Martingale Pricing and Single Index Models: Unified Approach with Esscher and Minimal Relative Entropy Measures

In this paper, we explore the connection between a single index model under the real-world probability measure and martingale pricing via minimal relative entropy or Esscher transform, within the context of a one-period market model, possibly incomplete, with multiple risky assets and a single risk-free asset. The minimal relative entropy martingale measure and the Esscher martingale measure coincide in such a market, provided they both exist. From their Radon–Nikodym derivative, we derive a portfolio of risky assets in a natural way, termed portfolio G. Our analysis shows that pricing using the Esscher or minimal relative entropy martingale measure is equivalent to a single index model (SIM) incorporating portfolio G. In the special case of elliptical returns, portfolio G coincides with the classical tangency portfolio. Furthermore, in the case of jointly normal returns, Esscher or minimal relative entropy martingale measure pricing is equivalent to CAPM pricing.