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 It\^o'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.
Keywords
Cite
@article{arxiv.1904.11565,
title = {The Black-Scholes Equation in Presence of Arbitrage},
author = {Simone Farinelli and Hideyuki Takada},
journal= {arXiv preprint arXiv:1904.11565},
year = {2021}
}
Comments
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