From Bachelier to Dupire via Optimal Transport

TL;DR

This paper reviews the historical development of mathematical finance from Bachelier to Dupire, emphasizing the role of optimal transport in understanding arbitrage-free models and volatility surfaces.

Contribution

It highlights the connection between classical finance models and modern optimal transport theory, providing a unified perspective on their development.

Findings

Optimal transport offers insights into arbitrage-free model calibration.

Connections between Bachelier, Dupire, and modern mathematical tools are elucidated.

The survey clarifies the mathematical foundations of volatility surface modeling.

Abstract

Famously mathematical finance was started by Bachelier in his 1900 PhD thesis where - among many other achievements - he also provides a formal derivation of the Kolmogorov forward equation. This forms also the basis for Dupire's (again formal) solution to the problem of finding an arbitrage free model calibrated to the volatility surface. The later result has rigorous counterparts in the theorems of Kellerer and Lowther. In this survey article we revisit these hallmarks of stochastic finance, highlighting the role played by some optimal transport results in this context.