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Practical Applications and Testing

  • David NicolayEmail author
Chapter
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Part of the Springer Finance book series (FINANCE)

Abstract

In this chapter we turn to even more practical considerations, by applying ACE results to some popular stochastic (instantaneous) volatility models, namely the SABR and FL-SV classes. We start by discussing the financial, practical and numerical issues involved. We then derive the chaos dynamics of each model, up to the third layer, stressing the technical benefits of staying model-generic and of exploiting induction. We can then express the desired static IATM differentials, which we subsequently use in either direct or inverse mode. In inverse fashion, we use those quantities to illustrate an “intuitive” model re-parametrisation of the generic SABR class. In direct mode, we test the flexibility and quality of static smile approximations provided by ACE for the CEV-SABR model, compared to Hagan et al’s benchmark.

Keywords

Stochastic Volatility Stochastic Volatility Model Central Configuration Chaos Dynamic Negative Density 
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. 1.
    Hagan, P.S., Kumar, D., Lesniewski, A.S., Woodward, D.E.: Managing smile risk. Wilmott Mag. 1, 84–108 (2002)Google Scholar
  2. 2.
    Osajima, Y.: The asymptotic expansion formula of implied volatility for dynamic SABR model and FX hybrid model. University of Tokyo, Graduate school of mathematical sciences, Report (2006)Google Scholar
  3. 3.
    Fouque, J.-P., Papanicolau, G., Sircar, K.R.: Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  4. 4.
    Fournie, E., Lebuchoux, J., Touzi, N.: Small noise expansion and importance sampling. Asymptotic Anal. 14(4), 361–376 (1997)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Beresticki, H., Busca, J., Florent, I.: Asymptotics and calibration of local volatility models. Quant. Financ. 2, 61–69 (2002)CrossRefGoogle Scholar
  6. 6.
    Henry-Labordere, P.: A general asymptotic implied volatility for stochastic volatility models. Technical report, Barclays Capital (Apr 2005)Google Scholar
  7. 7.
    Durrleman, V.: From implied to spot volatilities. Finance Stochast. 14(2), 157–177 (2006) (Springer, Cambridge)Google Scholar
  8. 8.
    Liu, E.H.L.: Fundamental Methods of Numerical Extrapolation with Applications. Mitopencourseware. Massachusetts Institute of Technology, Cambridge (2006)Google Scholar
  9. 9.
    Lee, R.W.: The moment formula for implied volatility at extreme strikes. Math. Financ. 14(3), 469–480 (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    Benaim, S., Friz, P.: Regular variation and smile asymptotics. Math. Financ. 19, 1–12 (2009)Google Scholar
  11. 11.
    Dragulescu, A.A., Yakovenko, V.M.: Probability distribution of returns in the Heston model with stochastic volatility. Quant. Financ. 2, 443–453 (2002)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Fouque, J.-P., Papanicolau, G., Sircar, K.R.: Financial modeling in a fast mean-reverting stochastic volatility environment. Asia-Pacific Financ. Markets 6, 37–48 (1999)CrossRefzbMATHGoogle Scholar
  13. 13.
    Fouque, J.-P., Papanicolau, G., Sircar, K.R.: Mean-reverting stochastic volatility. SIAM J. Control Optim. 31, 470–493 (2000)Google Scholar
  14. 14.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C, 2nd edn. Cambridge University Press, New York (1992)zbMATHGoogle Scholar
  15. 15.
    Andersen, L.B.G., Piterbarg, V.V.: Moment explosions in stochastic volatility models. Financ. Stochast. 11, 29–50 (2006)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Benhamou, E., Croissant, O.: Local time for the SABR model. Connection with the complex Black-Scholes and application to CMS and spread options. Working paper, IXIS CIB, Sep (2007)Google Scholar
  17. 17.
    Obloj, J.: Fine-tune tour smile. Correction to Hagan & al. Technical report,Imperial College, London (Mar 2008)Google Scholar
  18. 18.
    Beresticki, H., Busca, J., Florent, I.: Computing the implied volatility in stochastic volatility models. Commun. Pure Appl. Math. LVII, 1352–1373 (2004)CrossRefGoogle Scholar
  19. 19.
    Andersen, L., Brotherton-Ratcliffe, R.: Extended libor market models with stochastic volatility. Technical report, Bank of America (2001)Google Scholar
  20. 20.
    Andersen, L., Andreasen, J.: Volatile volatilities. Risk Mag. 15, 163–168 (2002)Google Scholar
  21. 21.
    Andersen, L., Andreasen, J.: Volatility skews and extension of the Libor market model. Appl. Math. Financ. 7, 1–32 (2000)CrossRefzbMATHGoogle Scholar
  22. 22.
    Piterbarg, V.V.: Stochastic volatility model with time-dependent skew. Appl. Math. Financ. 12, 147–185 (2005)Google Scholar
  23. 23.
    Piterbarg, V.V.: Markovian projection for volatility calibration. Risk Mag. 20, 84–89 (2007)Google Scholar
  24. 24.
    Andreasen, J.: Back to the future. Risk Mag. 18, 104–109 (2005)Google Scholar
  25. 25.
    Hagan, P.S., Woodward, D.E.: Equivalent black volatilities. Appl. Math. Financ. 6, 147–157 (1999)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.LondonUK

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