Second order finite volume methods with IMEX time-marching for linear and nonlinear parabolic PDE problems in option pricing

This article deals with the development of  second order numerical schemes for solving option pricing problems, given by linear or nonlinear  parabolic  partial differential equations (PDEs), with nonlinearities in the source and/or convection terms. These equations will be discretized using second order finite volume  Implicit-Explicit (IMEX) Runge-Kutta schemes.

The here proposed numerical schemes have several advantages. First of all, they are able to reach high order,  not only in the presence of  nonregular initial conditions, the usual situation in finance, but also in the case of nonlinear advection and/or reaction terms, which appear in many recent and important PDEs in finance.

Furthermore, the proposed schemes combine explicit and implicit time discretizations in a highly efficient way. They allow to take large time steps, overcoming the strict stability condition imposed for the diffusion terms in explicit schemes. Also, the here proposed numerical schemes offer highly-accurate and non oscillatory approximations of option prices and their Greeks.

Finally the developed numerical methods rely in a very general methodology, as they make use of well established techniques in the finite volume literature, such as numerical fluxes based in finite volume solvers, high order reconstruction operators or IMEX time marching schemes for stiff problems. This allows to apply these numerical schemes to a broad range of challenging state of the art problems in finance, given by nonlinear parabolic PDEs.