Option pricing on multiple assets (Q852003)

The paper combines ideas from stochastic differential equations and from geometry in the study of a multidimensional version of the Black-Scholes equations from mathematical economics. The approach is motivated by the need to introduce geometric invariants in the study of the multi-dimensional case, as the variable configurations in particular models naturally form non-linear manifold structures. Some basic assumptions are made, for example, short selling is permitted and the assets are divisible. After the authors consider the simplest model in the multiple asset case, the more elegant one is proposed which is called curvature arbitrage model. The purpose of the paper is threefold: the systematic approach to the multi-dimensional Black-Scholes equation is given, two fundamental curvature invariants from geometry are introduced and time asymptotics of the multi-dimensional Black-Scholes equation with the use of heat equation asymptotics.