Fast Fourier Transform (FFT)

Convolution-FFT for option pricing in the Heston model ArXiv ID: 2512.05326 “View on arXiv” Authors: Xiang Gao, Cody Hyndman 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. ...

NUFFT for the Fast COS Method ArXiv ID: 2507.13186 “View on arXiv” Authors: Fabien LeFloc’h Abstract The COS method is a very efficient way to compute European option prices under Lévy models or affine stochastic volatility models, based on a Fourier Cosine expansion of the density, involving the characteristic function. This note shows how to compute the COS method formula with a non-uniform fast Fourier transform, thus allowing to price many options of the same maturity but different strikes at an unprecedented speed. ...

Numerical analysis of American option pricing in a two-asset jump-diffusion model ArXiv ID: 2410.04745 “View on arXiv” Authors: Unknown Abstract This paper addresses an important gap in rigorous numerical treatments for pricing American options under correlated two-asset jump-diffusion models using the viscosity solution framework, with a particular focus on the Merton model. The pricing of these options is governed by complex two-dimensional (2-D) variational inequalities that incorporate cross-derivative terms and nonlocal integro-differential terms due to the presence of jumps. Existing numerical methods, primarily based on finite differences, often struggle with preserving monotonicity in the approximation of cross-derivatives, a key requirement for ensuring convergence to the viscosity solution. In addition, these methods face challenges in accurately discretizing 2-D jump integrals. We introduce a novel approach to effectively tackle the aforementioned variational inequalities while seamlessly handling cross-derivative terms and nonlocal integro-differential terms through an efficient and straightforward-to-implement monotone integration scheme. Within each timestep, our approach explicitly enforces the inequality constraint, resulting in a 2-D Partial Integro-Differential Equation (PIDE) to solve. Its solution is expressed as a 2-D convolution integral involving the Green’s function of the PIDE. We derive an infinite series representation of this Green’s function, where each term is non-negative and computable. This facilitates the numerical approximation of the PIDE solution through a monotone integration method. To enhance efficiency, we develop an implementation of this monotone scheme via FFTs, exploiting the Toeplitz matrix structure. The proposed method is proved to be both $\ell_{"\infty"} $-stable and consistent in the viscosity sense, ensuring its convergence to the viscosity solution of the variational inequality. Extensive numerical results validate the effectiveness and robustness of our approach. ...

A monotone piecewise constant control integration approach for the two-factor uncertain volatility model ArXiv ID: 2402.06840 “View on arXiv” Authors: Unknown Abstract Option contracts on two underlying assets within uncertain volatility models have their worst-case and best-case prices determined by a two-dimensional (2D) Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) with cross-derivative terms. This paper introduces a novel ``decompose and integrate, then optimize’’ approach to tackle this HJB PDE. Within each timestep, our method applies piecewise constant control, yielding a set of independent linear 2D PDEs, each corresponding to a discretized control value. Leveraging closed-form Green’s functions, these PDEs are efficiently solved via 2D convolution integrals using a monotone numerical integration method. The value function and optimal control are then obtained by synthesizing the solutions of the individual PDEs. For enhanced efficiency, we implement the integration via Fast Fourier Transforms, exploiting the Toeplitz matrix structure. The proposed method is $\ell_{"\infty"}$-stable, consistent in the viscosity sense, and converges to the viscosity solution of the HJB equation. Numerical results show excellent agreement with benchmark solutions obtained by finite differences, tree methods, and Monte Carlo simulation, highlighting its robustness and effectiveness. ...

Fast and General Simulation of Lévy-driven OU processes for Energy Derivatives ArXiv ID: 2401.15483 “View on arXiv” Authors: Unknown Abstract Lévy-driven Ornstein-Uhlenbeck (OU) processes represent an intriguing class of stochastic processes that have garnered interest in the energy sector for their ability to capture typical features of market dynamics. However, in the current state of play, Monte Carlo simulations of these processes are not straightforward for two main reasons: i) algorithms are available only for some specific processes within this class; ii) they are often computationally expensive. In this paper, we introduce a new simulation technique designed to address both challenges. It relies on the numerical inversion of the characteristic function, offering a general methodology applicable to all Lévy-driven OU processes. Moreover, leveraging FFT, the proposed methodology ensures fast and accurate simulations, providing a solid basis for the widespread adoption of these processes in the energy sector. Lastly, the algorithm allows an optimal control of the numerical error. We apply the technique to the pricing of energy derivatives, comparing the results with the existing benchmarks. Our findings indicate that the proposed methodology is at least one order of magnitude faster than the existing algorithms, while maintaining an equivalent level of accuracy. ...