Duplicating and Pricing Contingent Claims in Incomplete Markets

Abstract

The paper studies the problem of pricing contingent claims in the situation where the constraints imposed on an investor’s portfolios are important. There are two types of rule of constraint: under a rigid rule, an investor must strictly limit his portfolios inside the constraint; under an elastic rule, an investor pays a penalty when the constraint is violated. The central problem of pricing a contingent claim is to determine the initial investment required to duplicate the contingent claim. The following results are obtained: (i) under elastic rules, the cost to duplicate a contingent claim exists and is unique;(ii) this cost depends nonlinearly and convexly on the contingent claim;(iii) the cost under rigid rules resulting from passing the penalty to infinity is also a nonlinear and convex function of the contingent claim. Owing to this nonlinearity, the cost of duplication may be or may not be the nonarbitrage price of the contingent claim; this depends on how the market organizes the production of contingent claims. The conclusion that the cost of duplication is a convex function of the contingent claim provides an explanation for why the service of providing contingent claims is often a highly profitable business. The main mathematical tool in the analysis is backward stochastic differential equations (BSDEs). In fact the cost to duplicate a contingent claim is the solution of a BSDE in which the contingent claim is the terminal value.