The Annals of Applied Probability

Bubbles, convexity and the Black–Scholes equation

Erik Ekström and Johan Tysk

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Abstract

A bubble is characterized by the presence of an underlying asset whose discounted price process is a strict local martingale under the pricing measure. In such markets, many standard results from option pricing theory do not hold, and in this paper we address some of these issues. In particular, we derive existence and uniqueness results for the Black–Scholes equation, and we provide convexity theory for option pricing and derive related ordering results with respect to volatility. We show that American options are convexity preserving, whereas European options preserve concavity for general payoffs and convexity only for bounded contracts.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 4 (2009), 1369-1384.

Dates
First available in Project Euclid: 27 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1248700621

Digital Object Identifier
doi:10.1214/08-AAP579

Mathematical Reviews number (MathSciNet)
MR2538074

Zentralblatt MATH identifier
1219.91138

Subjects
Primary: 35K65: Degenerate parabolic equations 60G44: Martingales with continuous parameter
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 91B28

Keywords
Parabolic equations stochastic representation preservation of convexity local martingales

Citation

Ekström, Erik; Tysk, Johan. Bubbles, convexity and the Black–Scholes equation. Ann. Appl. Probab. 19 (2009), no. 4, 1369--1384. doi:10.1214/08-AAP579. https://projecteuclid.org/euclid.aoap/1248700621


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