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Convolution-FFT for option pricing in the Heston model

Xiang Gao, Cody Hyndman

TL;DR

This work develops a convolution-FFT method for European option pricing under the Heston model, leveraging a continuously differentiable joint characteristic function to avoid branch-cut discontinuities. By formulating pricing as a convolution with the log-price transition density on a truncated domain, the authors provide explicit analytical bounds for truncation and discretization errors and introduce damping and exponential shifting boundary controls to enhance stability. They implement two variants, CFFT-I and CFFT-II, and demonstrate through numerical experiments that the method achieves high accuracy with competitive or superior speed compared to existing FFT approaches. The framework offers a solid theoretical and practical foundation for efficient calibration and pricing, with potential extensions to multi-factor volatility models and more complex derivatives.

Abstract

We propose a convolution-FFT method for pricing European options under the Heston model that leverages a continuously differentiable representation of the joint characteristic function. Unlike existing Fourier-based methods that rely on branch-cut adjustments or empirically tuned damping parameters, our approach yields a stable integrand even under large frequency oscillations. Crucially, we derive fully analytical error bounds that quantify both truncation error and discretization error in terms of model parameters and grid settings. To the best of our knowledge, this is the first work to provide such explicit, closed-form error estimates for an FFT-based convolution method specialized to the Heston model. Numerical experiments confirm the theoretical rates and illustrate robust, high-accuracy option pricing at modest computational cost.

Convolution-FFT for option pricing in the Heston model

TL;DR

This work develops a convolution-FFT method for European option pricing under the Heston model, leveraging a continuously differentiable joint characteristic function to avoid branch-cut discontinuities. By formulating pricing as a convolution with the log-price transition density on a truncated domain, the authors provide explicit analytical bounds for truncation and discretization errors and introduce damping and exponential shifting boundary controls to enhance stability. They implement two variants, CFFT-I and CFFT-II, and demonstrate through numerical experiments that the method achieves high accuracy with competitive or superior speed compared to existing FFT approaches. The framework offers a solid theoretical and practical foundation for efficient calibration and pricing, with potential extensions to multi-factor volatility models and more complex derivatives.

Abstract

We propose a convolution-FFT method for pricing European options under the Heston model that leverages a continuously differentiable representation of the joint characteristic function. Unlike existing Fourier-based methods that rely on branch-cut adjustments or empirically tuned damping parameters, our approach yields a stable integrand even under large frequency oscillations. Crucially, we derive fully analytical error bounds that quantify both truncation error and discretization error in terms of model parameters and grid settings. To the best of our knowledge, this is the first work to provide such explicit, closed-form error estimates for an FFT-based convolution method specialized to the Heston model. Numerical experiments confirm the theoretical rates and illustrate robust, high-accuracy option pricing at modest computational cost.

Paper Structure

This paper contains 12 sections, 4 theorems, 104 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Heston model with characteristic function
  3. Heston's stochastic volatility model
  4. Convolution-FFT method
  5. Error analysis
  6. Boundary control for the CFFT method
  7. Numerical results
  8. Conclusion
  9. Appendix
  10. Proof of Theorem \ref{['thm_characteristic']}
  11. Proof of Proposition \ref{['Prop_limiting_charac']}
  12. Proof of Theorem \ref{['thm_error_bound']}

Key Result

Theorem 2.1

(Joint characteristic function) The characteristic function of the joint process $X_t = (x_t, v_t)^{\top}$ under $P_i$, with initial condition $X = (x, v)^{\top}$ and Fourier variables $U = (p, q)$, is given by where

Figures (6)

  • Figure 2.1: Heston's characteristic function
  • Figure 2.2: Joint characteristic function
  • Figure 4.3: Comparison of CFFT-I probabilities $P_1$ and $P_2$ with and without the shifting scheme.
  • Figure 4.4: Error of CFFT-I
  • Figure 4.5: Error of CFFT-II
  • ...and 1 more figures

Theorems & Definitions (9)

  • Definition 2.1
  • Theorem 2.1
  • Remark 3.1
  • Proposition 3.1: Asymptotic characteristic function
  • Theorem 3.1: Error of the convolution method
  • Corollary 3.1
  • proof
  • proof
  • proof