Geometric Arbitrage Theory and Market Dynamics
72 Pages Posted: 27 Mar 2008 Last revised: 19 Jul 2015
Date Written: July 18, 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:
• 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: Suggested Citation