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Analytic Bounds and Approximations for Annuities and Asian Options

18 Pages Posted: 12 Jan 2006 Last revised: 22 Apr 2009

Steven Vanduffel

Vrije Universiteit Brussel (VUB)

Luc Henrard

BNP Paribas

Jan Dhaene

Katholieke Universiteit Leuven - Faculty of Economics and Business

Emiliano Valdez

University of New South Wales (UNSW) - School of Actuarial Studies

Zhaoning Shang

KU Leuven

Date Written: February 3, 2005

Abstract

Even in case of the Brownian motion as most natural rate of return model it appears too difficult to obtain analytic expressions for most risk measures of constant continuous annuities. In literature so-called comonotonic approximations have been proposed but these still require the evaluation of integrals.

In this paper we show that these integrals can sometimes be computed, and we obtain explicit approximations for some popular risk measures for annuities.

Next, we show how these results can be used to obtain fully analytic expressions for lower and upper bounds for the price of a continuously sampled European-style Asian option with fixed exercise price.

These analytic lower bound prices are as sharp as those from Rogers & Shi (1995), if not sharper, but in contrast do not require any longer the evalution of a two-dimensional or a onedimensional integral.

Keywords: Asian option, closed- form, analytical, annuity, fast approximation, lognormal, maximal variance, conditional expectation

Suggested Citation

Vanduffel, Steven and Henrard, Luc and Dhaene, Jan and Valdez, Emiliano and Shang, Zhaoning, Analytic Bounds and Approximations for Annuities and Asian Options (February 3, 2005). Insurance: Mathematics and Economics, Vol. 42, No. 3, 2008. Available at SSRN: https://ssrn.com/abstract=875292

Steven Vanduffel (Contact Author)

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
Brussels, Brabant 1050
Belgium

HOME PAGE: http://www.stevenvanduffel.com

Luc Henrard

BNP Paribas ( email )

Montagne Du Parc 3
Belgium

Jan Dhaene

Katholieke Universiteit Leuven - Faculty of Economics and Business ( email )

Naamsestraat 69
Leuven, 3000
Belgium

Emiliano Valdez

University of New South Wales (UNSW) - School of Actuarial Studies ( email )

Sydney NSW 2052, ACT 2600
Australia

Zhaoning Shang

KU Leuven

Naamsestraat 69
Leuven, Vlaams Brabant 3000
Belgium

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