Explicit description of all deflators for market models under random horizon with applications to NFLVR

Abstract

This paper considers an initial market model, specified by its underlying assets $S$ and its flow of information $\mathbb F$, and an arbitrary random time $τ$ which might not be an $\mathbb F$-stopping time. As the death time and the default time (that $τ$ might represent) can be seen when they occur only, the progressive enlargement of $\mathbb F$ with $τ$ sounds tailor-fit for modelling the new flow of information $\mathbb G$ that incorporates both $\mathbb F$ and $τ$. In this setting of informational market, the first principal goal resides in describing as explicitly as possible the set of all deflators for $(S^τ, \mathbb G)$, while the second principal goal lies in addressing the No-Free-Lunch-with-Vanishing-Risk concept (NFLVR hereafter) for $(S^τ, \mathbb G)$. Besides this direct application to NFLVR, the set of all deflators constitutes the dual set of all "admissible" wealth processes for the stopped model $(S^τ,\mathbb G)$, and hence it is vital in many hedging and pricing related optimization problems. Thanks to the results of Choulli et al. [7], on martingales classification and representation for progressive enlarged filtration, our two main goals are fully achieved in different versions, when the survival probability never vanishes. The results are illustrated on the two particular cases when $(S,\mathbb F)$ follows the jump-diffusion model and the discrete-time model.