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December  2015, 7(4): 431-471. doi: 10.3934/jgm.2015.7.431

## Geometric arbitrage theory and market dynamics

 1 Core Dynamics GmbH, Scheuchzerstrasse 43, CH-8006, Zurich, Switzerland

Received  December 2011 Revised  August 2015 Published  October 2015

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.
Citation: Simone Farinelli. Geometric arbitrage theory and market dynamics. Journal of Geometric Mechanics, 2015, 7 (4) : 431-471. doi: 10.3934/jgm.2015.7.431
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