Fourier Neural Network Approximation of Transition Densities in Finance

TL;DR

This paper presents FourNet, a neural network method using Gaussian activation functions to accurately approximate transition densities in finance by leveraging known characteristic functions, with proven theoretical properties and practical applications.

Contribution

Introduces FourNet, a novel neural network architecture that approximates transition densities using Fourier transforms, with theoretical guarantees and practical demonstrations in finance models.

Findings

FourNet can approximate transition densities arbitrarily well in the $L_2$-sense.

The method effectively models complex financial processes like Lévy and Heston models.

The approach provides bounds on estimation error and nonnegativity in multiple dimensions.

Abstract

This paper introduces FourNet, a novel single-layer feed-forward neural network (FFNN) method designed to approximate transition densities for which closed-form expressions of their Fourier transforms, i.e. characteristic functions, are available. A unique feature of FourNet lies in its use of a Gaussian activation function, enabling exact Fourier and inverse Fourier transformations and drawing analogies with the Gaussian mixture model. We mathematically establish FourNet's capacity to approximate transition densities in the $L_2$-sense arbitrarily well with finite number of neurons. The parameters of FourNet are learned by minimizing a loss function derived from the known characteristic function and the Fourier transform of the FFNN, complemented by a strategic sampling approach to enhance training. We derive practical bounds for the $L_2$ estimation error and the potential pointwise loss of nonnegativity in FourNet for $d$-dimensions ($d\ge 1$), highlighting its robustness and applicability in practical settings. FourNet's accuracy and versatility are demonstrated through a wide range of dynamics common in quantitative finance, including L\'{e}vy processes and the Heston stochastic volatility models-including those augmented with the self-exciting Queue-Hawkes jump process.