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Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models

Michael Samet, Christian Bayer, Chiheb Ben Hammouda, Antonis Papapantoleon, Raúl Tempone

TL;DR

This work tackles the efficient pricing of European multi-asset options under multivariate models by marrying an optimal damping rule for Fourier integrals with hierarchical deterministic quadrature that is sparse and dimension-adaptive. The approach smooths the Fourier integrand and leverages sparsified, adaptive grids to combat high dimensionality, achieving accelerated convergence across GBM, VG, and NIG models. Empirical results show substantial speed-ups over the COS method in 2D and dramatic reductions in Monte Carlo cost up to six dimensions, highlighting practical benefits for calibration and pricing in Lévy-based frameworks. Overall, the paper delivers a scalable, accurate Fourier pricing framework for basket and rainbow options with strong implications for efficient multi-asset option analytics.

Abstract

Efficiently pricing multi-asset options is a challenging problem in quantitative finance. When the characteristic function is available, Fourier-based methods are competitive compared to alternative techniques because the integrand in the frequency space often has a higher regularity than that in the physical space. However, when designing a numerical quadrature method for most Fourier pricing approaches, two key aspects affecting the numerical complexity should be carefully considered: (i) the choice of damping parameters that ensure integrability and control the regularity class of the integrand and (ii) the effective treatment of high dimensionality. We propose an efficient numerical method for pricing European multi-asset options based on two complementary ideas to address these challenges. First, we smooth the Fourier integrand via an optimized choice of the damping parameters based on a proposed optimization rule. Second, we employ sparsification and dimension-adaptivity techniques to accelerate the convergence of the quadrature in high dimensions. The extensive numerical study on basket and rainbow options under the multivariate geometric Brownian motion and some Lévy models demonstrates the advantages of adaptivity and the damping rule on the numerical complexity of quadrature methods. Moreover, for the tested two-asset examples, the proposed approach outperforms the COS method in terms of computational time. Finally, we show significant speed-up compared to the Monte Carlo method for up to six dimensions.

Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models

TL;DR

This work tackles the efficient pricing of European multi-asset options under multivariate models by marrying an optimal damping rule for Fourier integrals with hierarchical deterministic quadrature that is sparse and dimension-adaptive. The approach smooths the Fourier integrand and leverages sparsified, adaptive grids to combat high dimensionality, achieving accelerated convergence across GBM, VG, and NIG models. Empirical results show substantial speed-ups over the COS method in 2D and dramatic reductions in Monte Carlo cost up to six dimensions, highlighting practical benefits for calibration and pricing in Lévy-based frameworks. Overall, the paper delivers a scalable, accurate Fourier pricing framework for basket and rainbow options with strong implications for efficient multi-asset option analytics.

Abstract

Efficiently pricing multi-asset options is a challenging problem in quantitative finance. When the characteristic function is available, Fourier-based methods are competitive compared to alternative techniques because the integrand in the frequency space often has a higher regularity than that in the physical space. However, when designing a numerical quadrature method for most Fourier pricing approaches, two key aspects affecting the numerical complexity should be carefully considered: (i) the choice of damping parameters that ensure integrability and control the regularity class of the integrand and (ii) the effective treatment of high dimensionality. We propose an efficient numerical method for pricing European multi-asset options based on two complementary ideas to address these challenges. First, we smooth the Fourier integrand via an optimized choice of the damping parameters based on a proposed optimization rule. Second, we employ sparsification and dimension-adaptivity techniques to accelerate the convergence of the quadrature in high dimensions. The extensive numerical study on basket and rainbow options under the multivariate geometric Brownian motion and some Lévy models demonstrates the advantages of adaptivity and the damping rule on the numerical complexity of quadrature methods. Moreover, for the tested two-asset examples, the proposed approach outperforms the COS method in terms of computational time. Finally, we show significant speed-up compared to the Monte Carlo method for up to six dimensions.
Paper Structure (21 sections, 3 theorems, 43 equations, 11 figures, 10 tables)

