## The use of stochastic collocation for sampling from expensive distributions with applications in finance

Master Thesis

2016

##### Publisher

University of Cape Town

##### Abstract
The pricing of financial derivatives using numerical methods often requires sampling from expensive distributions. These are distributions with inverse cumulative distribution functions that are difficult to evaluate, thus requiring significant computation time. To mitigate this, Grzelak et al. (2015) introduced the stochastic collocation Monte Carlo sampler. This sampling method is based on a generalisation of the stochastic collocation method of Mathelin and Hussaini (Mathelin andHussaini, 2003) which was introduced in the context of solving stochastic partial differential equations (Babuˇska et al., 2007; Loeven et al., 2007).The stochastic collocation Monte Carlo sampling method entails sampling from a cheaper distribution and then transforming the samples to obtain realisations from the expensive distribution. The function that transforms the quantiles of the cheap distribution to the corresponding quantiles of the expensive distribution is approximated using an interpolating polynomial of a prespecified degree. The points at which the interpolating polynomial is constructed to exactly match the true quantile-to-quantile transformation function are known as collocation points. Any number of realisations from the expensive distribution may be read off using the interpolating polynomial, leading to a significant reduction in computation time when compared to methods like the inverse transform method. This dissertation provides an overview of the stochastic collocation method, using distributions and models frequently encountered in finance as examples. Where possible, goodness of fit tests are performed. The major contribution of the dissertation is the investigation of the roots of Chebyshev polynomials of the first kind as collocation points, as opposed to Gaussian quadrature points used by Babuˇska et al. (2007), Loeven et al. (2007) and Grzelak et al. (2015). The roots of the Chebyshev polynomials are constrained to lie in a specified closed interval and hence are convenient to use when the statistic to be estimated does not depend on the entire distribution of interest, e.g. option prices or conditional expectations like expected shortfall.