JID:APNUM AID:2817 /FLA [m3G; v 1.132; Prn:23/04/2014; 15:23] P.1(1-17) Applied Numerical Mathematics ••• (••••) •••–••• Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum Particle methods for PDEs arising in financial modeling Shumo Cui a , Alexander Kurganov a,∗ , Alexei Medovikov b a Mathematics Department, Tulane University, New Orleans, LA 70118, USA b Susquehanna International Group, 401 City Avenue, Bala Cynwyd, PA 19131, USA article info abstract Article history: Available online xxxx Keywords: Convection–diffusion equations Anisotropic diffusion PDE-based bond pricing models Deterministic and stochastic particle methods Monte-Carlo method We numerically study convection–diffusion equations arising in financial modeling. We focus on the convection-dominated cases, in which the diffusion coefficients are relatively small. Both finite-difference and Monte-Carlo methods which are widely used in the problems of this kind might be inefficient due to severe restrictions on the meshsize and the number of realizations needed to achieve high resolution. We propose an alternative approach based on particle methods which have extremely low numerical diffusion and thus do not have the aforementioned restrictions. Our approach is based on the operator splitting: The hyperbolic steps are made using the method of characteristics, while the parabolic steps are performed using either a special discretization of the integral representation of the solution (which leads to a deterministic particle method) or a stochastic random walk approach. We apply the designed particle methods to a variety of test problems and the numerical results indicate high accuracy, efficiency and robustness of both the deterministic and stochastic methods. In addition, our numerical experiments clearly demonstrate that the deterministic particle method outperforms its stochastic counterpart.  2014 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction In this paper, we are interested in development of highly accurate and efficient numerical method for multidimensional convection–diffusion equations with strictly anisotropic diffusion acting in just one spatial direction. Such equations arise, for instance, in bond pricing models. We will consider a particular example of the following linear two-dimensional (2-D) convection–diffusion equation: u t + κ  ( ˆ x − x)u  x + xu y = σ 2 2 u xx , (1.1) subject to the singular initial condition u(x, y, 0) = δ(x − r 0 , y), (1.2) where δ stands for the Dirac delta-function and κ , ˆ x, σ and r 0 are positive constants. In Appendix A, we show how the initial value problem (IVP) (1.1), (1.2) can be derived from the Vasicek bond pricing model [31]. * Corresponding author. E-mail addresses: scui2@tulane.edu (S. Cui), kurganov@math.tulane.edu (A. Kurganov), Alexei.Medovikov@sig.com (A. Medovikov). URL: http://www.math.tulane.edu/~kurganov (A. Kurganov). http://dx.doi.org/10.1016/j.apnum.2014.04.005 0168-9274/ 2014 IMACS. Published by Elsevier B.V. All rights reserved.