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IMEX-RK finite volume methods for nonlinear 1d parabolic PDEs. Application to option pricing

J. G. López-Salas, M. Suárez-Taboada, M. J. Castro, A. M. Ferreiro-Ferreiro, J. A. García-Rodríguez

TL;DR

This paper develops a 2nd-order implicit-explicit Runge-Kutta finite-volume framework for solving nonlinear 1D parabolic PDEs in option pricing. By discretizing convection and source terms explicitly and diffusion implicitly, the scheme achieves high accuracy on non-smooth initial data and avoids fixed-point iterations, while maintaining computational efficiency. The authors demonstrate strong convergence and substantial time-step speedups over explicit methods on both linear barrier-options and nonlinear counterparty-risk models, with accurate Greeks. The approach provides a robust, extendable tool for efficient pricing and risk assessment in financial contexts with nonlinear PDEs.

Abstract

The goal of this paper is to develop 2nd order Implicit-Explicit Runge-Kutta (IMEX-RK) finite volume (FV) schemes for solving 1d parabolic PDEs for option pricing, with possible nonlinearities in the source and advection terms. The spatial semi-discretization of the advection is carried out by combining finite volume methods with 2nd order state reconstructions; while the diffusive terms are discretized using second-order finite differences. The time integration is performed by means of IMEX-RK time integrators: the advection is treated explicitly, and the diffusion, implicitly. The obtained numerical schemes have several advantages: they are computationally very efficient, thanks to the implicit discretization of the diffusion in the IMEX-RK time integrators, which allows to overcome the strict time step restriction; they yield second-order accuracy for even nonlinear problems and with non-regular initial conditions; and they can be extended to higher order.

IMEX-RK finite volume methods for nonlinear 1d parabolic PDEs. Application to option pricing

TL;DR

This paper develops a 2nd-order implicit-explicit Runge-Kutta finite-volume framework for solving nonlinear 1D parabolic PDEs in option pricing. By discretizing convection and source terms explicitly and diffusion implicitly, the scheme achieves high accuracy on non-smooth initial data and avoids fixed-point iterations, while maintaining computational efficiency. The authors demonstrate strong convergence and substantial time-step speedups over explicit methods on both linear barrier-options and nonlinear counterparty-risk models, with accurate Greeks. The approach provides a robust, extendable tool for efficient pricing and risk assessment in financial contexts with nonlinear PDEs.

Abstract

The goal of this paper is to develop 2nd order Implicit-Explicit Runge-Kutta (IMEX-RK) finite volume (FV) schemes for solving 1d parabolic PDEs for option pricing, with possible nonlinearities in the source and advection terms. The spatial semi-discretization of the advection is carried out by combining finite volume methods with 2nd order state reconstructions; while the diffusive terms are discretized using second-order finite differences. The time integration is performed by means of IMEX-RK time integrators: the advection is treated explicitly, and the diffusion, implicitly. The obtained numerical schemes have several advantages: they are computationally very efficient, thanks to the implicit discretization of the diffusion in the IMEX-RK time integrators, which allows to overcome the strict time step restriction; they yield second-order accuracy for even nonlinear problems and with non-regular initial conditions; and they can be extended to higher order.
Paper Structure (8 sections, 14 equations, 3 figures, 2 tables)

This paper contains 8 sections, 14 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Numerical methods. Finite volume IMEX Runge-Kutta
  3. Finite volume spatial semi-discretization
  4. IMEX Runge-Kutta time discretization
  5. Numerical experiments
  6. Linear example. Options under the Black-Scholes model
  7. Nonlinear example. Counterparty credit risk
  8. Conclusions

Figures (3)

  • Figure 1: Down-and-out call option prices, numerical errors and Greeks ($\Delta$, $\Gamma$) at $t=T$.
  • Figure 2: Call option prices with valuation adjustments at $t=T$. Exact value in continous line, numerical value with dots, for different values of the default intensities $\lambda_B$
  • Figure 3: $\Delta$, left, and $\Gamma$, right, with valuation adjustments at $t=T$. Exact value in continous line, numerical value with dots, for different values of the default intensities $\lambda_B$