Advertisement

Decomposition of the Pricing Formula for Stochastic Volatility Models Based on Malliavin-Skorohod Type Calculus

  • Josep VivesEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 158)

Abstract

The goal of this survey article is to present in detail a method that, for a financial derivative under a certain stochastic volatility model, allows to obtain a decomposition of its pricing formula that distinguishes clearly the impact of correlation and jumps. This decomposed pricing formula, usually called Hull and White type formula, can be potentially useful for model selection and calibration. The method is based on the obtention of an ad-hoc anticipating Itô formula.

Keywords

Hull and White type formula Malliavin-Skorohod calculus Stochastic volatility jump-diffusion models Derivative pricing Quantitative finance 

Mathematical Subject Classification

60H07 60H30 91G80 91G20 

Notes

Acknowledgments

This paper was partially written during a four month stage in Center for Advanced Study (CAS) at the Norwegian Academy of Science and Letters. I thank CAS for the support and the kind hospitality.

References

  1. 1.
    Alòs, E.: A generalization of the Hull and White formula with applications to option pricing approximation. Finance Stochast. 10(3), 353–365 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alòs, E., Nualart, D.: An extension of Itô formula for anticipating processes. J. Theor. Probab. 11(2), 493–514 (1998)CrossRefzbMATHGoogle Scholar
  3. 3.
    Alòs, E., León, J.A., Vives, J.: On the short-time behavior of the implied volatility for jump diffusion models with stochastic volatility. Finance Stochast. 11(4), 571–589 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alòs, E., León, J., Vives, J.: An anticipating Itô formula for Lévy processes. Lat. Am. J. Probab. Math. Stat. 4, 285–305 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Alòs, E., León, J.A., Pontier, M., Vives, J.: A Hull and White formula for a general stochastic volatility jump diffusion model with applications to the study of the short time behavior of the implied volatility. J. Appl. Math. Stoch. Anal. ID 359142, 17 (2008)Google Scholar
  6. 6.
    Cont, R., Tankov, P.: Financial modelling with jump processes. Chapman-Hall/CRC (2004)Google Scholar
  7. 7.
    Fouque, J.P., Papanicolaou, G., Sircar, R.: Derivatives in financial markets with stochastic volatility. Cambridge (2000)Google Scholar
  8. 8.
    Gulisashvili, A.: Analytically Tractable Stochastic Stock Price Models. Springer (2012)Google Scholar
  9. 9.
    Jafari, H., Vives, J.: A Hull and White formula for a stochastic volatility Lévy model with infinite activity. Commun. Stochast. Anal. 7(2), 321–336 (2013)Google Scholar
  10. 10.
    Nualart, D.: The Malliavin Calculus and Related Topics. Springer (2006)Google Scholar
  11. 11.
    Solé, J.L., Utzet, F., Vives, J.: Canonical Lévy process and Malliavin calculus. Stochast. Processes Appl. 117(2), 165–187 (2007)CrossRefzbMATHGoogle Scholar
  12. 12.
    Vives, J.: Malliavin calculus for Lévy processes: a survey. Proceedings of the 8th Conference of the ISAAC-2011. Rendiconti del Seminario Matematico di Torino, 71 (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Institute of Mathematics of the University of Barcelona (IMUB)University of BarcelonaBarcelonaSpain

Personalised recommendations