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April 2002 The minimum maximum of a continuous martingale with given initial and terminal laws
David G. Hobson, J. L. Pedersen
Ann. Probab. 30(2): 978-999 (April 2002). DOI: 10.1214/aop/1023481014

Abstract

Let $(M_t)_{0 \leq t \leq 1} be a continuous martingale with initial law $M_0 \sim \mu_0$, and terminal law $M_1 \sim \mu_1$, and let $S = \sup_{0 \leq t \leq 1} M_t$. In this paper we prove that there exists a greatest lower bound with respect to stochastic ordering of probability measures, on the law of $S$. We give an explicit construction of this bound. Furthermore a martingale is constructed which attains this minimum by solving a Skorokhod embedding problem. The form of this martingale is motivated by a simple picture. The result is applied to the robust hedging of a forward start digital option.

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David G. Hobson. J. L. Pedersen. "The minimum maximum of a continuous martingale with given initial and terminal laws." Ann. Probab. 30 (2) 978 - 999, April 2002. https://doi.org/10.1214/aop/1023481014

Information

Published: April 2002
First available in Project Euclid: 7 June 2002

zbMATH: 1016.60047
MathSciNet: MR1906424
Digital Object Identifier: 10.1214/aop/1023481014

Subjects:
Primary: 60G40 , 60G44
Secondary: 60G30 , 91B28

Keywords: martingale , Optimal stopping , option pricing , Skorokhod embedding problem

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 2 • April 2002
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