  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.
