Non-Arbitrage Under Additional Information for Thin Semimartingale Models

TL;DR

This paper investigates the impact of additional information, such as random times, on the No-Unbounded-Risk-with-Bounded-Profit (NUPBR) property in thin semimartingale models, extending previous work to more general jump structures.

Contribution

It extends the analysis of NUPBR under enlarged filtrations to models on thin predictable sets, providing explicit constructions of local martingale deflators in this context.

Findings

NUPBR property is preserved or affected by random time enlargements.

Explicit construction of local martingale deflators under enlarged filtrations.

Generalization to models with unordered jump times.

Abstract

This paper completes the two studies undertaken in \cite{aksamit/choulli/deng/jeanblanc2} and \cite{aksamit/choulli/deng/jeanblanc3}, where the authors quantify the impact of a random time on the No-Unbounded-Risk-with-Bounded-Profit concept (called NUPBR hereafter) when the stock price processes are quasi-left-continuous (do not jump on predictable stopping times). Herein, we focus on the NUPBR for semimartingales models that live on thin predictable sets only and the progressive enlargement with a random time. For this flow of information, we explain how far the NUPBR property is affected when one stops the model by an arbitrary random time or when one incorporates fully an honest time into the model. This also generalizes \cite{choulli/deng} to the case when the jump times are not ordered in anyway. Furthermore, for the current context, we show how to construct explicitly local martingale deflator under the bigger filtration from those of the smaller filtration.