Diffusion-based models for financial markets without martingale measures

TL;DR

This paper explores diffusion-based financial market models that lack an Equivalent Local Martingale Measure, demonstrating conditions under which markets remain viable, complete, and capable of meaningful valuation without traditional martingale measures.

Contribution

It establishes necessary and sufficient conditions for market viability and completeness in models without martingale measures, using the growth-optimal portfolio as a numeraire.

Findings

Markets can be viable without an equivalent martingale measure.

Contingent claims can be valued under the real-world measure.

Market completeness is possible without traditional martingale measures.

Abstract

We consider a general class of diffusion-based models and show that, even in the absence of an Equivalent Local Martingale Measure, the financial market may still be viable, in the sense that strong forms of arbitrage are excluded and portfolio optimisation problems can be meaningfully solved. Relying partly on the recent literature, we provide necessary and sufficient conditions for market viability in terms of the market price of risk process and martingale deflators. Regardless of the existence of a martingale measure, we show that the financial market may still be complete and contingent claims can be valued under the original (real-world) probability measure, provided we use as numeraire the Growth-Optimal Portfolio.