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# P1.T2.601. Variance reduction techniques (Brooks)

#### Nicole Seaman

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Learning objectives: Explain how to use antithetic variate technique to reduce Monte Carlo sampling error. Explain how to use control variates to reduce Monte Carlo sampling error and when it is effective. Describe the benefits of reusing sets of random number draws across Monte Carlo experiments and how to reuse them.

Questions:

601.1. Betty is an analyst using Monte Carlo simulation to price an exotic option. Her simulation consists of 10,000 replications where the key random variable is a random standard normal because the underlying process is geometric Brownian motion (GBM). For example, in Excel a random standard normal value is achieved with an inverse transformation of a random uniform variable by way of the nested function NORM.S.INV(RAND()). In this case, each random standard normal, z(i) = N(0,1), is the random draw that becomes an input into the option price function. Her simulation succeeds in producing an estimate for the option's price, but Betty is concerned the confidence interval around her estimate is too large. If her aim is to reduce the standard error, which of the following approaches is NEAREST to the antithetic variate technique?

a. She simulates 5,000 pairs of random z(i) and -z(i) such that each pair has perfectly negative covariance
b. She quadruples the number of replications which will reduce the standard error by 50% because the sqrt(four) is equal to two
c. She imposes a condition of i.i.d. (independence and identically distributed) on the series of z(i) which eliminates the covariance term
d. She introduces low-discrepancy sequencing with leads the Monte Carlos standard errors to be reduced in direct proportion to the number of replications rather than in proportion to the square root of the number of replications

601.2. Betty is an analyst using Monte Carlo simulation to price an Asian option. An Asian option is a a path-dependent option because its payoff depends on the arithmetic average of the price of the underlying asset during option's life. She successfully prices the option by using 10,000 replications; the simulated Asian price is denoted P(A). However, she wants to reduce the simulation's sampling error. Which of the following approaches is NEAREST to the control variate technique?

a. She simulates the value of path-dependency as an variable, V(PD), and adds this to the analytical solution to the value of a Asian option WITHOUT path dependency
b. She simulates the prices of a vanilla European option, P(BS), and also analytically prices the same, denoted P*(BS), she then estimates the Asian option price as given by [P(BS) + P*(BS)] - P(A)
c. She simulates the prices of a vanilla European option, P(BS), and also analytically prices the same, denoted P*(BS), she then estimates the Asian option price as given by P(A) + [P*(BS) - P(BS)]
d. In this scenario, she does not have a good control variate approach because she cannot find a control statistic that is highly correlated to her statistic of interest

601.3. According to Brooks, which of the following is TRUE about random number re-usage?

a. Random number re-usage is the best way to increase the accuracy of an estimate
b. Random number re-usage can reduce the variability of the difference in estimates across experiments
c. Random number re-usage is typically a high priority because it tends to greatly reduce computational time
d. Although random number re-usage is not advisable across experiments, it makes great sense within a Monte Carlo experiment