On the Existence of Martingale Measures in Jump Diffusion Market Models

TL;DR

This paper explores the existence of martingale measures in jump-diffusion market models, demonstrating cases where no equivalent local martingale measure exists despite satisfying weaker no-arbitrage conditions.

Contribution

It constructs explicit examples of jump-diffusion models satisfying NA1 but not NFLVR, showing the supermartingale deflator is not a true martingale.

Findings

Existence of models satisfying NA1 but not NFLVR

Supermartingale deflator is not a martingale in these models

Constraints on portfolios extend beyond standard admissibility

Abstract

In the context of jump-diffusion market models we construct examples that satisfy the weaker no-arbitrage condition of NA1 (NUPBR), but not NFLVR. We show that in these examples the only candidate for the density process of an equivalent local martingale measure is a supermartingale that is not a martingale, not even a local martingale. This candidate is given by the supermartingale deflator resulting from the inverse of the discounted growth optimal portfolio. In particular, we con- sider an example with constraints on the portfolio that go beyond the standard ones for admissibility.