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Numerical valuation of European options under two-asset infinite-activity exponential Lévy models

Massimiliano Moda, Karel J. in 't Hout, Michèle Vanmaele, Fred Espen Benth

TL;DR

This work develops a robust numerical framework for pricing European options on two assets when the underlying follows a 2D infinite-activity exponential Lévy process, with a focus on Normal Tempered Stable dynamics. It extends the 1D approach of Wang, Wan & Forsyth to handle the high-dimensional nonlocal integral via a tailored discretization and an FFT-based evaluator, paired with a semi-Lagrangian θ-time stepping and a fixed-point scheme for the jump term. The scheme uses a three-region integral partition, a nonuniform spatial grid, cell averaging for non-smooth payoffs, and Krylov-type iterative solvers to achieve stable, second-order convergence in finite-variation settings. The numerical experiments demonstrate accurate European option prices and convergence behavior across representative VG and NIG parameter sets, highlighting practical applicability for two-asset derivatives under Lévy models.

Abstract

We propose a numerical method for the valuation of European-style options under two-asset infinite-activity exponential Lévy models. Our method extends the effective approach developed by Wang, Wan & Forsyth (2007) for the 1-dimensional case to the 2-dimensional setting and is applicable for general Lévy measures under mild assumptions. A tailored discretization of the non-local integral term is developed, which can be efficiently evaluated by means of the fast Fourier transform. For the temporal discretization, the semi-Lagrangian theta-method is employed in a convenient splitting fashion, where the diffusion term is treated implicitly and the integral term is handled explicitly by a fixed-point iteration. Numerical experiments for put-on-the-average options under Normal Tempered Stable dynamics reveal favourable second-order convergence of our method whenever the exponential Lévy process has finite-variation.

Numerical valuation of European options under two-asset infinite-activity exponential Lévy models

TL;DR

This work develops a robust numerical framework for pricing European options on two assets when the underlying follows a 2D infinite-activity exponential Lévy process, with a focus on Normal Tempered Stable dynamics. It extends the 1D approach of Wang, Wan & Forsyth to handle the high-dimensional nonlocal integral via a tailored discretization and an FFT-based evaluator, paired with a semi-Lagrangian θ-time stepping and a fixed-point scheme for the jump term. The scheme uses a three-region integral partition, a nonuniform spatial grid, cell averaging for non-smooth payoffs, and Krylov-type iterative solvers to achieve stable, second-order convergence in finite-variation settings. The numerical experiments demonstrate accurate European option prices and convergence behavior across representative VG and NIG parameter sets, highlighting practical applicability for two-asset derivatives under Lévy models.

Abstract

We propose a numerical method for the valuation of European-style options under two-asset infinite-activity exponential Lévy models. Our method extends the effective approach developed by Wang, Wan & Forsyth (2007) for the 1-dimensional case to the 2-dimensional setting and is applicable for general Lévy measures under mild assumptions. A tailored discretization of the non-local integral term is developed, which can be efficiently evaluated by means of the fast Fourier transform. For the temporal discretization, the semi-Lagrangian theta-method is employed in a convenient splitting fashion, where the diffusion term is treated implicitly and the integral term is handled explicitly by a fixed-point iteration. Numerical experiments for put-on-the-average options under Normal Tempered Stable dynamics reveal favourable second-order convergence of our method whenever the exponential Lévy process has finite-variation.

Paper Structure

This paper contains 18 sections, 1 theorem, 73 equations, 4 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model framework
  3. Market model
  4. Initial boundary value problem for derivatives pricing
  5. Numerical scheme
  6. Integral discretization
  7. Spatial discretization
  8. Diffusion term
  9. Summation term
  10. Cell averaging
  11. Temporal discretization: the semi-Lagrangian $\theta$-method
  12. Numerical experiments
  13. Conclusions
  14. Acknowledgements
  15. $d$-dimensional Normal Tempered Stable process
  16. ...and 3 more sections

Key Result

Proposition A.1

Consider a Lévy measure $\ell$ over $\mathbb{R}^d_*=\mathbb{R}^d\setminus\{0\}$. Assume that there exist constants $A_{\ell}$ and $B_{\ell}$, and for any given $h>0$ a constant $C_{\ell}(h)$ such that Then, for a Normal Tempered Stable process these constants are given by

Figures (4)

  • Figure 1: Partition of the integration domain $R_{z}$
  • Figure 2: Diagram of the scheme used to approximate $\left(\mathcal{B}_{\omega}v\right)\left(\mathbf{x},t\right)$
  • Figure 3: European put-on-the-average option price and the Greeks Delta and Gamma for the parameter set NIG0
  • Figure 4: Total error in $\left[0,3K\right]\times\left[0,3K\right]$

Theorems & Definitions (2)

  • Proposition A.1
  • proof