Quantile mechanics II: changes of variables in Monte Carlo methods and GPU-optimised normal quantiles

Abstract

With financial modelling requiring a better understanding of model risk, it is helpful to be able to vary assumptions about underlying probability distributions in an efficient manner, preferably without the noise induced by resampling distributions managed by Monte Carlo methods. This article presents differential equations and solution methods for the functions of the form Q(x) = F −1 (G(x)), where F and G are cumulative distribution functions. Such functions allow the direct recycling of Monte Carlo 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 may also be regarded as providing both analytical and numerical bases for doing more precise Cornish-Fisher transformations. Examples are given of equations for converting normal samples to Student t, and converting exponential to hyperbolic, variance gamma and normal. In the case of the normal distribution, the change of variables employed allows the sampling to take place to good accuracy based on a single rational approximation over a very wide range of the sample space. The avoidance of any branching statement is of use in optimal GPU computations as it avoids the effect of warp divergence, and we give examples of branch-free normal quantiles that offer performance improvements in a GPU environment, while retaining the best precision characteristics of well-known methods. We also offer models based on a low-probability of warp divergence. Comparisons of new and old forms are made on the Nvidia Quadro 4000, GTX 285 and 480, and Tesla C2050 GPUs. We argue that in single-precision mode, the change-of-variables approach offers performance competitive with the fastest existing scheme while substantially improving precision, and that in double-precision mode, this approach offers the most GPU-optimal Gaussian quantile yet, and without compromise on precision for Monte Carlo applications, working twice as fast as the CUDA 4 library function with increased precision.

Efficient and Accurate Parallel Inversion of the Gamma Distribution

“…The idea of using quantile mechanics to convert samples from one distribution to another was initiated in [26]. Let G be the CDF of an intermediary distribution and let q be the quantile function of the target distribution (with PDF f ).…”

“…Variate recycling was used in [26] to create GPU-optimized algorithms for Φ −1 . A Laplace double-exponential intermediary distribution was used, so the relevant mapping for 1/2 ≤ u < 1 is…”

“…The normal quantile function Φ −1 is a standard inclusion in most numerical libraries, including Intel's Math Kernel Library (MKL), NAG's numerical libraries, Boost's C++ Math Toolkit, and NVIDIA's CUDA Math Library. Standalone open-source implementations of Φ −1 are also freely available (e.g., [30] or [26]). 3 The normal distribution does not possess a shape parameter -that is, a parameter which neither shifts nor scales the distribution.…”

“…There is existing work looking at GPU-optimized algorithms for the normal quantile function in [26]. However, we are not aware of any such research or commercial solutions for the gamma distribution.…”

“…When the shape parameter α is fixed, it opens the possibility of precomputing a fast approximation to q α . We remodel the gamma quantile function for optimal evaluation on GPUs and other parallel architectures by using ideas from [26]. More specifically, we leverage quantile mechanics, the differential approach to quantile functions, with tailored changes of variable to facilitate fast and accurate computation of the gamma quantile function.…”