Abstract:

The fundamental theorem of asset pricing (FTAP) relates the existence of an element in the set (EMM) of equivalent sigma-martingale measures to a no arbitrage condition, the "no free lunch with vanishing risk"-condition (NFLVR). The latter is equivalent to the classical "no arbitrage"-condition (NA) and the "no unbounded profit with bounded risk"-condition (NUPBR). For continuous semimartingales, (NUPBR) is equivalent to the "structure conditon" (SC) which allows for an explicit characterization of the elements in (EMM). But even more important, it provides a natural candidate for an equivalent sigma-martingale measure, the so-called minimal martingale measure (MMM). Unfortunately, for non-continuous semimartingales the (MMM) is, if it exists, in general only a signed measure. Hence, the following natural questions arise: Does there exist a natural candidate for an equivalent sigma-martingale measure if the semimartingale is not continuous? Does there exist a characterization of the elements in (EMM)? Moreover, is the natural candidate, as in the case of a continuous semimartingale, related to a particular structure condition on the underlying semimartingale? The aim of the talk is to answer these questions for quasi-left-continuous semimartingales in a rather basic/didactic way that could be part of a course on continuous time mathematical finance.