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Finance and Stochastics

, Volume 11, Issue 4, pp 571–589 | Cite as

On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility

  • Elisa Alòs
  • Jorge A. León
  • Josep Vives
Article

Abstract

In this paper we use Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be a diffusion or a Markov process, as the examples in Sect. 7 show. This expression depends on the derivative of the volatility in the sense of Malliavin calculus.

Keywords

Black-Scholes formula Derivative operator Itô’s formula for the Skorohod integral Jump-diffusion stochastic volatility model 

JEL

G12 G13 

Mathematics Subject Classification (2000)

91B28 91B70 60H07 

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References

  1. 1.
    Alòs, E.: A generalization of the Hull and White formula with applications to option pricing approximation. Finance Stoch. 10, 353–365 (2006) zbMATHCrossRefGoogle Scholar
  2. 2.
    Alòs, E., Nualart, D.: An extension of Itô’s formula for anticipating processes. J. Theor. Probab. 11, 493–514 (1998) zbMATHCrossRefGoogle Scholar
  3. 3.
    Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29, 766–801 (2001) zbMATHCrossRefGoogle Scholar
  4. 4.
    Bakshi, G., Cao, C., Chen, Z.: Empirical performance of alternative option pricing models. J. Finance 52, 2003–2049 (1997) CrossRefGoogle Scholar
  5. 5.
    Ball, C., Roma, A.: Stochastic volatility option pricing. J. Finance Quant. Anal. 29, 589–607 (1994) CrossRefGoogle Scholar
  6. 6.
    Barndorff-Nielsen, O.E., Shephard, N.: Modelling by Lévy processes for financial econometrics. In: Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S.I. (eds.) Lévy Processes: Theory and Applications, pp. 283–318. Birkhäuser, Basel (2001) Google Scholar
  7. 7.
    Barndorff-Nielsen, O.E., Shephard, N.: Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. Roy. Stat. Soc. Ser. B Stat. Methodol. 64, 253–280 (2002) zbMATHCrossRefGoogle Scholar
  8. 8.
    Bates, D.S.: Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options. Rev. Finance Stud. 9, 69–107 (1996) CrossRefGoogle Scholar
  9. 9.
    Carr, P., Wu, L.: The finite moment log stable process and option pricing. J. Finance 58, 753–778 (2003) CrossRefGoogle Scholar
  10. 10.
    Comte, F., Renault, E.: Long memory in continuous-time stochastic volatility models. Math. Finance 8, 291–323 (1998) zbMATHCrossRefGoogle Scholar
  11. 11.
    Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000) zbMATHCrossRefGoogle Scholar
  12. 12.
    Fouque, J.-P., Papanicolaou, G., Sircar, R., Solna, K.: Singular perturbations in option pricing. SIAM J. Appl. Math. 63, 1648–1665 (2003) zbMATHCrossRefGoogle Scholar
  13. 13.
    Fouque, J.-P., Papanicolaou, G., Sircar, R., Solna, K.: Maturity cycles in implied volatility. Finance Stoch. 8, 451–477 (2004) zbMATHCrossRefGoogle Scholar
  14. 14.
    Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Finance Stud. 6, 327–343 (1993) CrossRefGoogle Scholar
  15. 15.
    Hull, J., White, A.: The pricing of options on assets with stochastic volatilities. J. Finance 42, 281–300 (1987) CrossRefGoogle Scholar
  16. 16.
    Jacod, J., Protter, P.: Risk neutral compatibility with option prices. Preprint (2006). http://legacy.orie.cornell.edu/~protter/finance.html
  17. 17.
    Lee, R.W.: Implied volatility: statics, dynamics, and probabilistic interpretation. In: Baeza-Yates, R., Glaz, J., Gzyl, H., et al. (eds.) Recent Advances in Applied Probability, pp. 241–268. Springer, Berlin (2004) Google Scholar
  18. 18.
    Lewis, A.L.: Option Valuation Under Stochastic Volatility with Mathematica Code. Finance Press, Newport Beach (2000) zbMATHGoogle Scholar
  19. 19.
    Medvedev, A., Scaillet, O.: Approximation and calibration of short-term implied volatilities under jump-diffusion stochastic volatility. Rev. Finance Stud. 20(2), 427–459 (2007) CrossRefGoogle Scholar
  20. 20.
    Nualart, D.: The Malliavin Calculus and Related Topics. Springer, Berlin (1995) zbMATHGoogle Scholar
  21. 21.
    Renault, E., Touzi, N.: Option hedging and implied volatilities in a stochastic volatility model. Math. Finance 6, 279–302 (1996) zbMATHCrossRefGoogle Scholar
  22. 22.
    Schweizer, M., Wissel, J.: Term structures of implied volatilities: absence of arbitrage and existence results. Math. Finance (2006, to appear) Google Scholar
  23. 23.
    Scott, L.O.: Option pricing when the variance changes randomly: theory, estimation and an application. J. Finance Quant. Anal. 22, 419–438 (1987) CrossRefGoogle Scholar
  24. 24.
    Stein, E.M., Stein, J.C.: Stock price distributions with stochastic volatility: an analytic approach. Rev. Finance Stud. 4, 727–752 (1991) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Dpt. d’Economia i EmpresaUniversitat Pompeu FabraBarcelonaSpain
  2. 2.Control AutomáticoCINVESTAV-IPNMéxicoMexico
  3. 3.Dpt. de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra (Barcelona)Spain
  4. 4.Dpt. Probabilitat, Lògica i EstadísticaUniversitat de BarcelonaBarcelonaSpain

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