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

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  • Stylianos Xanthopoulos

    (Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, 832 00 Samos, Greece)

Abstract

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.

Suggested Citation

  • Stylianos Xanthopoulos, 2024. "Martingale Pricing and Single Index Models: Unified Approach with Esscher and Minimal Relative Entropy Measures," JRFM, MDPI, vol. 17(10), pages 1-15, October.
    • Handle: RePEc:gam:jjrfmx:v:17:y:2024:i:10:p:446-:d:1490958
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