No-arbitrage for informational discrete time market models
Abstract This paper focuses on the stability of no-arbitrage, for discrete time market models, under additional uncertainty generated by a random time . At the practical level, this random time represents the death time, the default time of a firm, or any occurrence time of an event that might affect the market somehow. We address the no-arbitrage issue for the resulting new flow of information (filtration) which makes the random time either a nontrivial stopping time (progressive enlargement) or a known time from the beginning (initial enlargement). Our main conclusions are twofold. On the one hand, for a fixed initial market S, we completely and precisely characterize the interplay between S and such that the no-arbitrage is preserved for the new market model. On the other hand, we give the necessary and sufficient conditions on to ensure the preservation of the no-arbitrage under the additional uncertainty of for any market. Two concrete examples are presented to illustrate the results.