Maximum Entropy Estimates for Risk-Neutral Probability Measures with Non-Strictly-Convex Data

  • Christopher Bose
  • Rua Murray


This article investigates use of the Principle of Maximum Entropy for approximation of the risk-neutral probability density on the price of a financial asset as inferred from market prices on associated options. The usual strict convexity assumption on the market-price to strike-price function is relaxed, provided one is willing to accept a partially supported risk-neutral density. This provides a natural and useful extension of the standard theory. We present a rigorous analysis of the related optimization problem via convex duality and constraint qualification on both bounded and unbounded price domains. The relevance of this work for applications is in explaining precisely the consequences of any gap between convexity and strict convexity in the price function. The computational feasibility of the method and analytic consequences arising from non-strictly-convex price functions are illustrated with a numerical example.


Financial mathematics Risk-neutral probability density Maximum entropy method Moment constraint Lagrangian duality 



The authors thank an anonymous referee who pointed out an error in our use of Köthe spaces and conjugation past the integral in an earlier version of this paper. The first author acknowledges the hospitality of the Department of Mathematics and Statistics, University of Canterbury, New Zealand during sabbatical leave, where much of this work was carried out. This work has been supported by a grant from the Natural Sciences and Engineering Research Council of Canada.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.Department of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand

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