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The Black–Scholes equation in the presence of arbitrage

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  • Simone Farinelli
  • Hideyuki Takada

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.

Suggested Citation

  • Simone Farinelli & Hideyuki Takada, 2022. "The Black–Scholes equation in the presence of arbitrage," Quantitative Finance, Taylor & Francis Journals, vol. 22(12), pages 2155-2170, December.
    • Handle: RePEc:taf:quantf:v:22:y:2022:i:12:p:2155-2170
    DOI: 10.1080/14697688.2022.2117075
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    References listed on IDEAS

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