Skip to main content
SpringerLink
Log in

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

  • Published:
Finance and Stochastics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alòs, E.: A generalization of the Hull and White formula with applications to option pricing approximation. Finance Stoch. 10, 353–365 (2006)

    Article  MATH  Google Scholar 

  2. Alòs, E., Nualart, D.: An extension of Itô’s formula for anticipating processes. J. Theor. Probab. 11, 493–514 (1998)

    Article  MATH  Google Scholar 

  3. Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29, 766–801 (2001)

    Article  MATH  Google Scholar 

  4. Bakshi, G., Cao, C., Chen, Z.: Empirical performance of alternative option pricing models. J. Finance 52, 2003–2049 (1997)

    Article  Google Scholar 

  5. Ball, C., Roma, A.: Stochastic volatility option pricing. J. Finance Quant. Anal. 29, 589–607 (1994)

    Article  Google Scholar 

  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. 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)

    Article  MATH  Google Scholar 

  8. Bates, D.S.: Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options. Rev. Finance Stud. 9, 69–107 (1996)

    Article  Google Scholar 

  9. Carr, P., Wu, L.: The finite moment log stable process and option pricing. J. Finance 58, 753–778 (2003)

    Article  Google Scholar 

  10. Comte, F., Renault, E.: Long memory in continuous-time stochastic volatility models. Math. Finance 8, 291–323 (1998)

    Article  MATH  Google Scholar 

  11. Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000)

    Article  MATH  Google Scholar 

  12. Fouque, J.-P., Papanicolaou, G., Sircar, R., Solna, K.: Singular perturbations in option pricing. SIAM J. Appl. Math. 63, 1648–1665 (2003)

    Article  MATH  Google Scholar 

  13. Fouque, J.-P., Papanicolaou, G., Sircar, R., Solna, K.: Maturity cycles in implied volatility. Finance Stoch. 8, 451–477 (2004)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  Google Scholar 

  16. Jacod, J., Protter, P.: Risk neutral compatibility with option prices. Preprint (2006). http://legacy.orie.cornell.edu/~protter/finance.html

  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. Lewis, A.L.: Option Valuation Under Stochastic Volatility with Mathematica Code. Finance Press, Newport Beach (2000)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  20. Nualart, D.: The Malliavin Calculus and Related Topics. Springer, Berlin (1995)

    MATH  Google Scholar 

  21. Renault, E., Touzi, N.: Option hedging and implied volatilities in a stochastic volatility model. Math. Finance 6, 279–302 (1996)

    Article  MATH  Google Scholar 

  22. Schweizer, M., Wissel, J.: Term structures of implied volatilities: absence of arbitrage and existence results. Math. Finance (2006, to appear)

  23. Scott, L.O.: Option pricing when the variance changes randomly: theory, estimation and an application. J. Finance Quant. Anal. 22, 419–438 (1987)

    Article  Google Scholar 

  24. Stein, E.M., Stein, J.C.: Stock price distributions with stochastic volatility: an analytic approach. Rev. Finance Stud. 4, 727–752 (1991)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dpt. d’Economia i Empresa, Universitat Pompeu Fabra, c/Ramon Trias Fargas, 25–27, 08005, Barcelona, Spain

    Elisa Alòs

  2. Control Automático, CINVESTAV-IPN, Apartado Postal 14-740, 07000, México, D.F., Mexico

    Jorge A. León

  3. Dpt. de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain

    Josep Vives

  4. Dpt. Probabilitat, Lògica i Estadística, Universitat de Barcelona, Gran Via, 585, 08007, Barcelona, Spain

    Josep Vives

Authors
  1. Elisa Alòs
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jorge A. León
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Josep Vives
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elisa Alòs.

Additional information

E. Alòs’ research is supported by grants MEC FEDER MTM 2006 06427 and SEJ2006-13537.

J.A. León’s research is partially supported by the CONACyT grant 45684-F.

J. Vives’ research is supported by grant MEC FEDER MTM 2006 06427.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alòs, E., León, J.A. & Vives, J. On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch 11, 571–589 (2007). https://doi.org/10.1007/s00780-007-0049-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00780-007-0049-1

Keywords

JEL

Mathematics Subject Classification (2000)

Search

Navigation