No arbitrage and the existence of ACLMMs in general diffusion models

TL;DR

This paper explores the relationship between no arbitrage conditions and the existence of absolutely continuous local martingale measures in diffusion market models, establishing precise conditions for their equivalence.

Contribution

It provides a rigorous characterization of when no arbitrage holds in diffusion models, including conditions involving scale functions and boundary behaviors.

Findings

NA is equivalent to ACLMM plus regularity and boundary conditions for finite horizons

For infinite horizons, NA is equivalent to ACLMM without additional conditions

Counterexamples demonstrate the sharpness of the finite horizon characterization

Abstract

In a seminal paper, F. Delbaen and W. Schachermayer proved that the classical NA ("no arbitrage") condition implies the existence of an "absolutely continuous local martingale measure" (ACLMM). It is known that in general the existence of an ACLMM alone is not sufficient for NA. In this paper we investigate how close these notions are for single asset general diffusion market models. We show that NA is equivalent to the existence of an ACLMM plus a mild regularity condition on the scale function and the absence of reflecting boundaries. For infinite time horizon scenarios, the regularity assumption and the requirement on the boundaries can be dropped, showing equivalence between NA and the existence of an ACLMM. By means of counterexamples, we show that our characterization of NA for finite time horizons is sharp in the sense that neither the regularity condition on the scale function nor the absence of reflecting boundaries can be dropped.