Pricing European Options Under Stochastic Volatilities Models

  • Betuel CanhangaEmail author
  • Anatoliy Malyarenko
  • Jean-Paul Murara
  • Sergei Silvestrov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 178)


Interested by the volatility behavior, different models have been developed for option pricing. Starting from constant volatility model which did not succeed on capturing the effects of volatility smiles and skews; stochastic volatility models appear as a response to the weakness of the constant volatility models. Constant elasticity of volatility, Heston, Hull and White, Schöbel–Zhu, Schöbel–Zhu–Hull–White and many others are examples of models where the volatility is itself a random process. Along the chapter we deal with this class of models and we present the techniques of pricing European options. Comparing single factor stochastic volatility models to constant factor volatility models it seems evident that the stochastic volatility models represent nicely the movement of the asset price and its relations with changes in the risk. However, these models fail to explain the large independent fluctuations in the volatility levels and slope. Christoffersen et al. (Manag Sci 22(12):1914–1932, 2009, [4]) proposed a model with two-factor stochastic volatilities where the correlation between the underlying asset price and the volatilities varies randomly. In the last section of this chapter we introduce a variation of Chiarella and Ziveyi model, which is a subclass of the model presented in [4] and we use the first order asymptotic expansion methods to determine the price of European options.


Financial markets Option pricing Stochastic volatilities Asymptotic expansion 



This work was partially supported by Swedish SIDA Foundation International Science Program. Betuel Canhanga and Jean-Paul Murara thanks Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University for creating excellent research and educational environment.


  1. 1.
    Andersen, L., Piterbarg, V.: Interest Rate Modeling. Volume 1: Foundations and Vanilla Models. Atlantic Financial Press, Boston (2010)Google Scholar
  2. 2.
    Carr, P.P., Madan, D.B.: Option valuation using fast fourier transform. J. Comput. Financ. 2, 61–73 (1999)CrossRefGoogle Scholar
  3. 3.
    Chiarella, C., Ziveyi, J.: American option pricing under two stochastic volatility processes. J. Appl. Math. Comput. 224, 283–310 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Christoffersen, P., Heston, S., Jacobs, K.: The shape and term structure of index option smirk: why multifactor stochastic volatilities models work so well. Manag. Sci. 22(12), 1914–1932 (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cox, J., Ross, S.: The valuation of options alternative stochastic processes. J. Financ. Econ. 3, 145–166 (1976)CrossRefGoogle Scholar
  6. 6.
    Detemple, J., Tian, W.: The valuation of American options for a class of diffusion processes. Manag. Sci. 48(7), 917–937 (2002)CrossRefzbMATHGoogle Scholar
  7. 7.
    Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fang, F., Oosterlee, C.W.: A Novel Option Pricing Method based on Fourier-Cosine Series Expansions. Technical Report 08-02, Delft University Technical, The Netherlands. (2008)
  9. 9.
    Feller, W.: Two singular diffusion problems. Ann. Math. 54, 173–182 (1951)Google Scholar
  10. 10.
    Grzelak, L., Oosterlee, C.W, Van, V.: Extension of stochastic volatility models with Hull–White interest rate process. Report 08-04, Delft University of technology, The Netherlands (2008)Google Scholar
  11. 11.
    Managing smile risk: Hegan, P.S., Kumar, D., Lesniewski, S.S., Woodward. D.E. Wilmott Mag. 1, 84–108 (2003)Google Scholar
  12. 12.
    Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bonds and currency options. Rev. Financ. Stud. 6, 327–343 (1993)CrossRefGoogle Scholar
  13. 13.
    Ilham, A., Sircar, R.: Optimal static-dynamic hedges for barrier options. Math. Financ. 16, 359–385 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jourdain, B., Sbai, M.: Higher order discretization schemes for stochastic volatility models. AeXiv e-prrint. (2011)
  15. 15.
    Kijima, M.: Stochastic Processes with Applications to Finance, 2nd edn. Capman and Hall/CRC, New York (2013)zbMATHGoogle Scholar
  16. 16.
    Musiela, M., Rutkowski, M.: Martingale Methods in Financial Modelling, 2nd edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  17. 17.
    Romano, M., Touzi, N.: Contingent claim and market completeness in a stochastic volatility model. Math. Financ. 7(4), 399–412 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schroder, M.: Computing the constant elasticity of variance option princing formula. J. Financ. 44, 211–219 (1989)CrossRefGoogle Scholar
  19. 19.
    Zhang, N.: Properties of the SABR model. Departement of Mathematics. Uppsala University. (2011)

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Betuel Canhanga
    • 1
    • 2
    Email author
  • Anatoliy Malyarenko
    • 2
  • Jean-Paul Murara
    • 2
    • 3
  • Sergei Silvestrov
    • 2
  1. 1.Faculty of Sciences, Department of Mathematics and Computer SciencesEduardo Mondlane UniversityMaputoMozambique
  2. 2.Division of Applied Mathematics, The School of EducationCulture and Communication, Mälardalen UniversityVästeråsSweden
  3. 3.Department of Applied Mathematics, School of Sciences, College of Science and TechnologyUniversity of RwandaKigaliRwanda

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