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Parameter estimates of Heston stochastic volatility model with MLE and consistent EKF algorithm

  • Ximei Wang
  • Xingkang He
  • Ying Bao
  • Yanlong Zhao
Research Paper
  • 74 Downloads

Abstract

Heston model is the most famous stochastic volatility model in finance. This paper considers the parameter estimation problem of Heston model with both known and unknown volatilities. First, parameters in equity process and volatility process of Heston model are estimated separately since there is no explicit solution for the likelihood function with all parameters. Second, the normal maximum likelihood estimation (NMLE) algorithm is proposed based on the Itô transformation of Heston model. The algorithm can reduce the estimate error compared with existing pseudo maximum likelihood estimation. Third, the NMLE algorithm and consistent extended Kalman filter (CEKF) algorithm are combined in the case of unknown volatilities. As an advantage, CEKF algorithm can apply an upper bound of the error covariance matrix to ensure the volatilities estimation errors to be well evaluated. Numerical simulations illustrate that the proposed NMLE algorithm works more efficiently than the existing pseudo MLE algorithm with known and unknown volatilities. Therefore, the upper bound of the error covariance is illustrated. Additionally, the proposed estimation method is applied to American stock market index S&P 500, and the result shows the utility and effectiveness of the NMLE-CEKF algorithm.

Keywords

Heston model stochastic volatility model parameter estimation normal maximum likelihood estimation pseudo maximum likelihood estimation consistent extended Kalman filter 

