Integral Transforms of Distribution Densities

  • Archil Gulisashvili
Part of the Springer Finance book series (FINANCE)

Abstract

Analytical tractability is a desirable property of stochastic stock price models. Informally speaking, a stochastic model is analytically tractable if various important characteristics of the model can be represented explicitly or asymptotically in terms of standard functions of mathematical analysis. Classical stochastic volatility models (Hull-White, Stein-Stein, Heston) are analytically tractable. In this chapter, explicit formulas are obtained for Laplace transforms of mixing densities and Mellin transforms of stock price densities in classical stochastic volatility models. For example, an alternative proof of an explicit formula for the Laplace transform of the distribution density of an integrated geometric Brownian motion due to L. Alili and J.C. Gruet is given. Chapter 4 also contains an explicit formula for the stock price density in the correlated Hull-White model with driftless volatility obtained by Y. Maghsoodi. In addition, Chap. 4 provides explicit formulas for the Mellin transform of the stock price density in the correlated Heston and Stein-Stein models.

Keywords

Brownian Motion Explicit Formula Laplace Transform Moment Generate Function Standard Brownian Motion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [AG97]
    Alili, L., Gruet, J. C., An explanation of a generalized Bboujerol’s identity in terms of hyperbolic geometry, in: Yor, M. (Ed.), Exponential Functionals and Principal Values Related to Brownian Motion, pp. 15–33, Biblioteca de la Revista Matemàtica Ibero-Americana, Madrid, 2007. Google Scholar
  2. [BRY04]
    Barrieu, P., Rouault, A., Yor, M., A study of the Hartman–Watson distribution motivated by numerical problems related to the pricing of Asian options, Journal of Applied Probability 41 (2004), pp. 1049–1058. MathSciNetMATHCrossRefGoogle Scholar
  3. [Bou83]
    Bougerol, Ph., Exemples des théorèmes locaux sur les groupes résolubles, Annales de l’Institut Henry Poincaré 19 (1983), pp. 369–391. MathSciNetMATHGoogle Scholar
  4. [dBRF-CU10]
    del Baño Rollin, S., Ferreiro-Castilla, A., Utzet, F., On the density of log-spot in Heston volatility model, Stochastic Processes and Their Applications 120 (2010), pp. 2037–2062. MathSciNetMATHCrossRefGoogle Scholar
  5. [FS09]
    Flajolet, P., Sedgewick, R., Analytic Combinatorics, Cambridge University Press, Cambridge, 2009. MATHCrossRefGoogle Scholar
  6. [Ger11]
    Gerhold, S., The Hartman–Watson distribution revisited: asymptotics for pricing Asian options, Journal of Applied Probability 48 (2011), pp. 892–899. MathSciNetMATHCrossRefGoogle Scholar
  7. [GS10a]
    Gulisashvili, A., Stein, E. M., Asymptotic behavior of distribution densities in models with stochastic volatility, I, Mathematical Finance 20 (2010), pp. 447–477. MathSciNetMATHCrossRefGoogle Scholar
  8. [GvC06]
    Gulisashvili, A., van Casteren, J. A., Non-Autonomous Kato Classes and Feynman–Kac Propagators, World Scientific, Singapore, 2006. MATHCrossRefGoogle Scholar
  9. [HW74]
    Hartman, P., Watson, G. S., “Normal” distribution functions on spheres and the modified Bessel functions, Annals of Probability 2 (1974), pp. 593–607. MathSciNetMATHCrossRefGoogle Scholar
  10. [Hes93]
    Heston, S. L., A closed-form solution for options with stochastic volatility, with applications to bond and currency options, Review of Financial Studies 6 (1993), pp. 327–343. CrossRefGoogle Scholar
  11. [Jef96]
    Jefferies, B., Evolution Processes and the Feynman–Kac Formula, Kluwer Academic, Dordrecht, 1996. MATHGoogle Scholar
  12. [K-R11]
    Keller-Ressel, M., Moment explosions and long-term behavior of affine stochastic volatility models, Mathematical Finance 21 (2011), pp. 73–98. MathSciNetMATHCrossRefGoogle Scholar
  13. [Mag07]
    Maghsoodi, Y., Exact solution of a martingale stochastic volatility option problem and its empirical evaluation, Mathematical Finance 17 (2007), pp. 249–265. MathSciNetMATHCrossRefGoogle Scholar
  14. [MY05a]
    Matsumoto, H., Yor, M., Exponential functionals of Brownian motion, I: probability laws at fixed time, Probability Surveys 2 (2005), pp. 312–347. MathSciNetMATHCrossRefGoogle Scholar
  15. [MY05b]
    Matsumoto, H., Yor, M., Exponential functionals of Brownian motion, II: some related diffusion processes, Probability Surveys 2 (2005), pp. 348–384. MathSciNetMATHCrossRefGoogle Scholar
  16. [Øks03]
    Øksendal, B., Stochastic Differential Equations. An Introduction with Applications, 6th ed., Springer, Berlin, 2003. Google Scholar
  17. [PK01]
    Paris, R. B., Kaminski, D., Asymptotics and Mellin–Barnes Integrals, Cambridge University Press, Cambridge, 2001. MATHCrossRefGoogle Scholar
  18. [PY82]
    Pitman, J., Yor, M., A decomposition of Bessel bridges, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 59 (1982), pp. 425–457. MathSciNetMATHCrossRefGoogle Scholar
  19. [RY04]
    Revuz, D., Yor, M., Continuous Martingales and Brownian Motion, Springer, Berlin, 2004. Google Scholar
  20. [SZ99]
    Schöbel, R., Zhu, J., Stochastic volatility with an Ornstein–Uhlenbeck process: an extension, European Finance Review 3 (1999), pp. 23–46. MATHCrossRefGoogle Scholar
  21. [SS91]
    Stein, E. M., Stein, J., Stock price distributions with stochastic volatility: an analytic approach, Review of Financial Studies 4 (1991), pp. 727–752. CrossRefGoogle Scholar
  22. [Wen90]
    Wenocur, M. L., Ornstein–Uhlenbeck process with quadratic killing, Journal of Applied Probability 27 (1990), pp. 707–712. MathSciNetMATHCrossRefGoogle Scholar
  23. [Yor80]
    Yor, M., Loi de l’indice du lacet Brownien, et distribution de Hartman–Watson, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 53 (1980), pp. 71–95. MathSciNetMATHCrossRefGoogle Scholar
  24. [Yor92a]
    Yor, M., On some exponential functionals of Brownian motion, Advances in Applied Probability 24 (1992), pp. 509–531. MathSciNetMATHCrossRefGoogle Scholar
  25. [Yor92b]
    Yor, M., Sur les lois des fonctionells exponentielles du mouvement brownien, considérées en certain instants aléatoires, Comptes Rendus de l’Académie des Sciences de Paris 314 (1992), pp. 951–956. MathSciNetMATHGoogle Scholar
  26. [Yor01]
    Yor, M., Exponential Functionals of Brownian Motion and Related Processes, Springer, Berlin, 2001. MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Archil Gulisashvili
    • 1
  1. 1.Department of MathematicsOhio UniversityAthensUSA

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