Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T04:09:10.840Z Has data issue: false hasContentIssue false
  • English
  • Français

The log-normal approximation in financial and other computations

Part of: Markov processes Limit theorems

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]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alili, L., Dufresne, D. and Yor, M. (1997). Sur l'identité de Bougerol pour les fonctionelles exponentielles du mouvement brownien avec drift. In Exponential Functionals and Principal Values Related to Brownian Motion. Biblioteca de la Revista Matemática Iberoamericana, Madrid, pp. 3–14.Google Scholar
Barrieu, P., Rouault, A. and Yor, M. (2004). A study of the Hartman–Watson distribution motivated by numerical problems related to the pricing of Asian options. To appear in J. Appl. Prob. 41, No. 4.Google Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.Google Scholar
Bougerol, P. (1983). Exemples de théorèmes locaux sur les groupes résolubles. Ann. Inst. H. Poincaré B (N.S.) 19, 369–391.Google Scholar
Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion. Biblioteca de la Revista Matemática Iberoamericana, Madrid, pp. 73–121.Google Scholar
Comtet, A., Monthus, C. and Yor, M. (1998). Exponential functionals of Brownian motion and disordered systems. J. Appl. Prob. 35, 255–271. (Reproduced in Yor (2001).)Google Scholar
Crow, E. L. and Shimizu, K. (eds) (1988). Lognormal Distributions: Theory and Applications. Marcel Dekker, New York.Google Scholar
Dennis, B. and Patil, G. P. (1988). Applications in ecology. In Lognormal Distributions: Theory and Applications, eds Crow, E. L. and Shimizu, K., Marcel Dekker, New York, pp. 303–330.Google Scholar
Dufresne, D. (1989). Weak convergence of random growth processes with applications to insurance. Insurance Math. Econom. 8, 187–201.Google Scholar
Dufresne (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J. 1990, 39–79.Google Scholar
Dufresne (2000). Laguerre series for Asian and other options. Math. Finance 10, 407–428.Google Scholar
Dufresne, D. (2001a). An affine property of the reciprocal Asian process. Osaka J. Math. 38, 379–381.Google Scholar
Dufresne, D. (2001b). The integral of geometric Brownian motion. Adv. Appl. Prob. 33, 223–241.Google Scholar
Durrett, R. (1982). A new proof of Spitzer's result on the winding of two-dimensional Brownian motion. Ann. Prob. 10, 244–246.Google Scholar
Fu, M. C., Madan, D. B. and Wang, T. (1999). Pricing continuous Asian options: a comparison of Monte Carlo and Laplace transform inversion methods. J. Comput. Finance 2, 49–74.Google Scholar
Geman, H. and Yor, M. (1993). Bessel processes, Asian options and perpetuities. Math. Finance 3, 349–375.Google Scholar
Lebedev, N. N. (1972). Special Functions and Their Applications. Dover, New York.Google Scholar
Linetsky, V. (2001). Exact pricing of Asian options: an application of spectral theory. To appear in Operat. Res. Google Scholar
Ramakrishnan, A. (1954). A stochastic model of a fluctuating density field. Astrophys. J. 119, 682–685.Google Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, New York.Google Scholar
Rogers, L. G. C. and Shi, Z. (1995). The value of an Asian option. J. Appl. Prob. 32, 1077–1088.Google Scholar
Schröder, M., (2002). On the valuation of arithmetic-average Asian options: Laguerre series and theta integrals. Preprint, Department of Mathematics, University of Mannheim.Google Scholar
Slimane, B. S. (2001). Bounds on the distribution of a sum of independent lognormal random variables. IEEE Trans. Commun. 49, 975–978.Google Scholar
Su, Y. and Fu, M. C. (2000). Importance sampling in derivative securities pricing. In Proc. 2000 Winter Simulation Conf., eds Joines, J. A. et al., Society for Computer Simulation International, San Diego, CA, pp. 587–596.Google Scholar
Taleb, N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. John Wiley, New York.Google Scholar
Vázquez-Abad, F. and Dufresne, D. (1998). Accelerated simulation for pricing Asian options. In Proc. 1998 Winter Simulation Conf., eds Medeiros, D. J. et al., IEEE, Los Alamitos, CA, pp. 1493–1500.Google Scholar
Yor, M. (1992). On some exponential functionals of Brownian motion. Adv. Appl. Prob. 24, 509–531. (Reproduced in Yor (2001).)Google Scholar
Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, New York.Google Scholar