Mathematical Finance (q-fin.MF)

Abstract: Consider a single asset financial market whose discounted asset price process is a stochastic integral with respect to a continuous regular strong Markov semimartingale (a so-called general diffusion semimartingale) that is parameterized by a scale function and a speed measure. In a previous paper, we established a characterization of the no free lunch with vanishing risk (NFLVR) condition for a canonical framework of such a financial market in terms of the scale function and the speed measure. Ioannis Karatzas (personal communication) asked us whether it is also possible to prove a characterization for the weaker no unbounded profit with bounded risk (NUPBR) condition, which is the main question we treat in this paper. Here, we do not restrict our attention to canonical frameworks but we allow a general setup with a general filtration that preserves the strong Markov property. Our main results are precise characterizations of NUPBR and NFLVR which only depend on the scale function and the speed measure. In particular, we prove that NUPBR forces the scale function to be continuously differentiable with absolutely continuous derivative. The latter extends our previous result, that, in the canonical framework, NFLVR implies such a property, in two directions (a weaker no-arbitrage notion and a more general framework). We also make the surprising observation that NUPBR and NFLVR are equivalent whenever finite boundary points are accessible for the driving diffusion.

Abstract: We introduce the notions of Collective Arbitrage and of Collective Super-replication in a setting where agents are investing in their markets and are allowed to cooperate through exchanges. We accordingly establish versions of the fundamental theorem of asset pricing and of the pricing-hedging duality. Examples show the advantage of our approach.