Pricing without martingale measure

TL;DR

This paper introduces a novel approach to financial asset pricing that replaces martingale measures with convex duality, leading to a new no-arbitrage condition called Absence of Immediate Profit and providing numerical insights.

Contribution

It proposes a convex duality framework for super-replication pricing, replacing traditional martingale measure duality, and introduces the AIP condition as a weaker no-arbitrage criterion.

Findings

Super-replication prices expressed via Fenchel conjugate.

AIP condition characterized and related to classical no-arbitrage.

Numerical illustrations demonstrate the approach's potential.

Abstract

For several decades, the no-arbitrage (NA) condition and the martingale measures have played a major role in the financial asset's pricing theory. We propose a new approach for estimating the super-replication cost based on convex duality instead of martingale measures duality: Our prices will be expressed using Fenchel conjugate and bi-conjugate. The super-hedging problem leads endogenously to a weak condition of NA called Absence of Immediate Profit (AIP). We propose several characterizations of AIP and study the relation with the classical notions of no-arbitrage. We also give some promising numerical illustrations.