Geometric Arbitrage Theory and Market Dynamics Reloaded
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 Dynamics to be satisfied.