Geometric arbitrage theory and market dynamics

Simone  Farinelli

doi:10.3934/jgm.2015.7.431

Article Contents

2015, Volume 7, Issue 4: 431-471. Doi: 10.3934/jgm.2015.7.431

This issue Previous Article Symmetry reduction, integrability and reconstruction in $k$-symplectic field theory Next Article A unifying mechanical equation with applications to non-holonomic constraints and dissipative phenomena

Geometric arbitrage theory and market dynamics

Simone Farinelli^1,

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

Received: November 30, 2011

Revised: July 31, 2015

Published: November 2015

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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.

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Mathematics Subject Classification: 62D05, 58J65.

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