[Paper Review] Arbitrage theory without a numéraire

Michael R. Tehranchi|arXiv (Cornell University)|Oct 11, 2014

Stochastic processes and financial applications18 references3 citations

TL;DR

This paper develops a discrete-time arbitrage theory without assuming the existence of a numéraire asset, establishing fundamental theorems of asset pricing using martingale deflators as the core tool. The key contribution is a numéraire-free characterization of no-arbitrage and super-replication, with applications to price bubbles and a discrete-time analogue of absolute vs. relative arbitrage.

ABSTRACT

This note develops an arbitrage theory for a discrete-time market model without the assumption of the existence of a numéraire asset. Fundamental theorems of asset pricing are stated and proven in this context. The distinction between the notions of investment-consumption arbitrage and pure-investment arbitrage provide a discrete-time analogue of the distinction between the notions of absolute arbitrage and relative arbitrage in the continuous-time theory. Applications to the modelling of bubbles is discussed.

Motivation & Objective

• To establish a fundamental theorem of asset pricing in discrete-time markets without assuming the existence of a numéraire asset.
• To characterize no-arbitrage conditions using martingale deflators instead of equivalent martingale measures, which depend on a numéraire.
• To provide a discrete-time analogue of the continuous-time distinction between absolute (investment-consumption) and relative (pure-investment) arbitrage.
• To formalize the concept of a price bubble in discrete time, consistent with continuous-time definitions.
• To show that the minimal super-replication cost of a contingent claim is characterized by the expectation of the claim under a martingale deflator, even without a numéraire.

Proposed method

• Uses the concept of a martingale deflator as the primary dual variable in optimal investment problems, independent of any numéraire.
• Applies generalised conditional expectations and discrete-time local martingale theory to handle non-integrable or pathologically defined random variables.
• Employs a utility maximisation framework inspired by Rogers (2002) to derive the fundamental theorem of asset pricing without convex analysis or separation theorems.
• Uses Kabanov’s theorem to show that a local martingale deflator implies the existence of a true martingale deflator under mild conditions.
• Applies measurable selection theorems (e.g., Proposition 7.10) to ensure measurability of optimizers in stochastic control problems.
• Introduces a measurable version of the Bolzano–Weierstrass theorem (Proposition 7.11) to handle convergence of sequences of random variables in the proof.

Experimental results

Research questions

• RQ1 Can the fundamental theorem of asset pricing be reformulated without assuming the existence of a numéraire?
• RQ2 How does the distinction between investment-consumption and pure-investment arbitrage in discrete time mirror the continuous-time notions of absolute and relative arbitrage?
• RQ3 Can a price bubble be meaningfully defined in a discrete-time market without a numéraire?
• RQ4 What is the role of the martingale deflator in characterizing no-arbitrage and minimal super-replication costs in the absence of a numéraire?
• RQ5 Under what conditions does a local martingale deflator imply the existence of a true martingale deflator in discrete time?

Key findings

• The existence of a martingale deflator is equivalent to the absence of arbitrage in a discrete-time market, even without assuming a numéraire.
• The minimal super-replication cost of a contingent claim is given by the supremum of the expected payoff under all martingale deflators.
• A discrete-time price bubble can exist when the market lacks a numéraire, analogous to the continuous-time definition of a bubble as a deviation from intrinsic value.
• The distinction between investment-consumption and pure-investment arbitrage provides a discrete-time analogue of absolute and relative arbitrage in continuous-time models.
• When the market is complete and arbitrage-free, a risk-free numéraire must exist, recovering the classical result under the new framework.
• Kabanov’s theorem ensures that a local martingale deflator can be transformed into a true martingale deflator under the given conditions, enabling the use of standard martingale techniques.