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Stochastic Volatility and Jumps Driven by Continuous Time Markov Chains

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  • Kyriakos Chourdakis

    (Queen Mary, University of London)

Abstract

This paper considers a model where there is a single state variable that drives the state of the world and therefore the asset price behavior. This variable evolves according to a multi-state continuous time Markov chain, as the continuous time counterpart of the Hamilton (1989) model. It derives the moment generating function of the asset log-price difference under very general assumptions about its stochastic process, incorporating volatility and jumps that can follow virtually any distribution, both of them being driven by the same state variable. For an illustration, the extreme value distribution is used as the jump distribution. The paper shows how GMM and conditional ML estimators can be constructed, generalizing Hamilton's filter for the continuous time case. The risk neutral process is constructed and contigent claim prices under this specification are derived, in the lines of Bakshi and Madan (2000). Finally, an empirical example is set up, to illustrate the potential benefits of the model.

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File URL: http://www.econ.qmul.ac.uk/papers/doc/wp430.pdf
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Bibliographic Info

Paper provided by Queen Mary, University of London, School of Economics and Finance in its series Working Papers with number 430.

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Date of creation: Dec 2000
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Handle: RePEc:qmw:qmwecw:wp430

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Related research

Keywords: Option pricing; Markov chain; Moment generating function;

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References

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  1. Schwert, G William, 1990. "Stock Volatility and the Crash of '87," Review of Financial Studies, Society for Financial Studies, vol. 3(1), pages 77-102.
  2. Lo, Andrew W & Wang, Jiang, 1995. " Implementing Option Pricing Models When Asset Returns Are Predictable," Journal of Finance, American Finance Association, vol. 50(1), pages 87-129, March.
  3. David S. Bates, . "Testing Option Pricing Models," Rodney L. White Center for Financial Research Working Papers 14-95, Wharton School Rodney L. White Center for Financial Research.
  4. Allan Timmermann, 1999. "Moments of Markov Switching Models," FMG Discussion Papers dp323, Financial Markets Group.
  5. Charles Quanwei Cao & Gurdip S. Bakshi & Zhiwu Chen, 1997. "Empirical Performance of Alternative Option Pricing Models," Yale School of Management Working Papers ysm65, Yale School of Management.
  6. Stephen G. Cecchetti & Pok-sang Lam & Nelson C. Mark, 1990. "Mean Reversion in Equilibrium Asset Prices," NBER Working Papers 2762, National Bureau of Economic Research, Inc.
  7. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
  8. Bakshi, Gurdip & Madan, Dilip, 2000. "Spanning and derivative-security valuation," Journal of Financial Economics, Elsevier, vol. 55(2), pages 205-238, February.
  9. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. " Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, vol. 52(5), pages 2003-49, December.
  10. Charles Quanwei Cao & Gurdip S. Bakshi & Zhiwu Chen, 1997. "Empirical Performance of Alternative Option Pricing Models," Yale School of Management Working Papers ysm54, Yale School of Management.
  11. Brennan, M J, 1979. "The Pricing of Contingent Claims in Discrete Time Models," Journal of Finance, American Finance Association, vol. 34(1), pages 53-68, March.
  12. Jin-Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32.
  13. Naik, Vasanttilak, 1993. " Option Valuation and Hedging Strategies with Jumps in the Volatility of Asset Returns," Journal of Finance, American Finance Association, vol. 48(5), pages 1969-84, December.
  14. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
  15. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-84, March.
  16. David S. Bates, 1995. "Testing Option Pricing Models," NBER Working Papers 5129, National Bureau of Economic Research, Inc.
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