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A MEAN–VARIANCE BOUND FOR A THREE-PIECE LINEAR FUNCTION

Published online by Cambridge University Press:  22 October 2007

Karthik Natarajan  and
Zhou Linyi
Karthik Natarajan
Affiliation:
Department of MathematicsNational University of SingaporeSingapore117543 E-mail: matkbn@nus.edu.sg; minniezhou@hotmail.com
Zhou Linyi
Affiliation:
Department of MathematicsNational University of SingaporeSingapore117543 E-mail: matkbn@nus.edu.sg; minniezhou@hotmail.com
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Abstract

In this article, we derive a tight closed-form upper bound on the expected value of a three-piece linear convex function E[max(0, X, mX − z)] given the mean μ and the variance σ2 of the random variable X. The bound is an extension of the well-known mean–variance bound for E[max(0, X)]. An application of the bound to price the strangle option in finance is provided.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 21 , Issue 4 , October 2007 , pp. 611 - 621
Copyright
Copyright © Cambridge University Press 2007

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References

1.Gallego, G. & Moon, I. (1993). The distribution free newsboy problem: Review and extensions. Journal of the Operational Research Society 44(8): 825–834.CrossRefGoogle Scholar
2.Isii, K. (1963). On the sharpness of Chebyshev-type inequalities. Annals of the Institute of Statistical Mathematics 14: 185–197.CrossRefGoogle Scholar
3.Lo, A. (1987). Semiparametric upper bounds for options prices and expected payoffs. Journal of Financial Economics 19: 373–388.CrossRefGoogle Scholar
4.Popescu, I. (2007). Robust mean–covariance solutions for stochastic optimization and applications. Operations Research 55(1): 98–112.CrossRefGoogle Scholar
5.Rogosinsky, W.W. (1958). Moments of non-negative mass. Proceedings of the Royal Society London A 245: 1–27.Google Scholar
6.Scarf, H. (1958). A min-max solution of an inventory problem. Studies in the mathematical theory of inventory and production. Stanford, CA: Stanford University Press, pp. 201–209.Google Scholar
7.van der Vlerk, M.H. (2005). On multiple simple recourse models. Mathematical Methods of Operations Research 62(2): 225–242.CrossRefGoogle Scholar