[Paper Review] Diffusion-based models for financial markets without martingale measures

Claudio Fontana, Wolfgang J. Runggaldier|arXiv (Cornell University)|Sep 20, 2012

Stochastic processes and financial applications48 references18 citations

TL;DR

This paper establishes that financial markets modeled as diffusions can remain viable and complete even without an Equivalent Local Martingale Measure (ELMM), by using the Growth-Optimal Portfolio (GOP) as a numeraire. It proves that contingent claims can be priced and hedged under the real-world probability measure via GOP-discounted expectations, offering a robust alternative to risk-neutral pricing in models with bubbles or absent ELMMs.

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.

Motivation & Objective

• To establish conditions under which financial markets remain viable—free of strong arbitrage—even when no Equivalent Local Martingale Measure (ELMM) exists.
• To demonstrate that market completeness and contingent claim valuation are possible without an ELMM, relying instead on the Growth-Optimal Portfolio (GOP) as a numeraire.
• To provide a real-world pricing framework for contingent claims using GOP-discounted expectations under the physical measure.
• To unify and extend results from the Benchmark Approach and Stochastic Portfolio Theory in diffusion-based models.
• To show that utility indifference pricing coincides with real-world pricing under the GOP numeraire, particularly for logarithmic utility.

Proposed method

• The authors analyze a class of diffusion-based financial models under minimal technical assumptions, using Itô processes and local martingale deflators as key tools.
• They define and analyze weaker no-arbitrage conditions than NFLVR, particularly focusing on the absence of arbitrage of the first kind and increasing profit.
• The Growth-Optimal Portfolio (GOP) is constructed explicitly as the unique strategy maximizing long-term growth, and shown to generate the numeraire portfolio.
• The reciprocal of a martingale deflator is identified as the GOP value process, linking no-arbitrage conditions to the GOP structure.
• The paper uses utility maximization and duality methods, particularly for logarithmic utility, to derive explicit pricing formulas.
• It extends results to incomplete markets under Markovian assumptions, using the Markov property of the GOP to derive hedging strategies via Itô’s formula.

Experimental results

Research questions

• RQ1 Can financial markets be viable and free of strong arbitrage even when no Equivalent Local Martingale Measure (ELMM) exists?
• RQ2 Is market completeness still achievable in the absence of an ELMM, and if so, under what conditions?
• RQ3 Can contingent claims be priced and hedged meaningfully under the real-world probability measure without risk-neutral measures?
• RQ4 What is the role of the Growth-Optimal Portfolio (GOP) as a numeraire in real-world pricing and hedging?
• RQ5 Does the utility indifference price coincide with the real-world GOP-discounted expectation for general utility functions?

Key findings

• The financial market is viable—free of arbitrage of the first kind and increasing profit—if and only if the market price of risk process is square-integrable.
• The Growth-Optimal Portfolio (GOP) exists and is unique, and its value process serves as the numeraire portfolio, enabling real-world pricing.
• Contingent claims can be valued as the GOP-discounted expectation of their payoff under the physical probability measure, even without an ELMM.
• For logarithmic utility, the log-utility indifference price equals the real-world price: $ p_{\log}(H) = \mathbb{E}\left[\frac{H}{V_T^{\pi^*}}\right] $, providing a universal pricing formula.
• The GOP-discounted value process is a martingale under the physical measure, and its reciprocal is a martingale deflator, linking no-arbitrage to market structure.
• In Markovian settings, the GOP-based pricing and hedging framework extends to incomplete markets via the Markov property and Itô’s formula, enabling explicit strategy construction.