On optimal arbitrage

TL;DR

This paper characterizes the optimal arbitrage opportunities in a Markovian financial market model using PDEs, linking the solutions to local martingale properties and extending to non-Markovian cases.

Contribution

It introduces a PDE-based method to identify the best arbitrage in Markovian models and extends the analysis to non-Markovian scenarios.

Findings

The optimal arbitrage can be characterized by the smallest positive solution to a PDE.

The solution relates to properties of strict local martingales.

A method to generate the best arbitrage strategy is provided.

Abstract

In a Markovian model for a financial market, we characterize the best arbitrage with respect to the market portfolio that can be achieved using nonanticipative investment strategies, in terms of the smallest positive solution to a parabolic partial differential inequality; this is determined entirely on the basis of the covariance structure of the model. The solution is intimately related to properties of strict local martingales and is used to generate the investment strategy which realizes the best possible arbitrage. Some extensions to non-Markovian situations are also presented.