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Antithetic Variable Technique

The philosophy underlying this technique is motivated by the idea that in simulating a path to evaluate the outcome of the experiment, one can simultaneously simulate a parallel path so as to arrive at an additional outcome by simply using the already generated number. To understand this, consider 2

two variables Z1 and Z2. The variance of the average of Z| + Z2 (denoted by is given by the formula

where Cov (Z1, Z2) represents the covariance between Z| and Z2. Since this covariance is of the form (where represents the correlation between Z1 and Z2), it can be seen that is minimized when Thus,is minimized if both and are perfectly negatively correlated.

In practice, it is often the case that one is interested in generating values of some function of the random variable (e.g., the value of a derivative on option maturity date) as opposed to simply generating a variable of a pdf. To apply the antithetic variable technique, one first needs to generate variables to arrive at the derivative value . Once that is done, one needs to generate another set (i.e., antithetic) of variables in parallel to obtain a derivative value . Taking the average, one is able to obtain a value of Going through a similar analysis, it can be seen that the variance ofis minimized when the correlation between and is – 1. In particular, as can be seen in Ross (2007), as long as this function is a monotonic function of the variables generated, this is akin to finding negatively correlated random variables.

Given the above backdrop, I will now illustrate this concept with an example. Consider the generation of a European-style call option value discussed earlier, where was generated using the formula , where are all given constants and represents the normal random variate. To use the antithetic variable technique to generate a derivative value that is perfectly negatively correlated to the initial derivative value, given that the derivative value is a monotonic function of the simulated normal variable, it suffices to generate another normal variable that is perfectly negatively correlated to z1 (i.e., ). Table 4.13 shows the implementation of this idea.

Control Variable Technique

The philosophy underlying the motivation of this technique revolves around the idea that in order to reduce the variance associated with simulating an

TABLE 4.13 Using the Antithetic Method to Value a European-Style Option

TABLE 4.14 Using the Control Variate Method to Value a European-Style Option

outcome of an experiment, one should simulate the outcome of an experiment whose analytical solution is known (and as perfectly correlated as possible). To understand this, suppose that one wants to estimate the outcome of the experiment g(X) that is based on the generated variable X.

Since the use of simulation to determine the outcome is equivalent to finding the value of E[g(X)], this method dictates the use of some function f(X) whose expected value E[f(X)] is known. To find the simulated value of g(X) one needs to be able to simulate the value g(X) – [f(X) – E (f(X))]. This can be understood by looking at the variance of g (X) – f (X) + E (f(X)) and realizing that this is given by the expression Var [g(X)] + Var (f(X)) – 2Cov(f(X), g(X)) which is minimized when g(X) and f(X) are perfectly positively correlated. The reader is referred to Ross (2007) for details.

Table 4.14 shows the application of this method, when the stock price itself is used as a control variate (i.e., and

 
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