Journal of Applied Mathematics and Stochastic Analysis
Volume 2008 (2008), Article ID 359142, 17 pages
doi:10.1155/2008/359142
Research Article

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

Elisa Alòs,1 Jorge A. León,2 Monique Pontier,3 and Josep Vives4

1Departament d'Economia i Empresa, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27, 08005 Barcelona, Spain
2Departamento de Control Automático, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (CINVESTAV-IPN), Apartado Postal 14-740, CP 07000 México D.F., Mexico
3Institut Mathématique de Toulouse, Université de Toulouse, 31062 Toulouse cedex 9, France
4Departament de Probabilitat, Lògica i Estadística, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain

Received 1 April 2008; Revised 3 September 2008; Accepted 25 November 2008

Academic Editor: Wenbo Li

Copyright © 2008 Elisa Alòs et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. L. Lewis, Option Valuation under Stochastic Volatility: With Mathematica Code, Finance Press, Newport Beach, Calif, USA, 2000. View at Zentralblatt MATH · View at MathSciNet
  2. R. W. Lee, “Implied volatility: statics, dynamics, and probabilistic interpretation,” in Recent Advances in Applied Probability, pp. 241–268, Springer, New York, NY, USA, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. Medvedev and O. Scaillet, “A simple calibration procedure of stochastic volatility models with jumps by short term asymptotics,” Discussion Paper HEC, Genève and FAME, Université de Genève, Genève, Switzerland, 2004.
  4. D. Bates, “Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options,” The Review of Financial Studies, vol. 9, no. 1, pp. 69–107, 1996. View at Publisher · View at Google Scholar
  5. O. E. Barndorff-Nielsen and N. Shephard, “Modelling by Lévy processes for financial econometrics,” in Lévy Processes: Theory and Applications, pp. 283–318, Birkhäuser, Boston, Mass, USA, 2001. View at Zentralblatt MATH · View at MathSciNet
  6. O. E. Barndorff-Nielsen and N. Shephard, “Econometric analysis of realized volatility and its use in estimating stochastic volatility models,” Journal of the Royal Statistical Society. Series B, vol. 64, no. 2, pp. 253–280, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. P. Carr and L. Wu, “The finite moment log stable process and option pricing,” The Journal of Finance, vol. 58, no. 2, pp. 753–778, 2003. View at Publisher · View at Google Scholar
  8. E. Alòs, J. A. León, and J. Vives, “On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility,” Finance and Stochastics, vol. 11, no. 4, pp. 571–589, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. E. Alòs, “A generalization of the Hull and White formula with applications to option pricing approximation,” Finance and Stochastics, vol. 10, no. 3, pp. 353–365, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000. View at Zentralblatt MATH · View at MathSciNet
  11. G. Bakshi, C. Cao, and Z. Chen, “Empirical performance of alternative option pricing models,” The Journal of Finance, vol. 52, no. 5, pp. 2003–2049, 1997. View at Publisher · View at Google Scholar
  12. D. Duffie, J. Pan, and K. Singleton, “Transform analysis and asset pricing for affine jump-diffusions,” Econometrica, vol. 68, no. 6, pp. 1343–1376, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. L. Solé, F. Utzet, and J. Vives, “Canonical Lévy process and Malliavin calculus,” Stochastic Processes and Their Applications, vol. 117, no. 2, pp. 165–187, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. A. Løkka, “Martingale representation of functionals of Lévy processes,” Stochastic Analysis and Applications, vol. 22, no. 4, pp. 867–892, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. E. Petrou, “Malliavin calculus in Lévy spaces and applications to finance,” Electronic Journal of Probability, vol. 13, no. 27, pp. 852–879, 2008. View at MathSciNet
  16. J. Gatheral, The Volatility Surface: A Practitioner's Guide, John Wiley & Sons, New York, NY, USA, 2006.
  17. K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, vol. 68 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1999. View at Zentralblatt MATH · View at MathSciNet
  18. D. Nualart and W. Schoutens, “Chaotic and predictable representations for Lévy processes,” Stochastic Processes and Their Applications, vol. 90, no. 1, pp. 109–122, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. K. Itô, “Spectral type of the shift transformation of differential processes with stationary increments,” Transactions of the American Mathematical Society, vol. 81, pp. 253–263, 1956. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. D. Nualart, The Malliavin Calculus and Related Topics, Probability and Its Applications, Springer, New York, NY, USA, 1995. View at Zentralblatt MATH · View at MathSciNet
  21. E. Alòs, J. A. León, and J. Vives, “An anticipating Itô formula for Lévy processes,” ALEA: Latin American Journal of Probability and Mathematical Statistics, vol. 4, pp. 285–305, 2008.
  22. G. Di Nunno, T. Meyer-Brandis, B. Øksendal, and F. Proske, “Malliavin calculus and anticipative Itô formulae for Lévy processes,” Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 8, no. 2, pp. 235–258, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. A. Álvarez, Modélisation de séries financières, estimation, ajustement de modèles et test d'hypothèses, M.S. thesis, Université Paul Sabatier Toulouse III, Toulouse, France, 2007.
  24. F. Espinosa and J. Vives, “A volatility-varying and jump-diffusion Merton type model of interest rate risk,” Insurance: Mathematics & Economics, vol. 38, no. 1, pp. 157–166, 2006. View at Zentralblatt MATH · View at MathSciNet