Eco-mputational Finance: Differential Equations for Monte Carlo Recycling

This article presents differential equations and solution methods for the functions of the form A(z) = F−1(G(z)), where F and G are cumulative distribution functions. Such functions allow the direct recycling of samples from one distribution into samples from another. The method may be developed analytically for certain special cases, and illuminate the idea that it is a more precise form of the traditional Cornish-Fisher expansion. In this manner the model risk of distributional risk may be assessed free of the Monte Carlo noise associated with resampling. The method developed here may also be regarded as providing analytical and numerical bases for doing a more precise form of Cornish-Fisher expansion. Examples are given of equations for converting normal samples to Student t, and converting exponential to hyperbolic and variance gamma.