References

  1. 1.
    Black F, Scholes M. The pricing of options and corporate liabilities. J Polit Econ, 1973, 81: 637–654MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Hull J. Options, futures, and other derivatives. In: Asset Pricing. Berlin: Springer, 2009. 9–26Google Scholar
  3. 3.
    Gallant A R, Tauchen G. Which moments to match? Economet Theory, 1996, 12: 657–681MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fama E F. The behavior of stock-market prices. J Bus, 1965, 38: 34–105CrossRefGoogle Scholar
  5. 5.
    Cont R. Empirical properties of asset returns: stylized facts and statistical issues. Quant Financ, 2001, 1: 223–236CrossRefGoogle Scholar
  6. 6.
    Engle R F, Ng V K. Measuring and testing the impact of news on volatility. J Financ, 1993, 48: 1749–1778CrossRefGoogle Scholar
  7. 7.
    Engle R F, Patton A J. What good is a volatility model. Quant Financ, 2001, 1: 237–245CrossRefGoogle Scholar
  8. 8.
    Hull J, White A. The pricing of options on assets with stochastic volatilities. J Financ, 1987, 42: 281–300CrossRefzbMATHGoogle Scholar
  9. 9.
    Wiggins J B. Option values under stochastic volatility: theory and empirical estimates. J Financ Econ, 1987, 19: 351–372CrossRefGoogle Scholar
  10. 10.
    Heston S L. A closed-form solution for options with stochastic volatilities with applications to bond and currency options. Rev Financ Stud, 1993, 6: 327–343CrossRefGoogle Scholar
  11. 11.
    Moodley N. The Heston model: a practical approach with matlab code. Johannesburg: University of the Witwatersrand, 2005Google Scholar
  12. 12.
    Dupire B. Pricing and hedging with smiles. In: Mathematics of Derivative Securities. Cambridge: Cambridge University Press, 1997. 103–111zbMATHGoogle Scholar
  13. 13.
    Weron R, Wystup U. Heston’s Model and the Smile. Berlin: Springer, 2005. 161–181Google Scholar
  14. 14.
    Tang C Y, Chen S X. Parameter estimation and bias correction for diffusion processes. J Economet, 2009, 149: 65–81MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ait-Sahalia Y, Kimmel R. Maximum likelihood estimation of stochastic volatility models. J Financ Econ, 2007, 83: 413–452CrossRefGoogle Scholar
  16. 16.
    Ren P. Parametric estimation of the Heston model under the indirect observability framework. Dissertation for Ph.D. Degree. Houston: University of Houston, 2014Google Scholar
  17. 17.
    Jacquier E, Polson N G, Rossi P E. Bayesian analysis of stochastic volatility models. J Bus Econ Stat, 1994, 20: 69–87MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nelson D B. The time series behavior of stock market volatility and returns. Dissertation for Ph.D. Degree. Cambridge: MIT, 1988Google Scholar
  19. 19.
    Javaheri A, Lautier D, Galli A. Filtering in finance. Wilmott, 2003, 3: 67–83CrossRefGoogle Scholar
  20. 20.
    Aihara S I, Bagchi A, Saha S. On parameter estimation of stochastic volatility models from stock data using particle filter-application to AEX index. Int J Innov Comput Inf Control, 2009, 5: 17–27Google Scholar
  21. 21.
    Li J. An unscented Kalman smoother for volatility extraction: evidence from stock prices and options. Comput Stat Data Anal, 2013, 58: 15–26MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pitt M K, Shephard N. Filtering via simulation: auxiliary particle filters. J Am Stat Assoc, 1999, 94: 590–599MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hu J, Wang Z D, Shen B, et al. Quantised recursive filtering for a class of nonlinear systems with multiplicative noises and missing measurements. Int J Control, 2013, 86: 650–663MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hu J, Wang Z D, Liu S, et al. A variance-constrained approach to recursive state estimation for time-varying complex networks with missing measurements. Automatica, 2016, 64: 155–162MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Jiang Y G, Xue W C, Huang Y, et al. The consistent extended Kalman filter. In: Proceedings of Chinese Control Conference (CCC), Chengdu, 2014. 6838–6843Google Scholar
  26. 26.
    He X K, Jing Y G, Xue W C, et al. Track correlation based on the quasi-consistent extended Kalman filter. In: Proceedings of the 34th Chinese Control Conference (CCC), Hangzhou, 2015. 2150–2155Google Scholar
  27. 27.
    Jiang Y G, Huang Y, Xue W C, et al. On designing consistent extended Kalman filter. J Syst Sci Complex, 2017, 30: 751–764MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Shreve S E. Stochastic Calculus for Finance II: Continuous-time Models. New York: Springer Science & Business Media, 2004zbMATHGoogle Scholar
  29. 29.
    Cox J C, Ingersoll J E, Ross S A. A theory of the term structure of interest rates. Econometrica, 1985, 53: 385–407MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nykvist J. Time consistency in option pricing models. Dissertation for Master Degree. Stockholm: Royal Institute of Technology, 2009Google Scholar
  31. 31.
    Hamilton J D. Time Series Analysis. Princeton: Princeton University Press, 1994zbMATHGoogle Scholar
  32. 32.
    Stein E M, Stein J C. Stock price distributions with stochastic volatility: an analytic approach. Rev Financ Stud, 1991, 4: 727–752CrossRefGoogle Scholar
  33. 33.
    Kladívko K. Maximum likelihood estimation of the Cox-Ingersoll-Ross process: the matlab implementation. Technical Computing Prague, 2007. http://www2.humusoft.cz/www/papers/tcp07/kladivko.pdfGoogle Scholar
  34. 34.
    Simon D. Optimal State Estimation: Kalman, H1, and Nonlinear Approaches. Hoboken: John Wiley & Sons Inc, 2006. 395–407CrossRefGoogle Scholar
  35. 35.
    Andersen L B G. Simple and efficient simulation of the Heston stochastic volatility model. J Comput Financ, 2008, 11Google Scholar
  36. 36.
    Lord R, Koekkoek R, Dijk D V. A comparison of biased simulation schemes for stochastic volatility models. Quant Financ, 2010, 10: 177–194MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Tang C Y, Chen S X. Parameter estimation and bias correction for diffusion processes. J Economet, 2009, 149: 65–81MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Exchange C B O. The CBOE volatility index-VIX. White Paper, 2009. www.cboe.com/micro/vix/vixwhite.pdfGoogle Scholar
  39. 39.
    Christensen B J, Prabhala N R. The relation between implied and realized volatility. J Financ Econ, 1998, 50: 125–150CrossRefGoogle Scholar
  40. 40.
    Zhu S H, Pykhtin M. A guide to modeling counterparty credit risk. Social Sci Elect Pub, 2008, 1: 16–22Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Ximei Wang
    • 1
    • 2
  • Xingkang He
    • 1
    • 2
  • Ying Bao
    • 3
  • Yanlong Zhao
    • 1
    • 2
  1. 1.Key Laboratory of System and ControlAcademy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.Risk Management DepartmentIndustrial and Commercial Bank of ChinaBeijingChina

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