Geometric Arbitrage Theory and Market Dynamics

72 Pages Posted: 27 Mar 2008 Last revised: 19 Jul 2015

Date Written: July 18, 2015

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:

• Write arbitrage as curvature of a principal fibre bundle. • Parameterize arbitrage strategies by its holonomy. • Give the Fundamental Theorem of Asset Pricing a differential homotopic characterization. • Characterize Geometric Arbitrage Theory by five principles and show they they are consistent with the classical theory of stochastic finance. • 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. • 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, Market Model, Stochastic Finance

JEL Classification: C62, C68, G12, G13

Suggested Citation

Farinelli, Simone, Geometric Arbitrage Theory and Market Dynamics (July 18, 2015). Available at SSRN: https://ssrn.com/abstract=1113292 or http://dx.doi.org/10.2139/ssrn.1113292

Simone Farinelli (Contact Author)

Core Dynamics GmbH ( email )

Scheuzerstrasse 43
Zurich, 8006
Switzerland

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