Geometric arbitrage theory and market dynamics
Pages: 431 - 471,
Simone Farinelli - Core Dynamics GmbH, Scheuchzerstrasse 43, CH-8006, Zurich, Switzerland (email)
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
$\bullet$ 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;
Available Online: October 2015.