Arbitrage-Free Pricing Before and Beyond Probabilities

TL;DR

This paper challenges the traditional link between no-arbitrage conditions and risk-neutral probabilities, showing that prices can be linear forms not corresponding to probabilities and exploring conditions for martingality in stochastic volatility models.

Contribution

It provides counterexamples to the classical theorem, clarifies the nature of arbitrage-free prices, and offers methods to recover martingality in complex models.

Findings

Prices can be linear forms not representing probabilities.

Arbitrage freedom relates to the continuity of pricing forms.

Martingality can be restored using advanced mathematical tools.

Abstract

"Fundamental theorem of asset pricing" roughly states that absence of arbitrage opportunity in a market is equivalent to the existence of a risk-neutral probability. We give a simple counterexample to this oversimplified statement. Prices are given by linear forms which do not always correspond to probabilities. We give examples of such cases. We also show that arbitrage freedom is equivalent to the continuity of the pricing linear form in the relevant topology. Finally we analyze the possible loss of martingality of asset prices with lognormal stochastic volatility. For positive correlation martingality is lost when the financial process is modelled through standard probability theory. We show how to recover martingality using the appropriate mathematical tools.