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General Properties of Option Prices

Authors

  • YAACOV Z. BERGMAN,

  • BRUCE D. GRUNDY,

  • ZVI WIENER

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    • Bergman is from the School of Business and the Center for Rationality and Interactive Decision Theory, Hebrew University. Grundy is from the Wharton School. Wiener is from the School of Business, Hebrew University, and was a visitor at the Wharton School when much of this article was written. This article is a substantially revised version of an earlier work previously circulated as Theory of Rational Option Pricing: II. The authors are grateful for helpful discussions with Kaushik Amin, Giovanni Barone-Adesi, Avi Bick, Peter Carr, Domenico Cuoco, Sanjiv Das, Darrell Duffle, Phil Dybvig, Mark Garman, Steve Grenadier, Sanford Grossman, Jon Ingersoll, Alexander Nabutovsky, Steve Ross, and Mark Rubinstein, and for the comments of workshop participants at the Wharton School, Berkeley, Carnegie-Mellon, Chicago, Harvard, Hebrew, Houston, Maryland, and Tel-Aviv Universities, Hong Kong University of Science & Technology, Washington University in St. Louis, the NBER Financial Risk Assessment and Management Conference, the Sixth Annual Conference in Financial Economics & Accounting, the 1995 European Finance Association Meetings, the 1996 American Finance Association Meetings, the 1996 American Mathematical Society Meetings, and the Sixth Summer Institute on Game Theory at SUNY Stony Brook. The authors gratefully acknowledge financial support from the H. Krueger Center for Finance at Hebrew University (Bergman and Wiener), a Batterymarch Fellowship (Grundy), and the Geewax-Terker Program in Financial Instruments (Grundy), and a Rothschild Fellowship (Wiener).


ABSTRACT

When the underlying price process is a one-dimensional diffusion, as well as in certain restricted stochastic volatility settings, a contingent claim's delta is bounded by the infimum and supremum of its delta at maturity. Further, if the claim's payoff is convex (concave), the claim's price is a convex (concave) function of the underlying asset's value. However, when volatility is less specialized, or when the underlying process is discontinuous or non-Markovian, a call's price can be a decreasing, concave function of the underlying price over some range, increasing with the passage of time, and decreasing in the level of interest rates.

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