Asymptotic Analysis of Stock Price Distributions

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

Abstract

Chapter 6 focuses on the asymptotics of stock price densities in classical stochastic volatility models. Sharp asymptotic formulas with relative error estimates for stock price densities in the uncorrelated Hull-White, Stein-Stein, and Heston models due to E.M. Stein and the author are presented in this chapter. The proofs use the asymptotic formulas for mixing distributions and the Abelian theorem for the integral transforms with log-normal kernels established in Chap.  5. Extensions of the asymptotic formulas for the stock price density to the case of the correlated Heston and Stein-Stein models are also presented. The asymptotic behavior of the stock price density in the correlated Hull-White model remains a mystery.

Keywords

Saddle Point Stock Price Asymptotic Formula Stochastic Volatility Model Local Expansion 
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. [AP07]
    Andersen, L. B. G., Piterbarg, V. V., Moment explosions in stochastic volatility models, Finance and Stochastics 11 (2007), pp. 29–50. MathSciNetMATHCrossRefGoogle Scholar
  2. [BA88a]
    Ben Arous, G., Dévepoppement asymptotique du noyau de la chaler hypoelliptique hors du cut-locus, Annales Scientifiques de l’École Normale Supérieure 4 (1988), pp. 307–331. MathSciNetGoogle Scholar
  3. [BA88b]
    Ben Arous, G., Methods de Laplace et de la phase stationnaire sur l’espace de Wiener, Stochastics 25 (1988), pp. 125–153. MathSciNetMATHCrossRefGoogle Scholar
  4. [BH95]
    Bleistein, N., Handelsman, R. A., Asymptotic Expansions of Integrals, Holt, Rinehart and Winston, New York, 1995. Google Scholar
  5. [dBru81]
    de Bruijn, N. G., Asymptotic Methods in Analysis, Dover, New York, 1981. MATHGoogle Scholar
  6. [DFJV11]
    Deuschel, J.-D., Friz, P. K., Jacquier, A., Violante, S., Marginal density expansions for diffusions and stochastic volatility, preprint, 2011; available at arXiv:1111.2462.
  7. [DY02]
    Drăgulescu, A. A., Yakovenko, V. M., Probability distribution of returns in the Heston model with stochastic volatility, Quantitative Finance 2 (2002), pp. 443–453. MathSciNetGoogle Scholar
  8. [FS09]
    Flajolet, P., Sedgewick, R., Analytic Combinatorics, Cambridge University Press, Cambridge, 2009. MATHCrossRefGoogle Scholar
  9. [FGGS11]
    Friz, P., Gerhold, S., Gulisashvili, A., Sturm, S., On refined volatility smile expansion in the Heston model, Quantitative Finance 11 (2011), pp. 1151–1164. MathSciNetCrossRefGoogle Scholar
  10. [FK-R10]
    Friz, P., Keller-Ressel, M., Moment explosions, in: Cont, R. (Ed.), Encyclopedia of Quantitative Finance, pp. 1247–1253, Wiley, Chichester, 2010. Google Scholar
  11. [GK10]
    Glasserman, P., Kim, K.-K., Moment explosions and stationary distributions in affine diffusion modles, Mathematical Finance 20 (2010), pp. 1–33. MathSciNetMATHCrossRefGoogle Scholar
  12. [GS06]
    Gulisashvili, A., Stein, E. M., Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: the Hull–White model, Comptes Rendus de l’Académie des Sciences de Paris, Série I 343 (2006), pp. 519–523. MathSciNetMATHGoogle Scholar
  13. [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
  14. [GS10b]
    Gulisashvili, A., Stein, E. M., Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models, Applied Mathematics and Optimization 61 (2010), pp. 287–315. MathSciNetMATHCrossRefGoogle Scholar
  15. [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
  16. [LM07]
    Lions, P.-L., Musiela, M., Correlations and bounds for stochastic volatility models, Annales de l’Institut Henry Poincaré 24 (2007), pp. 1–16. MathSciNetMATHCrossRefGoogle Scholar
  17. [Luc07]
    Lucic, V., On singularities in the Heston models, working paper, 2007. Google Scholar
  18. [Mil06]
    Miller, P. D., Applied Asymptotic Analysis, American Mathematical Society, Providence, 2006. MATHGoogle Scholar
  19. [Mur84]
    Murray, J. D., Asymptotic Analysis, Springer, New York, 1984. MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

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

Personalised recommendations