Foreign Exchange Option Symmetry and a Coordinate-Free Description of a Multiple Currency Market in Terms of Differential Geometry on Graphs

In this article we present a framework describing the world foreign exchange market in terms of differential geometry on graphs, that is, in terms of vector lattice bundles on graphs and connections on these bundles. This framework is based on the concept of financial equivalence introduced by Kholodnyi and Price in 1996 for a two currency and a multiple currency foreign exchange markets. Such a framework allows for a coordinate-free description of financial phenomena in the world foreign exchange market and for detecting, analyzing, and utilizing fundamental symmetries and related arbitrage opportunities in this market. The framework presented in this article is not limited to foreign exchange markets and can be applied to any financial market with exchange of an arbitrary number of arbitrary underlying securities