Finance and Stochastics

, Volume 22, Issue 2, pp 281–295 | Cite as

A risk-neutral equilibrium leading to uncertain volatility pricing

  • Johannes Muhle-Karbe
  • Marcel Nutz


We study the formation of derivative prices in an equilibrium between risk-neutral agents with heterogeneous beliefs about the dynamics of the underlying. Under the condition that short-selling is limited, we prove the existence of a unique equilibrium price and show that it incorporates the speculative value of possibly reselling the derivative. This value typically leads to a bubble; that is, the price exceeds the autonomous valuation of any given agent. Mathematically, the equilibrium price operator is of the same nonlinear form that is obtained in single-agent settings with worst-case aversion against model uncertainty. Thus, our equilibrium leads to a novel interpretation of this price.


Heterogeneous beliefs Equilibrium Derivative price bubble Uncertain volatility model Nonlinear expectation 

Mathematics Subject Classification (2010)

91B51 91G20 93E20 

JEL Classification

D52 G12 G13 D53 



The authors are most grateful to José Scheinkman for the stimulating discussions that have initiated this work. They would also like to thank the Editors and the referees for their detailed and constructive remarks which have greatly improved this paper.

The second author was partially supported by an Alfred P. Sloan Fellowship and NSF Grant DMS-1512900.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.Departments of Statistics and MathematicsColumbia UniversityNew YorkUSA

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