Efficient Simulation of Stochastic Differential Equations Based on Markov Chain Approximations With Applications

49 Pages Posted: 11 Sep 2020

See all articles by Zhenyu Cui

Zhenyu Cui

Stevens Institute of Technology - School of Business

Justin Kirkby

Georgia Institute of Technology - The H. Milton Stewart School of Industrial & Systems Engineering (ISyE)

Duy Nguyen

Marist College - Department of Mathematics

Date Written: August 2, 2020

Abstract

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases.

Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme.

Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms \textit{both biased and unbiased} simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory. Note: this is an earlier version of the work "Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models based on Markov Chain Approximations".

Keywords: Simulation; SABR; stochastic local volatility; Markov chain; stochastic differential equation; finance

JEL Classification: G13, C15, C02

Suggested Citation

Cui, Zhenyu and Kirkby, Justin and Nguyen, Duy, Efficient Simulation of Stochastic Differential Equations Based on Markov Chain Approximations With Applications (August 2, 2020). Available at SSRN: https://ssrn.com/abstract=3665661 or http://dx.doi.org/10.2139/ssrn.3665661

Zhenyu Cui

Stevens Institute of Technology - School of Business ( email )

Hoboken, NJ 07030
United States

HOME PAGE: http://sites.google.com/site/zhenyucui86/publications

Justin Kirkby (Contact Author)

Georgia Institute of Technology - The H. Milton Stewart School of Industrial & Systems Engineering (ISyE) ( email )

765 Ferst Drive
Atlanta, GA 30332-0205
United States

Duy Nguyen

Marist College - Department of Mathematics ( email )

NY
United States

HOME PAGE: http://sites.google.com/site/nducduy/

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