[Paper Review] A Finite Element Framework for Option Pricing with the Bates Model

Edie Miglio, Carlo Sgarra|ArXiv.org|Dec 16, 2008

Stochastic processes and financial applications15 references3 citations

TL;DR

This paper presents a finite element method (FEM) for pricing European options under the Bates stochastic volatility model with jumps, formulating the pricing problem as a parabolic partial-integro-differential equation (PIDE). The method achieves high accuracy and robustness, with numerical results showing strong agreement (within 0.1% deviation) with the benchmark Fast Fourier Transform (FFT) method, validating its effectiveness for complex option pricing under jump-diffusion dynamics.

ABSTRACT

In the present paper we present a finite element approach for option pricing in the framework of a well-known stochastic volatility model with jumps, the Bates model. In this model the asset log-returns are assumed to follow a jump-diffusion model where the jump component consists of a Levy process of compound Poisson type, while the volatility behavior is described by a stochastic differential equation of CIR type, with a mean-reverting drift term and a diffusion component correlated with that of the log-returns. Like in all the Levy models, the option pricing problem can be formulated in terms of an integro-differential equation: for the Bates model the unknown F(S, V, t) (the option price) of the pricing equation depends on three independent variables and the differential operator part turns out to be of parabolic kind, while the nonlocal integral operator is calculated with respect to the Levy measure of the jumps. In this paper we will present a variational formulation of the problem suitable for a finite element approach. The numerical results obtained for european options will be compared with those obtained with different methods.

Motivation & Objective

• To develop a stable and accurate finite element method for solving the partial-integro-differential equation (PIDE) arising in option pricing under the Bates model.
• To address the challenge of numerically solving a three-dimensional PIDE with non-local integral operators due to Lévy jumps and stochastic volatility.
• To provide a variational formulation suitable for finite element discretization, enabling flexible and adaptive spatial approximation.
• To validate the proposed FEM approach against established FFT-based methods for European options under various parameter sets.
• To lay the foundation for extending the method to American option pricing via free boundary problems.

Proposed method

• The authors derive a variational formulation of the PIDE governing option prices in the Bates model, transforming the problem into a weak form suitable for Galerkin finite element methods.
• The spatial domain is discretized using piecewise polynomial finite elements on an unstructured triangular mesh with 2634 elements and 1398 nodes.
• The time discretization employs an implicit scheme to ensure stability for the parabolic component of the PIDE.
• The non-local integral operator, arising from compound Poisson jumps, is approximated using quadrature rules over the Lévy measure.
• The resulting linear system is solved iteratively, with the method designed for efficiency and robustness across diverse parameter regimes.
• The approach is tested on European call options with maturity up to 3 years and strike prices between 80 and 120.

Experimental results

Research questions

• RQ1 Can a finite element method be effectively formulated and implemented for the PIDE arising in the Bates model with jumps and stochastic volatility?
• RQ2 How does the accuracy and convergence of the FEM compare to the benchmark FFT method for European option pricing?
• RQ3 What is the impact of key model parameters—especially volatility diffusion (θ), correlation (ρ), and jump intensity (λ)—on the implied volatility surface?
• RQ4 How computationally efficient is the FEM approach for pricing options under the Bates model?
• RQ5 Can the finite element framework be extended to American option pricing via free boundary problems?

Key findings

• The finite element method produces option prices that show excellent agreement with FFT results, with a maximum relative deviation of less than 0.1% across tested parameter sets.
• The method successfully captures the volatility smile and skew features in the implied volatility surfaces, particularly for short maturities and negative correlation (ρ).
• The volatility surface exhibits pronounced skews for parameter sets S₂ and S₃, with the skew being less evident for S₁ and S₄, indicating sensitivity to the diffusion coefficient θ and correlation ρ.
• The method is robust across a wide range of parameters, with consistent performance observed across four distinct calibration sets (S₁ to S₄).
• Computational cost is moderate: approximately 3 minutes per option price and 3 hours for a full volatility surface on a 2GHz dual-core processor with 1MB RAM.
• The finite element approach demonstrates strong potential for extension to American option pricing, which is formulated as a free boundary problem for the same PIDE.