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

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

JRFM, 2024, vol. 17, issue 10, 1-15

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.

Keywords: esscher transform; relative entropy; martingale measure; SIM; CAPM; incomplete markets (search for similar items in EconPapers)
JEL-codes: C E F2 F3 G (search for similar items in EconPapers)
Date: 2024
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