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The log-normal approximation in financial and other computations

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

Daniel Dufresne
Daniel Dufresne*
Affiliation:
University of Melbourne
*
∗ Postal address: Centre for Actuarial Studies, Department of Economics, University of Melbourne, VIC 3010, Australia. Email address: [email protected]
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Abstract

Sums of log-normals frequently appear in a variety of situations, including engineering and financial mathematics. In particular, the pricing of Asian or basket options is directly related to finding the distributions of such sums. There is no general explicit formula for the distribution of sums of log-normal random variables. This paper looks at the limit distributions of sums of log-normal variables when the second parameter of the log-normals tends to zero or to infinity; in financial terms, this is equivalent to letting the volatility, or maturity, tend either to zero or to infinity. The limits obtained are either normal or log-normal, depending on the normalization chosen; the same applies to the reciprocal of the sums of log-normals. This justifies the log-normal approximation, much used in practice, and also gives an asymptotically exact distribution for averages of log-normals with a relatively small volatility; it has been noted that all the analytical pricing formulae for Asian options perform poorly for small volatilities. Asymptotic formulae are also found for the moments of the sums of log-normals. Results are given for both discrete and continuous averages. More explicit results are obtained in the case of the integral of geometric Brownian motion.

Keywords

Sums of log-normal variablesBrownian motionAsian optionbasket optionexponential functional of Brownian motion

MSC classification

Primary: 60J65: Brownian motion 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 747 - 773
Copyright
Copyright © Applied Probability Trust 2004 

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