Weighted Monte Carlo with Least Squares and Randomized Extended Kaczmarz for Option Pricing

31 Pages Posted: 18 Oct 2019 Last revised: 23 Oct 2019

See all articles by Damir Filipović

Damir Filipović

Ecole Polytechnique Fédérale de Lausanne; Swiss Finance Institute

Kathrin Glau

Queen Mary University of London

Yuji Nakatsukasa

University of Oxford

Francesco Statti

Ecole Polytechnique Fédérale de Lausanne

Date Written: September 30, 2019

Abstract

We propose a methodology for computing single and multi-asset European option prices, and more generally expectations of scalar functions of (multivariate) random variables. This new approach combines the ability of Monte Carlo simulation to handle high-dimensional problems with the efficiency of function approximation. Specifically, we first generalize the recently developed method for multivariate integration in [arXiv:1806.05492] to integration with respect to probability measures. The method is based on the principle “approximate and integrate” in three steps i) sample the integrand at points in the integration domain, ii) approximate the integrand by solving a least-squares problem, iii) integrate the approximate function. In high-dimensional applications we face memory limitations due to large storage requirements in step ii). Combining weighted sampling and the randomized extended Kaczmarz algorithm we obtain a new efficient approach to solve large-scale least-squares problems. Our convergence and cost analysis along with numerical experiments show the effectiveness of the method in both low and high dimensions, and under the assumption of a limited number of available simulations.

Keywords: Monte Carlo, Monte Carlo under budget constraints, variance reduction, multi-asset options, Kaczmarz algorithm, weighted sampling, large-scale least-squares problems

Suggested Citation

Filipovic, Damir and Glau, Kathrin and Nakatsukasa, Yuji and Statti, Francesco, Weighted Monte Carlo with Least Squares and Randomized Extended Kaczmarz for Option Pricing (September 30, 2019). Swiss Finance Institute Research Paper No. 19-54. Available at SSRN: https://ssrn.com/abstract=3471164 or http://dx.doi.org/10.2139/ssrn.3471164

Damir Filipovic (Contact Author)

Ecole Polytechnique Fédérale de Lausanne ( email )

Odyssea
Station 5
Lausanne, 1015
Switzerland

HOME PAGE: http://people.epfl.ch/damir.filipovic

Swiss Finance Institute

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

Kathrin Glau

Queen Mary University of London ( email )

Mile End Rd
Mile End Road
London, London E1 4NS
United Kingdom

Yuji Nakatsukasa

University of Oxford ( email )

Mansfield Road
Oxford, Oxfordshire OX1 4AU
United Kingdom

Francesco Statti

Ecole Polytechnique Fédérale de Lausanne ( email )

Station 5
Odyssea 1.04
1015 Lausanne, CH-1015
Switzerland

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