Skip to main content

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

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.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    L.B.G. Andersen and V.V. Piterbarg, Moment explosions in stochastic volatility models, Finance Stoch., 11:29–50, 2007.

    MathSciNet  Article  MATH  Google 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.

    Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    A. Borodin and P. Salminen, Handbook of Brownian Motion – Facts and Formulae, 2nd ed., Birkhäuser, Basel, 2002.

    Book  MATH  Google Scholar 

  6. 6.

    P. Carr and D. Madan, Option valuation using the fast Fourier transform, J. Comput. Finance, 2:61–73, 1998.

    Article  Google 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.

    MathSciNet  Article  MATH  Google 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.

    Article  MATH  Google 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.

    Article  Google 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.

    Article  Google Scholar 

  12. 12.

    A. Ronald Gallant and G. Tauchen, The relative efficiency of method of moments estimators, J. Econom., 6(2):149–172, 1999.

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    J. Gil-Pelaez, Note on the inversion theorem, Biometrika, 38(3-4):481–482, 1951.

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    A. Gulisashvili, Analytically Tractable Stochastic Stock Price Models, Springer, Berlin, Heidelberg, 2012.

    Book  MATH  Google Scholar 

  15. 15.

    P. Hagan, D. Kumar, A. Lesniewski, and D. Woodward, Managing smile risk, Wilmott Magazine, pp. 84–108, 2002.

  16. 16.

    L.P. Hansen, Large sample properties of generalized method of moments estimators, Econometrica, 50:1029–1054, 1982.

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finance, 42:281–300, 1987.

    Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    J. Jakubowski and M. Wiśniewolski, On matching diffusions, Laplace transforms and partial differential equations, Stochastic Processes Appl., 125:3663–3690, 2015.

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    M. Jeanblanc, M. Yor, and M. Chesney, Mathematical Methods for Financial Markets, Springer-Verlag, London, 2009.

    Book  MATH  Google Scholar 

  21. 21.

    I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1998.

    Book  MATH  Google Scholar 

  22. 22.

    E. Levy, Pricing European average rate currency options, J. Int. Money Finance, 11:474–491, 1992.

    MathSciNet  Article  Google Scholar 

  23. 23.

    G.D. Lin, Characterizations of distributions via moments, Sankhyā, Ser. A, 54:128–132, 1992.

    MathSciNet  MATH  Google Scholar 

  24. 24.

    R. Mansuy and M. Yor, Aspects of Brownian Motion, Springer-Verlag, Berlin, Heidelberg, 2008.

    Book  MATH  Google 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.

  27. 27.

    R. Rebonato, Volatility and Correlation: The Perfect Hedger and the Fox, 2nd ed., John Wiley & Sons, 2004.

  28. 28.

    D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Springer-Verlag, Berlin, Heidelberg, 2005.

    MATH  Google Scholar 

  29. 29.

    I. SenGupta, Pricing Asian options in financial markets using Mellin transform, Electron. J. Differ. Equ., 2014(234):1–9, 2014.

    MathSciNet  MATH  Google Scholar 

  30. 30.

    I. SenGupta, Generalized BN-S stochastic volatility model for option pricing, Int. J. Theor. Appl. Finance, 19(02):1650014, 2016.

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    R. Shöbel and J. Zhu, Stochastic volatility with an Ornstein–Uhlenbeck process: An extension, Eur. Finance Rev., 3:23–46, 1999.

    Article  MATH  Google Scholar 

  32. 32.

    C. Sin, Complications with stochastic volatility models, Adv. Appl. Probab., 30:256–268, 1998.

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    E. Stein and J. Stein, Stock price distributions with stochastic volatility: An analytic approach, Rev. Financ. Stud., 4:727–752, 1991.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jacek Jakubowski.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jakubowski, J., Michalik, Z. & Wiśniewolski, M. Moments and Mellin transform of the asset price in Stein and Stein model and option pricing. Lith Math J 58, 33–47 (2018). https://doi.org/10.1007/s10986-018-9380-9

Download citation

MSC

  • 60J70
  • 91G80
  • 60H30

Keywords

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