Non-Equivalent Martingale Measures: An Example

absence of arbitrage implies that there exists a linear functional that values all con-tingent claims. This functional may be represented in various forms: it may be a set of state prices, a stochastic process (the stochastic discount factor) or a probability measure (the risk-neutral measure). The Fundamental Theorem of Finance always holds without qualiÞcation in Þnite settings, but not necessarily in inÞnite settings. In inÞnite settings the requisite linear functional may not exist even in the absence of arbitrage (Kreps [5]). If it does exist, it may not be representable by a measure. For example, Back and Pliska [1], Gilles and LeRoy [3], [4], Werner [7] and others have analyzed examples in which the pricing measure is not countably additive, contrary to the deÞnition of a measure. In this paper we analyze an example in which there exists a valuation functional, and this functional is representable by a measure. However, that measure is not equivalent to the natural probability measure (in the sense that the two measures do not have the same sets of zero measure). The example is a general equilibrium representative-agent model, which has the convenient property that markets are ef-fectively complete under very weak assumptions. In such settings the zero portfolio is always optimal. Therefore one can analyze valuation without even specifying the traded securities. To establish notation we will begin with the Þnite case, in which there is no problem deriving the equivalent martingale measure