The Black-Scholes Equation in Presence of Arbitrage

32 Pages Posted: 20 Dec 2016 Last revised: 28 Jun 2019

Date Written: June 19, 2019


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.

Keywords: Geometric Arbitrage, Black-Scholes PDE, Expected Utility Maximization

JEL Classification: C02

Suggested Citation

Farinelli, Simone and Takada, Hideyuki, The Black-Scholes Equation in Presence of Arbitrage (June 19, 2019). Available at SSRN: or

Simone Farinelli (Contact Author)

Core Dynamics GmbH ( email )

Scheuzerstrasse 43
Zurich, 8006

Hideyuki Takada

Toho University ( email )

Room 4421
Miyama 2-2-1
Funabashi, Chiba 274-8510
(+81)-47-472-1856 (Phone)

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