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Research Papers

The Black–Scholes equation in the presence of arbitrage

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Pages 2155-2170 | Received 18 Aug 2021, Accepted 19 Aug 2022, Published online: 15 Sep 2022
 

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.

JEL Classification:

Acknowledgements

The authors wish to thank anonymous reviewers for comments on earlier version of this paper. We are grateful to the participants of the Quantitative Methods in Finance congress (Sydney, December 2017) and 10th World Congress of Bachelier Finance Society (Dublin, July 2018) for valuable discussions, especially for Stefan Tappe, suggesting the relation between NUPBR and ZC. We would like to extend our gratitude to Claudio Fontana, who highlighted that for a previous incorrect version of proposition 2.12 Bessel's processes, which satisfy NUPBR but not NFLVR as shown in Fontana (Citation2015) and in Fontana and Runggaldier (Citation2013), would have been a counterexample, thus leading to the current corrected version.

Disclosure statement

No potential conflict of interest was reported by the authors.

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