# 11-1 - exponentially with the dimensionality of the...

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#11. http://en.wikipedia.org/wiki/Monte_Carlo_methods_in_finance http://www.riskglossary.com/link/monte_carlo_method.htm We call [11] the crude Monte Carlo estimator. Formula [12] for its standard error is important for two reasons. First, it tells us that the standard error of a Monte Carlo analysis decreases with the square root of the sample size. If we quadruple the number of realizations used, we will half the standard error. Second, standard error does not depend upon the dimensionality of the integral [6]. Most techniques of numerical integration—such as the trapezoidal rule or Simpson's method—suffer from the curse of dimensionality. When generalized to multiple dimensions, the number of computations required to apply them increases
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Unformatted text preview: exponentially with the dimensionality of the integral. For this reason, such methods cannot be applied to integrals of more than a few dimensions. The Monte Carlo method does not suffer from the curse of dimensionality. It is as applicable to a 1000-dimensional integral as it is to a one-dimensional integral. While increasing the sample size is one technique for reducing the standard error of a Monte Carlo analysis, doing so can be computationally expensive. A better solution is to employ some technique of variance reduction. These techniques incorporate additional information about the analysis directly into the estimator. This allows them to make the Monte Carlo estimator more deterministic, and hence have a lower standard error....
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