Stochastic Volatility Models

  • Norbert Hilber
  • Oleg Reichmann
  • Christoph Schwab
  • Christoph Winter
Part of the Springer Finance book series (FINANCE)


In Sect.  4.5, we considered local volatility models as an extension of the Black–Scholes model. These models replace the constant volatility by a deterministic volatility function, i.e. the volatility is a deterministic function of s and t. In stochastic volatility (SV) models, the volatility is modeled as a function of at least one additional stochastic process. Such models can explain some of the empirical properties of asset returns, such as volatility clustering and the leverage effect. These models can also account for long term smiles and skews.


Brownian Motion Bilinear Form Option Price Stochastic Volatility American Option 
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  1. 2.
    Y. Achdou and N. Tchou. Variational analysis for the Black and Scholes equation with stochastic volatility. M2AN Math. Model. Numer. Anal., 36(3):373–395, 2002. MathSciNetzbMATHCrossRefGoogle Scholar
  2. 3.
    D. Applebaum. Lévy processes and stochastic calculus, volume 93 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2004. zbMATHCrossRefGoogle Scholar
  3. 6.
    C.A. Ball and A. Roma. Stochastic volatility option pricing. J. Financ. Quant. Anal., 29(4):589–607, 1994. CrossRefGoogle Scholar
  4. 10.
    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, 2001. MathSciNetzbMATHCrossRefGoogle Scholar
  5. 13.
    D.S. Bates. Jumps stochastic volatility: the exchange rate process implicit in Deutsche Mark options. Rev. Finance, 9(1):69–107, 1996. Google Scholar
  6. 14.
    D.S. Bates. Post-’87 crash fears in the S&P 500 futures option market. J. Econ., 94(1–2):181–238, 2000. MathSciNetzbMATHGoogle Scholar
  7. 16.
    F.E. Benth and M. Groth. The minimal entropy martingale measure and numerical option pricing for the Barndorff-Nielsen–Shephard stochastic volatility model. Stoch. Anal. Appl., 27(5):875–896, 2009. MathSciNetzbMATHCrossRefGoogle Scholar
  8. 37.
    P. Carr, H. Geman, D.B. Madan, and M. Yor. Stochastic volatility for Lévy processes. Math. Finance, 13(3):345–382, 2003. MathSciNetzbMATHCrossRefGoogle Scholar
  9. 67.
    J.-P. Fouque, G. Papanicolaou, R. Sircar, and K. Solna. Multiscale stochastic volatility asymptotics. Multiscale Model. Simul., 2(1):22–42, 2003. MathSciNetCrossRefGoogle Scholar
  10. 68.
    J.P. Fouque, G. Papanicolaou, and K.R. Sircar. Derivatives in financial markets with stochastic volatility. Cambridge University Press, Cambridge, 2000. zbMATHGoogle Scholar
  11. 79.
    S.L. Heston. A closed-form solution for options with stochastic volatility, with applications to bond and currency options. Rev. Finance, 6:327–343, 1993. Google Scholar
  12. 81.
    N. Hilber, A.M. Matache, and Ch. Schwab. Sparse wavelet methods for option pricing under stochastic volatility. J. Comput. Finance, 8(4):1–42, 2005. Google Scholar
  13. 90.
    S. Ikonen and J. Toivanen. Efficient numerical methods for pricing American options under stochastic volatility. Numer. Methods Partial Differ. Equ., 24(1):104–126, 2008. MathSciNetzbMATHCrossRefGoogle Scholar
  14. 145.
    R. Schöbel and J. Zhu. Stochastic volatility with an Ornstein–Uhlenbeck process: an extension. Eur. Finance Rev., 3:23–46, 1999. zbMATHCrossRefGoogle Scholar
  15. 150.
    L.O. Scott. Option pricing when the variance changes randomly: theory, estimation, and an application. J. Financ. Quant. Anal., 22:419–438, 1987. CrossRefGoogle Scholar
  16. 151.
    N. Shephard, editor. Stochastic volatility: selected readings. Oxford University Press, London, 2005. zbMATHGoogle Scholar
  17. 153.
    E.M. Stein and J.C. Stein. Stock price distributions with stochastic volatility: an analytic approach. Rev. Finance, 4(4):727–752, 1991. Google Scholar
  18. 166.
    R. Zvan, P.A. Forsyth, and K.R. Vetzal. Penalty methods for American options with stochastic volatility. J. Comput. Appl. Math., 91(2):199–218, 1998. MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Norbert Hilber
    • 1
  • Oleg Reichmann
    • 2
  • Christoph Schwab
    • 2
  • Christoph Winter
    • 3
  1. 1.Dept. for Banking, Finance, Insurance, School of Management and LawZurich University of Applied SciencesWinterthurSwitzerland
  2. 2.Seminar for Applied MathematicsSwiss Federal Institute of Technology (ETH)ZurichSwitzerland
  3. 3.Allianz Deutschland AGMunichGermany

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