Bounds for Probabilities of Extreme Events Defined by Two Random Variables

Variance, Vol. 4, No. 1, pp. 47-65, 2010

34 Pages Posted: 30 Jul 2007 Last revised: 28 Aug 2011

See all articles by Samuel H. Cox

Samuel H. Cox

University of Manitoba - Asper School of Business

Yijia Lin

University of Nebraska at Lincoln - Department of Finance

Ruilin Tian

North Dakota State University - Department of Accounting, Finance, and Information Systems

Luis Zuluaga

University of New Brunswick - Fredericton

Date Written: January 5, 2010

Abstract

This paper offers a methodology for calculating optimal bounds on tail risk probabilities by deriving upper and lower semiparametric bounds, given only the first two moments of the distribution. We apply this methodology to determine bounds for probabilities of two tail events. The first tail event occurs when two financial variables simultaneously have extremely low values. The second occurs when the sum of two financial variables takes a very low value. In both cases we are finding bounds for actual or physical probabilities of these events rather than probabilities for a pricing or risk neutral measure. We use sum of squares optimization programs to obtain the desired bounds. To illustrate our ideas, we present several numerical examples. This approach is suitable in the situations when it is difficult to make exact distributional assumptions due to, for instance, scarcity and/or high volatility of data. Even in the situations when distributional assumptions can be made, this approach can be used to check the consistency of those assumptions.

Keywords: Semiparametric bounds, joint tail probabilities, value at risk, moments, sum of square programming

JEL Classification: C10, G22

Suggested Citation

Cox, Samuel H. and Lin, Yijia and Tian, Ruilin and Zuluaga, Luis, Bounds for Probabilities of Extreme Events Defined by Two Random Variables (January 5, 2010). Variance, Vol. 4, No. 1, pp. 47-65, 2010, Available at SSRN: https://ssrn.com/abstract=1003723

Samuel H. Cox

University of Manitoba - Asper School of Business ( email )

181 Freedman Crescent
Winnipeg, Manitoba R3T 5V4
Canada

Yijia Lin

University of Nebraska at Lincoln - Department of Finance ( email )

Lincoln, NE 68588-0490
United States

Ruilin Tian (Contact Author)

North Dakota State University - Department of Accounting, Finance, and Information Systems ( email )

Fargo, ND
United States
7012316544 (Phone)
7012316545 (Fax)

Luis Zuluaga

University of New Brunswick - Fredericton ( email )

Bailey Drive
P.O. Box 4400
Fredericton NB E3B 5A3, New Brunswick E3B 5A3
Canada

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