The Journal of Geometric Mechanics (JGM)

Geometric arbitrage theory and market dynamics

Pages: 431 - 471, Volume 7, Issue 4, December 2015      doi:10.3934/jgm.2015.7.431

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Simone Farinelli - Core Dynamics GmbH, Scheuchzerstrasse 43, CH-8006, Zurich, Switzerland (email)

Abstract: 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.

Keywords:  Geometric arbitrage theory, arbitrage pricing, stochastic differential geometry.
Mathematics Subject Classification:  62D05, 58J65.

Received: December 2011;      Revised: August 2015;      Available Online: October 2015.