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 areconsistent with the classical theory of stochastic finance.• Derive for a closed market the equilibrium solution for market portfolio and dynamics inthe cases where: – Arbitrage is allowed but minimized. – Arbitrage is not allowed.• Prove that the no-free-lunch-with-vanishing-risk condition implies the zero curvaturecondition. 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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