Quantitative Fundamental Theorem of Asset Pricing

TL;DR

This paper develops a quantitative framework for asset pricing and hedging that accounts for small arbitrage and model uncertainty, extending classical theorems to more robust, real-world market conditions.

Contribution

It introduces a quantitative version of the Fundamental Theorem of Asset Pricing and Super-Replication Theorem that handles small arbitrage and model uncertainty.

Findings

Markets with small arbitrage still allow meaningful pricing and hedging.

Pricing measures approximate martingales with additional costs.

Robustness of arbitrage and pricing measures is established under Wasserstein distance.

Abstract

In this paper we provide a quantitative analysis to the concept of arbitrage, that allows to deal with model uncertainty without imposing the no-arbitrage condition. In markets that admit ``small arbitrage", we can still make sense of the problems of pricing and hedging. The pricing measures here will be such that asset price processes are close to being martingales, and the hedging strategies will need to cover some additional cost. We show a quantitative version of the Fundamental Theorem of Asset Pricing and of the Super-Replication Theorem. Finally, we study robustness of the amount of arbitrage and existence of respective pricing measures, showing stability of these concepts with respect to a strong adapted Wasserstein distance.