Minimal Martingale Measure, CAPM, and Representative Agent Pricing in Incomplete Markets

Abstract

The minimal martingale measure (MMM) was introduced and studied by Föllmer and Schweizer (1990) in the context of mean square hedging in incomplete markets. Recently, the theory of no-good-deal pricing gave further evidence that the MMM plays a prominent role in security valuation in an incomplete market when security prices follow a diffusion process. Namely, it was shown that the price defined by the MMM lies in the centre of no-good-deal price bounds. In the first part of the paper we examine the relationship between the MMM and the optimal portfolio problem in diffusion environment and show that the MMM arises in equilibrium with log-utility maximizing representative agent. A puzzling property of the MMM is that outside the diffusion environment it easily becomes negative. As we show in the second part of the paper this fact can be explained from the link between the MMM and the CAPM risk-neutral measure.

Keywords: minimalmartingale measure, optimal portfolio, hedging, CAPM

JEL Classification: D40, G11, G12