This paper contains 21 sections, 3 theorems, 43 equations, 11 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Problem Setting and Pricing Framework
  3. Multivariate Fourier Pricing Valuation Formula
  4. Payoffs and Multivariate Asset Models
  5. Payoffs and their Fourier Transforms
  6. Multivariate Models and the Corresponding Charactersitic Functions
  7. Methodology
  8. Motivation and Characterization of the Optimal Damping Rule
  9. Motivation of the Damping Rule
  10. Characterization of the Optimal Damping Rule
  11. Numerical evaluation of the inverse Fourier integrals using hierarchical deterministic quadrature methods
  12. Numerical Experiments and Results
  13. Combining the optimal damping heuristic rule with hierarchical deterministic quadrature methods
  14. Effect of sparsification and dimension-adaptivity
  15. Effect of the optimal damping rule
  16. ...and 6 more sections

Key Result

Proposition 2.4

We use Notation notation and suppose Assumptions ass:Assumptions on the payoff and ass:Assumptions on the distribution hold, and that $\delta_V = \delta_X \cap \delta_P \neq \emptyset$, then, for $\mathbf{R} \in \delta_V$, the option value is given by

Figures (11)

  • Figure 3.1: 1D illustration: (Left) Shape of the integrand w.r.t the damping parameter, R. (Right) $\mathcal{E}_{R}$ convergence w.r.t. $N$, using Gauss--Laguerre quadrature for the European put option under (a) GBM, (b) VG, and (c) NIG pricing models. The relative quadrature error $\mathcal{E}_{R}$ is defined as $\mathcal{E}_{R} = \frac{ \mid Q_{N}[g] - \text{Reference Value} \mid }{ \text{Reference Value}}$, where $Q_N$ is the quadrature estimator of \ref{['QOI']} based on the Gauss--Laguerre rule.
  • Figure 3.2: Effect of the damping parameters on the shape of the integrand in the case of 2D-basket put option under the VG model with parameters $\boldsymbol{\sigma} = (0.4,0.4)$, $\boldsymbol{\theta} = (-0.3,-0.3), \nu = 0.257$. (a) $\mathbf{R}=(0.2,0.2)$ (b) $\mathbf{R}=(1,1)$, (c) $\mathbf{R}=(2,2)$, (d) $\mathbf{R}=(3,3)$.
  • Figure 4.1: GBM: Convergence of the relative quadrature error, $\mathcal{E}_{R}$, w.r.t. $N$ for TP, SM and ASGQ methods for European $4$-asset options, when optimal damping parameters, $\mathbf{\overline{R}}$, are used.
  • Figure 4.2: VG: convergence of the relative quadrature error, $\mathcal{E}_{R}$, w.r.t. $N$ for TP, SM and ASGQ methods for European $4$-asset options, when optimal damping parameters, $\mathbf{\overline{R}}$, are used.
  • Figure 4.3: NIG: convergence of the relative quadrature error, $\mathcal{E}_{R}$, w.r.t. $N$ for TP, SM and ASGQ methods for European $4$-asset options, when optimal damping parameters, $\mathbf{\overline{R}}$, are used.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Proposition 2.4: Multivariate Fourier pricing valuation formula
  • proof
  • Remark 2.5: Connection to the valuation formula in eberlein2010analysis
  • Remark 2.6: Case of discontinuous payoffs
  • Example 2.7: Multivariate Geometric Brownian Motion (GBM)
  • Example 2.8: Multivariate Variance Gamma (VG) luciano2006multivariate
  • Example 2.9: Multivariate Normal Inverse Gaussian (NIG) barndorff1997normalbarndorff1977exponentially
  • Remark 2.10: About the strip of regularity
  • Remark 2.11: Efficient vectorized implementation for model calibration
  • Theorem 3.1
  • ...and 6 more