Operator splitting schemes for the two-asset Merton jump-diffusion model

TL;DR

The paper tackles the numerical valuation of rainbow options under a two-asset Merton jump-diffusion by solving a two-dimensional time-dependent PIDE with a nonlocal integral and a mixed-derivative term. It implements a MOL-based spatial discretization and FFT-enabled evaluation of the integral, then analyzes seven operator-splitting time-stepping schemes (IMEX and ADI) that treat the integral explicitly. The results show that all schemes except CNFE achieve second-order temporal convergence, with the two-step MCS2 scheme (θ=1/3) consistently providing the smallest temporal error constant, especially for ensembles with heavier jumps, and the total discretization error is typically dominated by spatial discretization. The study offers practical guidance for efficient and stable pricing of multi-asset jump-diffusion options, recommending the MCS2 scheme for two-dimensional PIDEs in financial contexts.

Abstract

This paper deals with the numerical solution of the two-dimensional time-dependent Merton partial integro-differential equation (PIDE) for the values of rainbow options under the two-asset Merton jump-diffusion model. Key features of this well-known equation are a two-dimensional nonlocal integral part and a mixed spatial derivative term. For its efficient and stable numerical solution, we study seven recent and novel operator splitting schemes of the implicit-explicit (IMEX) and the alternating direction implicit (ADI) kind. Here the integral part is always conveniently treated in an explicit fashion. The convergence behaviour and the relative performance of the seven schemes are investigated in ample numerical experiments for both European put-on-the-min and put-on-the-average options.

Figures (5)

• Figure 1: Spatial grid for the European put-on-the-min (left) and the European put-on-the-average (right) option if $m=50$, $K = 100$, $S_{\rm max} = 5K$.
• Figure 2: Payoff of the European put-on-the-min (left) and the European put-on-the-average (right) option if $K = 100$, $S_{\rm max} = 5K$.
• Figure 3: Temporal errors $\widehat{E}^{ROI}(150,N)$ of the seven operator splitting schemes in the case of the European put-on-the-min (left) and the put-on-the-average (right) option under the two-asset Merton model with parameter Set 1 (top), Set 2 (mid) and Set 3 (bottom) from Table \ref{['tabpars']}. The CNFI, IETR, MCS schemes are applied with $\Delta t = T/N$ and the CNFE, CNAB, MCS2, SC2A schemes with $\Delta t = T/(2N)$.
• Figure 4: Total errors $E^{ROI}(m,N)$ in the case of the European put-on-the-min option under the two-asset Merton model with parameter Set 1 (top), Set 2 (mid) and Set 3 (bottom) from Table \ref{['tabpars']}. The CNFI, IETR, MCS schemes are applied with $\Delta t = T/N$ and the CNFE, CNAB, MCS2, SC2A schemes with $\Delta t = T/(2N)$ and $N=\lceil m/3 \rceil$.
• Figure :