Lithuanian Mathematical Journal

, Volume 58, Issue 1, pp 33–47 | Cite as

Moments and Mellin transform of the asset price in Stein and Stein model and option pricing

  • Jacek Jakubowski
  • Zofia Michalik
  • Maciej Wiśniewolski
Article
  • 27 Downloads

Abstract

In this paper, we derive closed formulas for moments and Mellin transform of the asset price in the stochastic volatility Stein and Stein model. Next, we present applications of our results to pricing power and self-quanto options using numerical methods.

Keywords

Stein and Stein model correlated Brownian motions moments squared radial Ornstein–Uhlenbeck process Mellin transform fast Fourier transform power options self-quanto options 

MSC

60J70 91G80 60H30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L.B.G. Andersen and V.V. Piterbarg, Moment explosions in stochastic volatility models, Finance Stoch., 11:29–50, 2007.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    T. Andersen, H.-J. Chung, and B.E. Sorensen, Efficient method of moments estimation of a stochastic volatility model: A Monte Carlo study, J. Econom., 91:61–87, 1999.CrossRefMATHGoogle Scholar
  3. 3.
    T.G. Andersen and B.E. Sorensen, GMM estimation of a stochastic volatility model: A Monte Carlo study, J. Bus. Econ. Stat., 14:328–352, 1996.Google Scholar
  4. 4.
    O.E. Barndorff-Nielsen and N. Shephard, Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics, J. R. Stat. Soc. Ser. B Stat. Methodol., 63(2):167–241, 1999.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    A. Borodin and P. Salminen, Handbook of Brownian Motion – Facts and Formulae, 2nd ed., Birkhäuser, Basel, 2002.CrossRefMATHGoogle Scholar
  6. 6.
    P. Carr and D. Madan, Option valuation using the fast Fourier transform, J. Comput. Finance, 2:61–73, 1998.CrossRefGoogle Scholar
  7. 7.
    J.W. Cooley and J.W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comput., 19(90):297–301, 1965.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    M. D’Amico, G. Fusai, and A. Tagliani, Valuation of exotic options using moments, Oper. Res., Int. J., 2(2):157–186, 2002.CrossRefMATHGoogle Scholar
  9. 9.
    Eqworld – the world of mathematical equations, available from: http://eqworld.ipmnet.ru/index.htm.
  10. 10.
    S. Fadugba and C. Nwozo, Valuation of European call options via the fast Fourier transform and the improved Mellin transform, Journal of Mathematical Finance, 6(2):338–359, 2016.CrossRefGoogle Scholar
  11. 11.
    L. Fatone, F. Mariani, M.C. Recchioni, and F. Zirilli, The calibration of some stochastic volatility models used in mathematical finance, Open Journal of Applied Sciences, 92(1):23–33, 2014.CrossRefGoogle Scholar
  12. 12.
    A. Ronald Gallant and G. Tauchen, The relative efficiency of method of moments estimators, J. Econom., 6(2):149–172, 1999.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    J. Gil-Pelaez, Note on the inversion theorem, Biometrika, 38(3-4):481–482, 1951.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    A. Gulisashvili, Analytically Tractable Stochastic Stock Price Models, Springer, Berlin, Heidelberg, 2012.CrossRefMATHGoogle Scholar
  15. 15.
    P. Hagan, D. Kumar, A. Lesniewski, and D. Woodward, Managing smile risk, Wilmott Magazine, pp. 84–108, 2002.Google Scholar
  16. 16.
    L.P. Hansen, Large sample properties of generalized method of moments estimators, Econometrica, 50:1029–1054, 1982.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finance, 42:281–300, 1987.CrossRefMATHGoogle Scholar
  18. 18.
    J. Jakubowski and M. Wiśniewolski, On some Brownian functionals and their applications to moments in the lognormal stochastic volatility model, Stud. Math., 219:201–224, 2013.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    J. Jakubowski and M. Wiśniewolski, On matching diffusions, Laplace transforms and partial differential equations, Stochastic Processes Appl., 125:3663–3690, 2015.MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    M. Jeanblanc, M. Yor, and M. Chesney, Mathematical Methods for Financial Markets, Springer-Verlag, London, 2009.CrossRefMATHGoogle Scholar
  21. 21.
    I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1998.CrossRefMATHGoogle Scholar
  22. 22.
    E. Levy, Pricing European average rate currency options, J. Int. Money Finance, 11:474–491, 1992.MathSciNetCrossRefGoogle Scholar
  23. 23.
    G.D. Lin, Characterizations of distributions via moments, Sankhyā, Ser. A, 54:128–132, 1992.MathSciNetMATHGoogle Scholar
  24. 24.
    R. Mansuy and M. Yor, Aspects of Brownian Motion, Springer-Verlag, Berlin, Heidelberg, 2008.CrossRefMATHGoogle Scholar
  25. 25.
    S.E. Posner and M.A. Milevsky, Valuing exotic options by approximating the spd with higher moments, Journal of Financial Engineering, 7(2):109–125, 1998.Google Scholar
  26. 26.
    S. Raible, Lévy Processes in Finance: Theory, Numerics and Empirical Facts, PhD dissertation, Freiburg University, 2000.Google Scholar
  27. 27.
    R. Rebonato, Volatility and Correlation: The Perfect Hedger and the Fox, 2nd ed., John Wiley & Sons, 2004.Google Scholar
  28. 28.
    D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Springer-Verlag, Berlin, Heidelberg, 2005.MATHGoogle Scholar
  29. 29.
    I. SenGupta, Pricing Asian options in financial markets using Mellin transform, Electron. J. Differ. Equ., 2014(234):1–9, 2014.MathSciNetMATHGoogle Scholar
  30. 30.
    I. SenGupta, Generalized BN-S stochastic volatility model for option pricing, Int. J. Theor. Appl. Finance, 19(02):1650014, 2016.MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    R. Shöbel and J. Zhu, Stochastic volatility with an Ornstein–Uhlenbeck process: An extension, Eur. Finance Rev., 3:23–46, 1999.CrossRefMATHGoogle Scholar
  32. 32.
    C. Sin, Complications with stochastic volatility models, Adv. Appl. Probab., 30:256–268, 1998.MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    E. Stein and J. Stein, Stock price distributions with stochastic volatility: An analytic approach, Rev. Financ. Stud., 4:727–752, 1991.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Jacek Jakubowski
    • 1
  • Zofia Michalik
    • 1
  • Maciej Wiśniewolski
    • 1
  1. 1.Institute of MathematicsUniversity of WarsawWarszawaPoland

Personalised recommendations