Fourier Transform

Boundary error control for numerical solution of BSDEs by the convolution-FFT method ArXiv ID: 2512.24714 “View on arXiv” Authors: Xiang Gao, Cody Hyndman Abstract We first review the convolution fast-Fourier-transform (CFFT) approach for the numerical solution of backward stochastic differential equations (BSDEs) introduced in (Hyndman and Oyono Ngou, 2017). We then propose a method for improving the boundary errors obtained when valuing options using this approach. We modify the damping and shifting schemes used in the original formulation, which transforms the target function into a bounded periodic function so that Fourier transforms can be applied successfully. Time-dependent shifting reduces boundary error significantly. We present numerical results for our implementation and provide a detailed error analysis showing the improved accuracy and convergence of the modified convolution method. ...

Tensor train representations of Greeks for Fourier-based pricing of multi-asset options ArXiv ID: 2507.08482 “View on arXiv” Authors: Rihito Sakurai, Koichi Miyamoto, Tsuyoshi Okubo Abstract Efficient computation of Greeks for multi-asset options remains a key challenge in quantitative finance. While Monte Carlo (MC) simulation is widely used, it suffers from the large sample complexity for high accuracy. We propose a framework to compute Greeks in a single evaluation of a tensor train (TT), which is obtained by compressing the Fourier transform (FT)-based pricing function via TT learning using tensor cross interpolation. Based on this TT representation, we introduce two approaches to compute Greeks: a numerical differentiation (ND) approach that applies a numerical differential operator to one tensor core and an analytical (AN) approach that constructs the TT of closed-form differentiation expressions of FT-based pricing. Numerical experiments on a five-asset min-call option in the Black-Sholes model show significant speed-ups of up to about $10^{“5”} \times$ over MC while maintaining comparable accuracy. The ND approach matches or exceeds the accuracy of the AN approach and requires lower computational complexity for constructing the TT representation, making it the preferred choice. ...

Learning parameter dependence for Fourier-based option pricing with tensor trains ArXiv ID: 2405.00701 “View on arXiv” Authors: Unknown Abstract A long-standing issue in mathematical finance is the speed-up of option pricing, especially for multi-asset options. A recent study has proposed to use tensor train learning algorithms to speed up Fourier transform (FT)-based option pricing, utilizing the ability of tensor trains to compress high-dimensional tensors. Another usage of the tensor train is to compress functions, including their parameter dependence. Here, we propose a pricing method, where, by a tensor train learning algorithm, we build tensor trains that approximate functions appearing in FT-based option pricing with their parameter dependence and efficiently calculate the option price for the varying input parameters. As a benchmark test, we run the proposed method to price a multi-asset option for the various values of volatilities and present asset prices. We show that, in the tested cases involving up to 11 assets, the proposed method outperforms Monte Carlo-based option pricing with $10^6$ paths in terms of computational complexity while keeping better accuracy. ...

Introduction to Fast Fourier Transform inFinance ArXiv ID: ssrn-559416 “View on arXiv” Authors: Unknown Abstract The Fourier transform is an important tool in Financial Economics. It delivers real time pricing while allowing for a realistic structure of asset returns, taki Keywords: Fourier transform, asset pricing, financial economics, time series analysis, real-time pricing, Financial Derivatives Complexity vs Empirical Score Math Complexity: 8.0/10 Empirical Rigor: 3.0/10 Quadrant: Lab Rats Why: The paper involves advanced mathematical concepts like Fourier transforms, complex numbers, and convolution, but it is a conceptual pedagogical piece focusing on methodology rather than providing empirical data, backtests, or implementation details for real-world trading. flowchart TD A["Research Goal: Use Fourier Transform<br>for Real-Time Financial Pricing"] --> B["Key Methodology: Fast Fourier Transform<br>FFT Algorithm"] B --> C["Data Inputs: Asset Return Time Series<br>& Market Data"] C --> D["Computational Process: FFT of<br>Return Distributions to Price Derivatives"] D --> E["Key Findings: Efficient Real-Time Pricing<br>Model for Financial Derivatives"]