 # 5.8.3 Monte Carlo Simulation – Directly Modeling Relevant Random Vectors

###### 5.8.3  Monte Carlo Simulation – Directly Modeling Relevant Random Vectors

A Monte Carlo estimator need not employ a Un((0,1)n) sample. We saw this in Section 5.2.3 where we estimated the variance of a random variable Y that was a function of an N(0,1) random variable X. For that example, it was computationally convenient—and intuitively appealing—to directly model a realization {x, x, … , x[m]} of a sample {X, X, … , X[m]} for the non-uniform random variable X. Another example was in Section 1.7.4, where we used the Monte Carlo method to estimate thevalue-at-risk of a portfolio. There, we directly modeled a realization {1r, 1r, … , 1r[m]} of a sample {1R, 1R, … , 1R[m]} for 1R. Because they directly model relevant random vectors, such analyses are sometimes called simulations—Monte Carlo simulations.

Mathematically, such estimators are no different from the crude Monte Carlo estimator [5.34]. They simply incorporate a change of variables. As the intuitive nature of our earlier examples illustrates, such changes of variables can simplify problems that are inherently probabilistic.

Consider an n-dimensional random vector X with PDF ϕ. We need to calculate the expected value of E(q(X)) for some function q: n → . By [3.16], the expectation equals

[5.49]

To estimate the integral with the crude Monte Carlo estimator [5.34], we must convert the area of integration from n to (0,1)n. We do so with a differentiable function g: (0,1)n → n. By change of variables formula [2.168], the integral becomes

[5.50]

We obtain crude Monte Carlo estimator

[5.51]

This change of variables can be accomplished with any differentiable function g: (0,1)n → n, but suppose we employ a function g that is probability preserving in the sense defined earlier in this section. Two things happen. First, it can be shown that

[5.52]

so the crude Monte Carlo estimator [5.49] becomes

[5.53]

Second, as indicated by [5.43], if u[k] is a Un((0,1)n) pseudorandom vector, g(u[k]) is a pseudorandom vector x[k] for X. Substituting this into [5.51] yields the highly intuitive Monte Carlo estimator

[5.54]

To estimate E(q(X)), all we need do is generate pseudorandom vectors x[k], apply q to each, and average the results. The foregoing derivation demonstrates that this intuitive Monte Carlo estimator—or Monte Carlo simulation—is nothing more than a crude Monte Carlo estimator with a change of variables.