Table of Contents Author Guidelines Submit a Manuscript
Journal of Probability and Statistics
Volume 2017, Article ID 8690491, 19 pages
https://doi.org/10.1155/2017/8690491
Research Article

Gram-Charlier Processes and Applications to Option Pricing

1Faculty of Business Administration, University of Macau, Macau
2Montreal, QC, Canada

Correspondence should be addressed to Daniel Dufresne; ua.ude.bleminu@enserfud

Received 17 August 2016; Accepted 1 November 2016; Published 8 February 2017

Academic Editor: Aera Thavaneswaran

Copyright © 2017 Jean-Pierre Chateau and Daniel Dufresne. 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. P. A. Abken, D. B. Madan, and S. Ramamurtie, “Estimation of risk-neutral and statistical densities by Hermite polynomial approximation,” Working Paper 96-5, Federal Reserve Bank of Atlanta, 1996. View at Google Scholar
  2. D. Backus, S. Foresi, K. Li, and L. Wu, “Accounting for biases in Black-Scholes,” Working Paper, Stern School of Business, New York University, New York, NY, USA, 1997. View at Google Scholar
  3. J.-P. D. Chateau, “Marking-to-model credit and operational risks of loan commitments: a Basel-2 advanced internal ratings-based approach,” International Review of Financial Analysis, vol. 18, no. 5, pp. 260–270, 2009. View at Publisher · View at Google Scholar · View at Scopus
  4. J. D. Chateau, “Valuing european put options under skewness and increasing (excess) kurtosis,” Journal of Mathematical Finance, vol. 4, no. 3, pp. 160–177, 2014. View at Publisher · View at Google Scholar
  5. C. J. Corrado, “The hidden martingale restriction in Gram-Charlier option prices,” Journal of Futures Markets, vol. 27, no. 6, pp. 517–534, 2007. View at Publisher · View at Google Scholar · View at Scopus
  6. E. Jondeau and M. Rockinger, “Gram-Charlier densities,” Journal of Economic Dynamics and Control, vol. 25, no. 10, pp. 1457–1483, 2001. View at Publisher · View at Google Scholar · View at Scopus
  7. E. Jurczenko, B. Maillet, and B. Négrea, “Skewness and kurtosis implied by option prices: a second comment,” Discussion Paper of the LSE-FMG 419, 2002. View at Google Scholar
  8. E. Jurczenko, B. Maillet, and B. Negrea, “A note on skewness and kurtosis adjusted option pricing models under the Martingale restriction,” Quantitative Finance, vol. 4, no. 5, pp. 479–488, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. J. L. Knight and S. E. Satchell, Return Distributions in Finance, Butterworth Heineman, 2001.
  10. E. Schlögl, “Option pricing where the underlying assets follow a Gram/Charlier density of arbitrary order,” Journal of Economic Dynamics & Control, vol. 37, no. 3, pp. 611–632, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. A. León, J. Mencía, and E. Sentana, “Parametric properties of semi-nonparametric distributions, with applications to option valuation,” Journal of Business & Economic Statistics, vol. 27, no. 2, pp. 176–192, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. M. Hardy, Investment Guarantees: Modeling and Risk Management for Equity-Linked Life Insurance, John Wiley & Sons, New York, NY, USA, 2003.
  13. P. Gaillardetz and X. S. Lin, “Valuation of equity-linked insurance and annuity products with binomial models,” North American Actuarial Journal, vol. 10, no. 4, pp. 117–144, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. P. Boyle and W. Tian, “The design of equity-indexed annuities,” Insurance: Mathematics & Economics, vol. 43, no. 3, pp. 303–315, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. S. G. Kou, “Discrete barrier and lookback options,” in Handbooks in Operations Research and Management Science, vol. 15, chapter 8, pp. 343–373, 2007. View at Publisher · View at Google Scholar
  16. D. E. Barton and K. E. Dennis, “The conditions under which Gram-Charlier and Edgeworth curves are positive definite and unimodal,” Biometrika, vol. 39, no. 3-4, pp. 425–427, 1952. View at Publisher · View at Google Scholar · View at MathSciNet
  17. H. Cramér, On Some Classes of Series Used in Mathematical Statistics, Skandinaviske Matematiker Congres, Copenhagen, Denmark, 1925.
  18. W. Feller, An Introduction to Probability Theory and Its Applications, vol. 2, John Wiley & Sons, New York, NY, USA, 1971. View at MathSciNet
  19. J. Stoyanov, “Krein condition in probabilistic moment problems,” Bernoulli, vol. 6, no. 5, pp. 939–949, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  20. F. D. Rouah and G. Vainberg, Option Pricing Models and Volatility Using Excel-VBA, John Wiley & Sons, New York, NY, USA, 2007.
  21. L. S. Rompolis and E. Tzavalis, “Retrieving risk neutral densities based on risk neutral moments through a Gram-Charlier series expansion,” Mathematical and Computer Modelling, vol. 46, no. 1-2, pp. 225–234, 2007. View at Publisher · View at Google Scholar · View at Scopus
  22. E. B. Del Brio and J. Perote, “Gram-Charlier densities: maximum likelihood versus the method of moments,” Insurance: Mathematics and Economics, vol. 51, no. 3, pp. 531–537, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. X. S. Lin and K. S. Tan, “Valuation of equity-indexed annuities under stochastic interest rates,” North American Actuarial Journal, vol. 7, no. 4, pp. 72–91, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  24. S. R. Das and R. K. Sundaram, “Of smiles and smirks: a term structure perspective,” Journal of Financial and Quantitative Analysis, vol. 34, no. 2, pp. 211–239, 1999. View at Publisher · View at Google Scholar · View at Scopus
  25. J. E. Kolossa, Series Approximation Methods in Statistics, Springer, New York, NY, USA, 3rd edition, 2006.
  26. H. Cramér, Mathematical Methods of Statistics, Princeton University, Princeton, NJ, USA, 1946. View at MathSciNet
  27. E. B. Del Brio, T.-M. Ñíguez, and J. Perote, “Gram-Charlier densities: a multivariate approach,” Quantitative Finance, vol. 9, no. 7, pp. 855–868, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. E. B. Del Brio, T.-M. Ñíguez, and J. Perote, “Multivariate semi-nonparametric distributions with dynamic conditional correlations,” International Journal of Forecasting, vol. 27, no. 2, pp. 347–364, 2011. View at Publisher · View at Google Scholar · View at Scopus
  29. D. Dufresne, “Laguerre series for Asian and other options,” Mathematical Finance, vol. 10, no. 4, pp. 407–428, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  30. N. N. Lebedev, Special Functions and Their Applications, Dover, New York, NY, USA, 1972. View at MathSciNet
  31. D. Dufresne, “Fitting combinations of exponentials to probability distributions,” Applied Stochastic Models in Business and Industry, vol. 23, no. 1, pp. 23–48, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus