[Paper Review] Non-Arbitrage under a Class of Honest Times

Tahir Choulli, Anna Aksamit+2 more|arXiv (Cornell University)|Apr 1, 2014

Stochastic processes and financial applications8 references3 citations

TL;DR

This paper introduces a new class of honest times—random times that are not stopping times but preserve the No-Unbounded-Profit-with-Bounded-Risk (NUPBR) condition in financial markets after their occurrence. By characterizing pairs of initial market models and such honest times, the authors establish conditions under which the post-τ market remains arbitrage-free, and construct explicit local martingale deflators for the 'after-τ' dynamics in risk-neutralized models.

ABSTRACT

This paper quantifies the interplay between the non-arbitrage notion of No-Unbounded-Profit-with-Bounded-Risk (NUPBR hereafter) and additional information generated by a random time. This study complements the one of Aksamit/Choulli/Deng/Jeanblanc [1] in which the authors studied similar topics for the case of stopping at the random time instead, while herein we are concerned with the part after the occurrence of the random time. Given that all the literature -up to our knowledge- proves that the NUPBR notion is always violated after honest times that avoid stopping times in a continuous filtration, herein we propose a new class of honest times for which the NUPBR notion can be preserved for some models. For this family of honest times, we elaborate two principal results. The first main result characterizes the pairs of initial market and honest time for which the resulting model preserves the NUPBR property, while the second main result characterizes the honest times that preserve the NUPBR property for any quasi-left continuous model. Furthermore, we construct explicitly "the-after-tau" local martingale deflators for a large class of initial models (i.e. models in the small filtration) that are already risk-neutralized.

Motivation & Objective

• To resolve the open problem of whether NUPBR can be preserved after a random time τ that is not a stopping time.
• To identify a new class of honest times for which the NUPBR property holds in the post-τ market, even when τ avoids stopping times.
• To characterize the pairs of initial market models and honest times that preserve NUPBR after τ.
• To construct explicit 'after-τ' local martingale deflators for risk-neutralized models under the new class of honest times.
• To extend the applicability of NUPBR in models with extra information from random times, particularly in quasi-left-continuous settings.

Proposed method

• Introduces a new class of honest times that includes all stopping times and certain non-stopping times, ensuring semimartingale and NUPBR preservation after τ.
• Applies the Dellacherie-Mokobodski criterion to analyze the semimartingale property of S − S^τ under the enlarged filtration G.
• Uses the equivalence between NUPBR and the local integrability of gain processes to derive conditions on the compensator of the jump measure.
• Employs the Galtchouk-Kunita-Watanabe decomposition and the Girsanov-type transformation via the density process Z to relate the dynamics under P and Q.
• Applies Theorem A.1 to compare predictable characteristics of the post-τ process under different measures, ensuring equivalence of NUPBR conditions.
• Constructs explicit local martingale deflators via the functional K and the density process Z, ensuring the post-τ model remains arbitrage-free.

Experimental results

Research questions

• RQ1 Can the NUPBR condition be preserved after a random time τ that is not a stopping time?
• RQ2 What class of honest times allows the post-τ market (S − S^τ) to remain arbitrage-free?
• RQ3 Under what conditions on the initial market model and τ does the enlarged filtration G preserve NUPBR?
• RQ4 How can one explicitly construct local martingale deflators for the 'after-τ' dynamics in risk-neutralized models?
• RQ5 What is the role of the compensator of the jump measure and the density process Z in ensuring NUPBR after τ?

Key findings

• The paper identifies a new class of honest times, including all stopping times and certain non-stopping times, for which the NUPBR condition is preserved after τ.
• It characterizes the pairs (S, τ) such that the post-τ model (S − S^τ, G) satisfies NUPBR, providing a functional K observable from public information that determines the arbitrage-free property.
• For any quasi-left-continuous initial model, the NUPBR property is preserved after τ if and only if the compensator of the jump measure satisfies a specific integrability condition involving the density process Z.
• The authors construct explicit local martingale deflators for the 'after-τ' dynamics using the functional K and the density process Z, ensuring the post-τ market remains arbitrage-free.
• The equivalence between the NUPBR condition for (S − S^τ, G) and the local integrability of (1 − Z_−) · V under the initial filtration F is established, providing a tractable criterion.
• The proof relies on comparing predictable characteristics under equivalent measures and applying Theorem A.1 to show that the NUPBR condition is preserved when the compensator structure is compatible with the new information.