The Black–Scholes equation in the presence of arbitrage

Simone Farinelli and Hideyuki Takada

Quantitative Finance, 2022, vol. 22, issue 12, 2155-2170

Abstract: We apply Geometric Arbitrage Theory to obtain results in Mathematical Finance, which do not need stochastic differential geometry in their formulation. First, for generic market dynamics given by a subclass of multidimensional Itô processes we specify and prove the equivalence between No-Free-Lunch-with-Vanishing-Risk (NFLVR) and expected utility maximization. As a by-product, we provide a geometric characterization of the No-Unbounded-Profit-with-Bounded-Risk (NUPBR) condition given by the zero curvature (ZC) condition for this subclass of Itô processes. Finally, we extend the Black–Scholes partial differential equation to markets allowing arbitrage.

Date: 2022
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DOI: 10.1080/14697688.2022.2117075

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