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The Black-Scholes Equation in 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 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.

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  • Simone Farinelli & Hideyuki Takada, 2019. "The Black-Scholes Equation in Presence of Arbitrage," Papers 1904.11565, arXiv.org, revised Oct 2021.
    • Handle: RePEc:arx:papers:1904.11565
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    References listed on IDEAS

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    1. Hardy Hulley & Martin Schweizer, 2010. "M6 - On Minimal Market Models and Minimal Martingale Measures," Research Paper Series 280, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. Tomas Björk & Henrik Hult, 2005. "A note on Wick products and the fractional Black-Scholes model," Finance and Stochastics, Springer, vol. 9(2), pages 197-209, April.
    3. Kirill Ilinski, 1997. "Physics of Finance," Papers hep-th/9710148, arXiv.org.
    4. Simone Farinelli, 2009. "Geometric Arbitrage Theory and Market Dynamics Reloaded," Papers 0910.1671, arXiv.org, revised Jul 2021.
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