The Black-Scholes Equation in Presence of Arbitrage

Abstract

We apply Geometric Arbitrage Theory to obtain results in Mathematical Finance, which do not need stochastic differential geometry in their formulation. First, for a generic market dynamics given by a multidimensional Itô's process we specify and prove the equivalence between (NFLVR) and expected utility maximization. As a by-product we provide a geometric characterization of the (NUPBR) condition given by the zero curvature (ZC) condition. Finally, we extend the Black-Scholes PDE to markets allowing arbitrage.

The assumptions of Proposition 23 were corrected after Claudio Fontana provided us with a counterexample for the previous version of this proposition. arXiv admin note: substantial text overlap with arXiv:1509.03264, arXiv:1906.07164, arXiv:1406.6805, arXiv:0910.1671