An approximation pricing algorithm in an incomplete market: A differential geometric approach

The minimal distance equivalent martingale measure (EMM) defined in Goll and Rüschendorf (2001) is the arbitrage-free equilibrium pricing measure. This paper provides an algorithm to approximate its density and the fair price of any contingent claim in an incomplete market. We first approximate the infinite dimensional space of all EMMs by a finite dimensional manifold of EMMs. A Riemannian geometric structure is shown on the manifold. An optimization algorithm on the Riemannian manifold becomes the approximation pricing algorithm. The financial interpretation of the geometry is also given in terms of pricing model risk.Received: February 2004, Mathematics Subject Classification (2000): 62P05, 91B24, 91B28JEL Classification: G11, G12, G13Yuan Gao: Present address Block 617, Bukit Panjang Ring Road, 16-806,Singapore 670617. I am currently working in a major investment bank.This paper is based on parts of my doctoral dissertation Gao (2002),which isavailable upon request.Part of the research was done during my visit to HumboldtUniversity in 2002 and was partially supported by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 373. I am especially thankful to Professor Hans Föllmer for the invitation and helpful discussions.We would like to thank Professor Martin Schweizer,the associate editor and the referee for their constructive comments.