Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T14:07:58.741Z Has data issue: false hasContentIssue false
  • English
  • Français

On ladder height densities and Laguerre series in the study of stochastic functionals. II. Exponential functionals of brownian motion and asian option values

Part of: Probabilistic methods, simulation and stochastic differential equations Markov processes

Published online by Cambridge University Press:  08 September 2016

Michael Schröder
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we develop a constructive structure theory for a class of exponential functionals of Brownian motion which includes Asian option values. This is done in two stages of differing natures. As a first step, the functionals are represented as Laguerre reduction series obtained from main results of Schröder (2006), this paper's companion paper. These reduction series are new and given in terms of the negative moments of the integral of geometric Brownian motion, whose structure theory is developed in a second step. Providing a new angle on these processes, this is done by establishing connections with theta functions. Integral representations and computable formulae for the negative moments are thus derived and then shown to furnish highly efficient ways for computing the negative moments. Application of this paper's Laguerre reduction series in numerical examples suggests that one of the most efficient methods for the explicit valuation of Asian options is obtained. The paper also provides mathematical background results referred to in Schröder (2005c).

Keywords

Laguerre series methods for stochastic functionalsladder height densitymoment formulae for the integral of geometric Brownian motionexplicit valuation of an Asian option

MSC classification

Primary: 60J65: Brownian motion
Secondary: 65C50: Other computational problems in probability 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 4 , December 2006 , pp. 995 - 1027
Copyright
Copyright © Applied Probability Trust 2006 

References

Alili, L. (1995). Fonctionelles exponentielles et valeurs principales de mouvement Brownien. , Université Paris VI.Google Scholar
Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions (Encyclopaedia Math. Appl. 71). Cambridge University Press.Google Scholar
Carr, P. and Schröder, M. (2004). Bessel processes, the integral of geometric Brownian motion, and Asian options. Theory Prob. Appl. 48, 400–425.Google Scholar
Comtet, A. and Monthus, C. (1996). Diffusion in a one-dimesional random medium and hyperbolic Brownian motion. J. Phys. A 29, 1331–1345.CrossRefGoogle Scholar
Doetsch, G. (1971). Handbuch der Laplace-Transformation, Band I. Birkhäuser, Basel.Google Scholar
Donati-Martin, C., Matsumoto, H. and Yor, M. (2000a). On striking identities about the exponential functionals of the Brownian bridge and Brownian motion. Periodica Math. Hung. 41, 103–119.Google Scholar
Donati-Martin, C., Matsumoto, H. and Yor, M. (2000b). On the positive and negative moments of the integral of geometric Brownian motion. Statist. Prob. Lett. 49, 45–52.CrossRefGoogle Scholar
Donati-Martin, C., Matsumoto, H. and Yor, M. (2002). The law of geometric Brownian motion and its integral, revisited; application to conditional moments. In Mathematical Finance: Bachelier Congress 2000, eds Geman, H. et al., Springer, Berlin, pp. 221–243.CrossRefGoogle Scholar
Dufresne, D. (1989). Weak convergence of random growth processes with applications to insurance. Insurance Math. Econom. 8, 187–201.Google Scholar
Dufresne, D. (2000). Laguerre series for Asian and other options. Math. Finance 10, 407–428.Google Scholar
Dufresne, D. (2001). The integral of geometric Brownian motion. Adv. Appl. Prob. 33, 223–241.CrossRefGoogle Scholar
Dufresne, D. (2005). Bessel processes and Asian options. In Numerical Methods in Finance, eds Breton, M. and Ben-Ameur, H., Springer, New York, pp. 35–58.Google Scholar
Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1953). Higher Transcendental Functions, Vol. 2. McGraw-Hill, New York.Google Scholar
Geman, H. and Yor, M. (1993). Bessel processes, Asian options, and perpetuities. Math. Finance 3, 349–375.Google Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.Google Scholar
Lebedev, N. N. (1972). Special Functions and Their Applications. Dover, New York.Google Scholar
Linetsky, V. (2004). Spectral expansions for Asian (average price) options. Operat. Res. 52, 856–867.Google Scholar
McKean, H. P. Jr. (1956). Elementary solutions of certain parabolic partial differential equations. Trans. Amer. Math. Soc. 82, 519–548.Google Scholar
Monthus, C. and Comtet, A. (1994). On the flux in a one-dimensional disordered system. J. Phys. I (France) 4, 635–653.CrossRefGoogle Scholar
Mumford, D. (1983). Tata Lectures on Theta I (Progress Math. 28). Birkhäuser, Boston, MA.CrossRefGoogle Scholar
Mumford, D. (1984). Tata Lectures on Theta II (Progress Math. 43). Birkhäuser, Boston, MA.Google Scholar
Mumford, D. (1991). Tata Lectures on Theta III (Progress Math. 97). Birkhäuser, Boston, MA.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes in C, 2nd edn. Cambridge University Press.Google Scholar
Rogers, L. C. G. and Shi, Z. (1995). The value of an Asian option. J. Appl. Prob. 32, 1077–1088.Google Scholar
Sansone, G. (1991). Orthogonal Functions. Dover, New York.Google Scholar
Schröder, M. (2002). Mathematical ramifications of option valuation: the case of the Asian option. Habilitationsschrift, Universität Mannheim.Google Scholar
Schröder, M. (2003). On the integral of geometric Brownian motion. Adv. Appl. Prob. 35, 159–183.Google Scholar
Schröder, M. (2005a). A constructive Hartman–Watson approach to stochastic functionals, with applications to finance. To appear in Theory Prob. Appl. Google Scholar
Schröder, M. (2005b). Continuous-time methods in the study of discretely sampled functionals of Lévy processes. I. The positive process case. To appear in Adv. Appl. Prob. Google Scholar
Schröder, M. (2005c). Laguerre series in contingent claim valuation, with applications to Asian options. Math. Finance 15, 491–531.Google Scholar
Schröder, M. (2006). On ladder height densities and Laguerre series in the study of stochastic functionals. I. Basic methods and results. Adv. Appl. Prob. 38, 969–994.CrossRefGoogle Scholar
Vecer, J. (2001). A new PDE approach for pricing arithmetic average Asian options. J. Comput. Finance 4, 105–113.Google Scholar
Vecer, J. (2002). Unified Asian pricing. Risk 15, 113–116.Google Scholar
Vecer, J. and Xu, M. (2004). Pricing Asian options in a semimartingale model. Quant. Finance 4, 170–175.Google Scholar
Watson, G. N. (1944). A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press.Google Scholar
Yor, M. (1992a). On some exponential functionals of Brownian motion. Adv. Appl. Prob. 24, 509–531.Google Scholar
Yor, M. (1992b). Some Aspects of Brownian Motion. Part I. Birkhäuser, Basel.Google Scholar
Yor, M. (ed.) (1997). Exponential Functionals and Principal Values Related to Brownian Motion. Revista Mathemática Iberoamericana, Madrid.Google Scholar
Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin.Google Scholar