Riemannian Geometry-based Analysis Of Arbitrage

The significant breakthrough in modern financial theory are benefited from the perfect combinations of H. Markowitz's portfolio selection theory and M-M theory which M. Miller and F. Modigliani found and implied the no free lunch with vanishing risk (NFLVR) analysis. The no free lunch with vanishing risk (NFLVR) principle is nature characteristic of the financial market equilibrium, showing the independence of equilibrium price and investor's risk preference. This is all based on the Arrow-Debreu model which verifies the fundamental theorem of asset pricing.The no free lunch with vanishing risk (NFLVR) principle is the cornerstone of modern financial mathematics. The no-arbitrage principle can therefore be seen as the most important point.With the basic theory and methods in Riemannian geometry, S. Farinelli (2009) defined the formof connection and curvature equations with cash flow to study the characteristic of financial markets. The no free lunch with vanishing risk (NFLVR) principle was presented in real financial markets with a series of theoretical and practical economics meaning.We embed the classical theory of stochastic finance into a differential geometric framework called Geometric Arbitrage Theory. Based on gauge, we define the market fibre bundle as the fibre bundle of gauges, and identify the most general measure of arbitrage for any market model. Moreover, such measure has a geometrical interpretation as gauge connection.Firstly, we introduce a new connection. There exists zero curvature if and only if there is no arbitrage.Secondly, we discuss the NFLVR property corresponding to connection defined in the situation of portfolio returns. Four equivalent assertions of no-arbitrage principle are followed.As for application, we give an example as illustration that the essence of Geometric Arbitrage Theory is for the pricing of forward.