BFS 2002

Contributed Talk

Differential Geometry of Equivalent Martingale Measures in an Incomplete Market

Yuan Gao, Kian Guan Lim, Kah Hwa Ng

We consider the arbitrage-free equilibrium pricing problem in an incomplete market. We provide a method to find the minimal distance measure defined by Goll and R\"{u}schendorf (2001) or the maximum entropy measure over a finite dimensional subset of the set of all equivalent martingale measures. We use the cross entropy to measure the pseudo-distance between two equivalent martingale measures. We interpret the cross entropy as model risk and prove that it induces a Riemannian geometric structure. A numerical optimization algorithm on the Riemannian manifold is applied to solve the pricing problem.