The Transport-based Mesh-free Method (TMM) and its applications in finance: a review

TL;DR

This paper reviews the Transport-based Mesh-free Method (TMM), highlighting its efficiency and accuracy in financial risk measure computations, and discusses its theoretical foundations and practical applications in finance.

Contribution

It introduces the TMM approach, combining transportation and reproducing kernels, and demonstrates its effectiveness for fast, accurate financial risk calculations with sharp convergence rates.

Findings

TMM provides accurate risk measure computations in finance.

The method achieves sharp convergence rates and optimal computational times.

It offers insights into overcoming the curse of dimensionality in finance applications.

Abstract

We review a numerical technique, referred to as the Transport-based Mesh-free Method (TMM), and we discuss its applications to mathematical finance. We recently introduced this method from a numerical standpoint and investigated the accuracy of integration formulas based on the Monte-Carlo methodology: quantitative error bounds were discussed and, in this short note, we outline the main ideas of our approach. The techniques of transportation and reproducing kernels lead us to a very efficient methodology for numerical simulations in many practical applications, and provide some light on the methods used by the artificial intelligence community. For applications in the finance industry, our method allows us to compute many types of risk measures with an accurate and fast algorithm. We propose theoretical arguments as well as extensive numerical tests in order to justify sharp convergence rates, leading to rather optimal computational times. Cases of direct interest in finance support our claims and the importance of the problem of the curse of dimensionality in finance applications is briefly discussed.