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Geometric arbitrage theory and market dynamics

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  • We have embedded the classical theory of stochastic finance into a differential geometric framework called Geometric Arbitrage Theory and show that it is possible to:
        $\bullet$ Write arbitrage as curvature of a principal fibre bundle.
        $\bullet$ Parameterize arbitrage strategies by its holonomy.
        $\bullet$ Give the Fundamental Theorem of Asset Pricing a differential homotopic characterization.
        $\bullet$ Characterize Geometric Arbitrage Theory by five principles and show they are consistent with the classical theory of stochastic finance.
        $\bullet$ Derive for a closed market the equilibrium solution for market portfolio and dynamics in the cases where:
           - Arbitrage is allowed but minimized.
           - Arbitrage is not allowed.
        $\bullet$ Prove that the no-free-lunch-with-vanishing-risk condition implies the zero curvature condition. The converse is in general not true and additionally requires the Novikov condition for the instantaneous Sharpe Ratio to be satisfied.
    Mathematics Subject Classification: 62D05, 58J65.


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