Pricing without no-arbitrage condition in discrete time

TL;DR

This paper introduces a novel convex duality approach to price financial products in discrete time without relying on traditional no-arbitrage conditions, instead using a weaker condition called AIP.

Contribution

It develops a super-replication cost estimation method based on convex conjugates, avoiding martingale measures, and characterizes the weaker AIP condition.

Findings

Prices are finite under AIP, a weak no-arbitrage condition.

The approach uses Fenchel conjugate and bi-conjugate without no-arbitrage.

AIP is characterized and compared to traditional no-arbitrage conditions.

Abstract

In a discrete time setting, we study the central problem of giving a fair price to some financial product. For several decades, the no-arbitrage conditions and the martingale measures have played a major role for solving this problem. We propose a new approach for estimating the super-replication cost based on convex duality instead of martingale measures duality: The prices are expressed using Fenchel conjugate and bi-conjugate without using any no-arbitrage condition.The super-hedging problem resolution leads endogenously to a weak no-arbitrage condition called Absence of Instantaneous Profit (AIP) under which prices are finite. We study this condition in details, propose several characterizations and compare it to the no-arbitrage condition.