Multiplicative noise, fast convolution, and pricing

TL;DR

This paper presents a fast convolution algorithm for high-dimensional integrals to efficiently model multiplicative noise processes in finance, offering an alternative to Monte Carlo simulations for option pricing.

Contribution

It introduces a novel fast convolution method for high-dimensional integrals applied to multiplicative noise processes, improving computational efficiency in option pricing models.

Findings

Fast convolution outperforms Monte Carlo in speed and efficiency.

The method accurately reproduces statistical features of financial returns.

Applicable to quadratic and piecewise linear diffusion processes.

Abstract

In this work we detail the application of a fast convolution algorithm computing high dimensional integrals to the context of multiplicative noise stochastic processes. The algorithm provides a numerical solution to the problem of characterizing conditional probability density functions at arbitrary time, and we applied it successfully to quadratic and piecewise linear diffusion processes. The ability in reproducing statistical features of financial return time series, such as thickness of the tails and scaling properties, makes this processes appealing for option pricing. Since exact analytical results are missing, we exploit the fast convolution as a numerical method alternative to the Monte Carlo simulation both in objective and risk neutral settings. In numerical sections we document how fast convolution outperforms Monte Carlo both in velocity and efficiency terms.