and a barrier of $$55$ with an absolute error of $$0\mathrm{.}01$

using

a$)$ IID sampling,

b$)$ IID sampling with a control variate: the European call option,

c$)$ IID sampling with a control variate: the European put option,

d$)$ Integration lattice sampling, and

e$)$ Scramble digital net sampling.

Compare the performance of these five methods and attempt to explain intuitively why

certain methods perform better than others.