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Exploiting arbitrage requires short selling

Eckhard Platen, Stefan Tappe

TL;DR

The paper establishes that in markets modeled by nonnegative semimartingales, any exploitable arbitrage must be realized through short selling. Central to the result is the concept of an equivalent supermartingale deflator (ESMD) and a change of numéraire, which together link arbitrage opportunities to required short positions and underpin a general no-arbitrage-without-short-selling principle. The authors provide a robust framework that covers various no-arbitrage notions (NA, NA1, NFLVR) and extend from continuous-time semimartingale models to discrete-time settings, illustrating the core claim with explicit examples. Practically, this work offers theoretical justification for regulatory distinctions that permit or restrict short selling among financial intermediaries and highlights the bankruptcy risk associated with accessing arbitrage opportunities via short positions.

Abstract

We show that in a financial market given by semimartingales an arbitrage opportunity, provided it exists, can only be exploited through short selling. This finding provides a theoretical basis for differences in regulation for financial services providers that are allowed to go short and those without short sales. The privilege to be allowed to short sell gives access to potential arbitrage opportunities, which creates by design a bankruptcy risk.

Exploiting arbitrage requires short selling

TL;DR

The paper establishes that in markets modeled by nonnegative semimartingales, any exploitable arbitrage must be realized through short selling. Central to the result is the concept of an equivalent supermartingale deflator (ESMD) and a change of numéraire, which together link arbitrage opportunities to required short positions and underpin a general no-arbitrage-without-short-selling principle. The authors provide a robust framework that covers various no-arbitrage notions (NA, NA1, NFLVR) and extend from continuous-time semimartingale models to discrete-time settings, illustrating the core claim with explicit examples. Practically, this work offers theoretical justification for regulatory distinctions that permit or restrict short selling among financial intermediaries and highlights the bankruptcy risk associated with accessing arbitrage opportunities via short positions.

Abstract

We show that in a financial market given by semimartingales an arbitrage opportunity, provided it exists, can only be exploited through short selling. This finding provides a theoretical basis for differences in regulation for financial services providers that are allowed to go short and those without short sales. The privilege to be allowed to short sell gives access to potential arbitrage opportunities, which creates by design a bankruptcy risk.

Paper Structure

This paper contains 4 sections, 25 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Proofs of the results
  3. Financial market models in discrete time
  4. Examples

Key Result

Theorem 1.2

For every arbitrage portfolio $S^{\delta} \in \mathbb{P}_{{\rm sf}}(\mathbb{S})$ we have $S^{\delta} \notin \mathbb{P}_{{\rm sf}}^{\delta \geq 0}(\mathbb{S})$.

Theorems & Definitions (49)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 39 more