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

 dc.contributor.advisor McWalter, Thomas en_ZA dc.contributor.author Brand, Hilmarie en_ZA dc.date.accessioned 2016-07-28T13:33:48Z dc.date.available 2016-07-28T13:33:48Z dc.date.issued 2016 en_ZA dc.identifier.citation Brand, H. 2016. The use of stochastic collocation for sampling from expensive distributions with applications in finance. University of Cape Town. en_ZA dc.identifier.uri http://hdl.handle.net/11427/20998 dc.description.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. en_ZA dc.language.iso eng en_ZA dc.subject.other Mathematical Finance en_ZA dc.title The use of stochastic collocation for sampling from expensive distributions with applications in finance en_ZA dc.type Master Thesis uct.type.publication Research en_ZA uct.type.resource Thesis en_ZA dc.publisher.institution University of Cape Town dc.publisher.faculty Faculty of Commerce en_ZA dc.publisher.department Division of Actuarial Science en_ZA dc.type.qualificationlevel Masters dc.type.qualificationname MPhil en_ZA uct.type.filetype Text uct.type.filetype Image dc.identifier.apacitation Brand, H. (2016). The use of stochastic collocation for sampling from expensive distributions with applications in finance. (Thesis). University of Cape Town ,Faculty of Commerce ,Division of Actuarial Science. Retrieved from http://hdl.handle.net/11427/20998 en_ZA dc.identifier.chicagocitation Brand, Hilmarie. "The use of stochastic collocation for sampling from expensive distributions with applications in finance." Thesis., University of Cape Town ,Faculty of Commerce ,Division of Actuarial Science, 2016. http://hdl.handle.net/11427/20998 en_ZA dc.identifier.vancouvercitation Brand H. The use of stochastic collocation for sampling from expensive distributions with applications in finance. [Thesis]. University of Cape Town ,Faculty of Commerce ,Division of Actuarial Science, 2016 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/20998 en_ZA dc.identifier.ris TY - Thesis / Dissertation en_ZA AU - Brand, Hilmarie AB - 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. DA - 2016 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 2016 T1 - The use of stochastic collocation for sampling from expensive distributions with applications in finance TI - The use of stochastic collocation for sampling from expensive distributions with applications in finance UR - http://hdl.handle.net/11427/20998 ER -
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