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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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    1. Mark Loewenstein & Gregory A. Willard, 2000. "Local martingales, arbitrage, and viability," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 16(1), pages 135-161.
    2. Hugonnier, Julien & Prieto, Rodolfo, 2015. "Asset pricing with arbitrage activity," Journal of Financial Economics, Elsevier, vol. 115(2), pages 411-428.
    3. Claudio Fontana, 2015. "Weak And Strong No-Arbitrage Conditions For Continuous Financial Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(01), pages 1-34.
    4. Ioannis Karatzas & Constantinos Kardaras, 2007. "The numéraire portfolio in semimartingale financial models," Finance and Stochastics, Springer, vol. 11(4), pages 447-493, October.
    5. Simone Farinelli, 2009. "Geometric Arbitrage Theory and Market Dynamics Reloaded," Papers 0910.1671, arXiv.org, revised Jul 2021.
    6. Simone Farinelli & Hideyuki Takada, 2015. "Can You hear the Shape of a Market? Geometric Arbitrage and Spectral Theory," Papers 1509.03264, arXiv.org, revised Sep 2021.
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