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Numerical methods for solving PIDEs arising in swing option pricing under a two-factor mean-reverting model with jumps

Mustapha Regragui, Karel J. in 't Hout, Michèle Vanmaele, Fred Espen Benth

TL;DR

The paper addresses numerical valuation of swing options under a two-factor affine mean-reverting model with jumps, yielding a sequence of two-dimensional PIDEs with convection-dominated dynamics and a nonlocal jump term, under nonsmooth payoffs. It introduces and analyzes three second-order schemes: a semi-Lagrangian approach (SLCNFI) and two high-order semidiscretisation-based methods (CNFI and DIRKFI) that treat the integral term explicitly via fixed-point iterations, with Crank–Nicolson and L-stable DIRK time stepping. The authors prove stability and convergence results for the CNFI and DIRKFI schemes under Dirichlet boundaries and demonstrate second-order convergence through extensive experiments on European calls and swing options under Merton and Kou jumps, including challenging convection-dominated regimes. The findings show these schemes are robust and accurate for practical swing-option pricing in electricity markets, even with nonsmooth initial data and nonlocal terms, offering reliable tools for dynamic programming-based valuations.

Abstract

This paper concerns the numerical valuation of swing options with discrete action times under a linear two-factor mean-reverting model with jumps. The resulting sequence of two-dimensional partial integro-differential equations (PIDEs) are convection-dominated and possess a nonlocal integral term due to the presence of jumps. Further, the initial function is nonsmooth. We propose various second-order numerical methods that can adequately handle these challenging features. The stability and convergence of these numerical methods are analysed theoretically. By ample numerical experiments, we confirm their second-order convergence behaviour.

Numerical methods for solving PIDEs arising in swing option pricing under a two-factor mean-reverting model with jumps

TL;DR

The paper addresses numerical valuation of swing options under a two-factor affine mean-reverting model with jumps, yielding a sequence of two-dimensional PIDEs with convection-dominated dynamics and a nonlocal jump term, under nonsmooth payoffs. It introduces and analyzes three second-order schemes: a semi-Lagrangian approach (SLCNFI) and two high-order semidiscretisation-based methods (CNFI and DIRKFI) that treat the integral term explicitly via fixed-point iterations, with Crank–Nicolson and L-stable DIRK time stepping. The authors prove stability and convergence results for the CNFI and DIRKFI schemes under Dirichlet boundaries and demonstrate second-order convergence through extensive experiments on European calls and swing options under Merton and Kou jumps, including challenging convection-dominated regimes. The findings show these schemes are robust and accurate for practical swing-option pricing in electricity markets, even with nonsmooth initial data and nonlocal terms, offering reliable tools for dynamic programming-based valuations.

Abstract

This paper concerns the numerical valuation of swing options with discrete action times under a linear two-factor mean-reverting model with jumps. The resulting sequence of two-dimensional partial integro-differential equations (PIDEs) are convection-dominated and possess a nonlocal integral term due to the presence of jumps. Further, the initial function is nonsmooth. We propose various second-order numerical methods that can adequately handle these challenging features. The stability and convergence of these numerical methods are analysed theoretically. By ample numerical experiments, we confirm their second-order convergence behaviour.

Paper Structure

This paper contains 24 sections, 9 theorems, 108 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Swing option price modelling
  3. The electricity spot price
  4. Formulation of the problem
  5. Spatial discretisation
  6. Diffusion-reaction part
  7. Integral part
  8. Convection part
  9. Semi-Lagrangian method
  10. Semidiscrete approximation
  11. Temporal discretisation
  12. Temporal scheme for the semi-Lagrangian approach
  13. Temporal schemes for the semidiscretisation approach
  14. Convergence and stability analysis for the semidiscretisation approach
  15. Convergence of CNFI and DIRKFI
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\varepsilon_{\ell} = V^{n+1}-Y_{\ell}$ where $V^{n+1}$ is given by CNFII and $Y_{\ell}$ is given by CNfpi. Let where the weights $w_{-1},w_0,w_1,w_2$ satisfy W3. If $\kappa_x\frac{\Delta t}{2\Delta x}+\kappa_y\frac{\Delta t}{2\Delta y}< 1 + \frac{\Delta t}{2}r$, then the CNFI scheme CNfpi converges to the Crank--Nicolson scheme CNFII in the $\ell_{\infty}$-norm and

Figures (7)

  • Figure 1: Sample spatial grid for the parameter values $m_{1}=m_{2}=50,K=50, x_{\min} = -50,x_{\max}=200,y_{\min}=-150,y_{\max} = 150$, $d=10$.
  • Figure 2: European call option in the Merton-type jump model. Total discretisation errors of the SLCNFI, CNFI and DIRKFI schemes for $N= \left \lceil{\frac{m}{2}}\right \rceil$ and set 1 (top), set 2 (middle), set 3 (bottom). Added: dashed reference line for convergence order 2.
  • Figure 3: European call option in the Kou-type jump model. Total discretisation errors of the SLCNFI, CNFI and DIRKFI schemes for $N= \left \lceil{\frac{m}{2}}\right \rceil$ and set 1 (top), set 2 (middle), set 3 (bottom). Added: dashed reference line for convergence order 2.
  • Figure 4: European call option in the Merton-type jump model. Temporal discretisation errors of the CNFI and DIRKFI schemes for $m=200$ and set 1 (top), set 2 (middle), set 3 (bottom). Added: dashed reference line for convergence order 2.
  • Figure 5: European call option in the Kou-type jump model. Temporal discretisation errors of the CNFI and DIRKFI schemes for $m=200$ and set 1 (top), set 2 (middle), set 3 (bottom). Added: dashed reference line for convergence order 2.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • ...and 10 more