Advances in Applied Probability

The log-normal approximation in financial and other computations

Daniel Dufresne

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Adv. in Appl. Probab. Volume 36, Number 3 (2004), 747-773.

Dates
First available in Project Euclid: 31 August 2004

Permanent link to this document
http://projecteuclid.org/euclid.aap/1093962232

Digital Object Identifier
doi:10.1239/aap/1093962232

Mathematical Reviews number (MathSciNet)
MR2079912

Zentralblatt MATH identifier
1063.60115

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60F05: Central limit and other weak theorems

Keywords
Sums of log-normal variables Brownian motion Asian option basket option exponential functional of Brownian motion

Citation

Dufresne, Daniel. The log-normal approximation in financial and other computations. Adv. in Appl. Probab. 36 (2004), no. 3, 747--773. doi:10.1239/aap/1093962232. http://projecteuclid.org/euclid.aap/1093962232.


Export citation

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.
  • 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.
  • Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
  • Bougerol, P. (1983). Exemples de théorèmes locaux sur les groupes résolubles. Ann. Inst. H. Poincaré B (N.S.) 19, 369--391.
  • 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.
  • 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).)
  • Crow, E. L. and Shimizu, K. (eds) (1988). Lognormal Distributions: Theory and Applications. Marcel Dekker, New York.
  • Dennis, B. and Patil, G. P. (1988). Applications in ecology. In Lognormal Distributions: Theory and Applications, eds E. L. Crow and K. Shimizu, Marcel Dekker, New York, pp. 303--330.
  • Dufresne, D. (1989). Weak convergence of random growth processes with applications to insurance. Insurance Math. Econom. 8, 187--201.
  • Dufresne (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J. 1990, 39--79.
  • Dufresne (2000). Laguerre series for Asian and other options. Math. Finance 10, 407--428.
  • Dufresne, D. (2001a). An affine property of the reciprocal Asian process. Osaka J. Math. 38, 379--381.
  • Dufresne, D. (2001b). The integral of geometric Brownian motion. Adv. Appl. Prob. 33, 223--241.
  • Durrett, R. (1982). A new proof of Spitzer's result on the winding of two-dimensional Brownian motion. Ann. Prob. 10, 244--246.
  • 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.
  • Geman, H. and Yor, M. (1993). Bessel processes, Asian options and perpetuities. Math. Finance 3, 349--375.
  • Lebedev, N. N. (1972). Special Functions and Their Applications. Dover, New York.
  • Linetsky, V. (2001). Exact pricing of Asian options: an application of spectral theory. To appear in Operat. Res.
  • Ramakrishnan, A. (1954). A stochastic model of a fluctuating density field. Astrophys. J. 119, 682--685.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, New York.
  • Rogers, L. G. C. and Shi, Z. (1995). The value of an Asian option. J. Appl. Prob. 32, 1077--1088.
  • Schröder, M. (2002). On the valuation of arithmetic-average Asian options: Laguerre series and theta integrals. Preprint, Department of Mathematics, University of Mannheim.
  • Slimane, B. S. (2001). Bounds on the distribution of a sum of independent lognormal random variables. IEEE Trans. Commun. 49, 975--978.
  • Su, Y. and Fu, M. C. (2000). Importance sampling in derivative securities pricing. In Proc. 2000 Winter Simulation Conf., eds J. A. Joines et al., Society for Computer Simulation International, San Diego, CA, pp. 587--596.
  • Taleb, N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. John Wiley, New York.
  • Vázquez-Abad, F. and Dufresne, D. (1998). Accelerated simulation for pricing Asian options. In Proc. 1998 Winter Simulation Conf., eds D. J. Medeiros et al., IEEE, Los Alamitos, CA, pp. 1493--1500.
  • Yor, M. (1992). On some exponential functionals of Brownian motion. Adv. Appl. Prob. 24, 509--531. (Reproduced in Yor (2001).)
  • Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, New York.