The Stochastic Collocation Monte Carlo Sampler: Highly Efficient Sampling from 'Expensive' Distributions

Abstract

In this article we propose an efficient approach for inverting computationally expensive cumulative distribution functions. The collocation method, called the Stochastic Collocation Monte Carlo Sampler (SCMC Sampler), within the polynomial chaos expansion framework, allows us the generation of any number of Monte Carlo samples based on only a few inversions of the original distribution and independent samples from standard normals. We will show that with this path independent collocation approach the so-called exact simulation of the Heston stochastic volatility model, as proposed in (Broadie and Kaya, 2006), can be performed efficiently and accurately. We also show how to efficiently generate samples from the squared Bessel process and perform the exact simulation of the SABR model.

Keywords: Exact Sampling, Heston, Squared Bessel, SABR, Stochastic Collocation, Lagrange Interpolation, Monte Carlo

JEL Classification: C63, G12, G13