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Can You hear the Shape of a Market? Geometric Arbitrage and Spectral Theory

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  • Simone Farinelli
  • Hideyuki Takada

Abstract

Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the 'instantaneous arbitrage capability' generated by the market itself. The cashflow bundle is the vector bundle associated to this stochastic principal fibre bundle for the natural choice of the vector space fibre. The cashflow bundle carries a stochastic covariant differentiation induced by the connection on the principal fibre bundle. The link between arbitrage theory and spectral theory of the connection Laplacian on the vector bundle is given by the zero eigenspace resulting in a parametrization of all risk neutral measures equivalent to the statistical one. This indicates that a market satisfies the (NFLVR) condition if and only if $0$ is in the discrete spectrum of the connection Laplacian on the cash flow bundle or of the Dirac Laplacian of the twisted cash flow bundle with the exterior algebra bundle. We apply this result by extending Jarrow-Protter-Shimbo theory of asset bubbles for complete arbitrage free markets to markets not satisfying the (NFLVR). Moreover, by means of the Atiyah-Singer index theorem, we prove that the Euler characteristic of the asset nominal space is a topological obstruction to the the (NFLVR) condition, and, by means of the Bochner-Weitzenb\"ock formula, the non vanishing of the homology group of the cash flow bundle is revealed to be a topological obstruction to (NFLVR), too. Asset bubbles are defined, classified and decomposed for markets allowing arbitrage.

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  • 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.
    • Handle: RePEc:arx:papers:1509.03264
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    References listed on IDEAS

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    1. 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.
    2. Kirill Ilinski, 1997. "Physics of Finance," Papers hep-th/9710148, arXiv.org.
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    Cited by:

    1. 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.
    2. Simone Farinelli & Hideyuki Takada, 2019. "When Risks and Uncertainties Collide: Mathematical Finance for Arbitrage Markets in a Quantum Mechanical View," Papers 1906.07164, arXiv.org, revised Jan 2021.

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