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Maximum likelihood estimation of the double exponential jump-diffusion process

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Abstract

The double exponential jump-diffusion (DEJD) model, recently proposed by Kou (Manage Sci 48(8), 1086–1101, 2002) and Ramezani and Zeng (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=606361, 1998), generates a highly skewed and leptokurtic distribution and is capable of matching key features of stock and index returns. Moreover, DEJD leads to tractable pricing formulas for exotic and path dependent options (Kou and Wang Manage Sci 50(9), 1178–1192, 2004). Accordingly, the double exponential representation has gained wide acceptance. However, estimation and empirical assessment of this model has received little attention to date. The primary objective of this paper is to fill this gap. We use daily returns for the S&P-500 and the NASDAQ indexes and individual stocks, in conjunction with maximum likelihood estimation (MLE) to fit the DEJD model. We utilize the BIC criterion to assess the performance of DEJD relative to log-normally distributed jump-diffusion (LJD) and the geometric brownian motion (GBM). We find that DEJD performs better than these alternatives for both indexes and individual stocks.

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References

  • Aït-Sahalia Y. (2002) Maximum-likelihood estimation of discretely-sampled diffusions: a closed-form approximation approach. Econometrica 70, 223–262

    Article  Google Scholar 

  • Aït-Sahalia Y. (2004) Disentangling diffusion from jumps. J Financ Econ 74, 487–528

    Article  Google Scholar 

  • Aït-Sahalia Y., Hansen L.P. (2004) Handbook of Financial Econometrics. Amsterdam, North-Holland

    Google Scholar 

  • Andersen L., Andreasen J. (2000) Jump-diffusion processes: Volatility smile fitting and numerical methods for pricing. Rev Derivative Res 4, 231–262

    Article  Google Scholar 

  • Andersen T.G., Benzoni L., Lund J. (2000) An empirical investigation of continuous-time equity return models. J Finance 57, 1239–1284

    Article  Google Scholar 

  • Bates D.S. (2003a) Empirical option pricing: a retrospection. J Econom 116(1–2): 387–404

    Article  Google Scholar 

  • Bates D.S. (2003b) Maximum likelihood estimation of latent affine processes. Working Paper, University of Iowa

  • Broadie M., Yamamoto Y. (2003) Application of the fast gauss transform to option pricing. Manage Sci 49(8): 1071–1088

    Article  Google Scholar 

  • Carr P., Madan D. (1999) Option pricing using the fast fourier transform. J Comput Finance 2(4): 61–73

    Google Scholar 

  • Carr P., Wu L. (2004) Time-changed Lévy processes and option pricing. Financ Econ 17(1): 113–141

    Article  Google Scholar 

  • Chernov M., Gallant A.R., Ghysels E., Tauchen G. (2003) Alternative models for stock price dynamics. J Econom 116(1–2): 225–257

    Article  Google Scholar 

  • Cont R., Tankov P. (2003) Financial modelling with jump processes. London, Chapman & Hall / CRC, Financial Mathematic Series

    Google Scholar 

  • Cont R., Tankov P. (2004) Nonparametric calibration of jump-diffusion option pricing models. J Comput Finance 7(3): 1–49

    Google Scholar 

  • dH́alluin, Y., Forsyth, P., Vetzal, K.: Robust numerical methods for contingent claims under jump diffusion processes. Working Paper, School of Computer Science, University of Waterloo (2004)

  • Duffie D., Pan J., Singleton K.J. (2000) Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6): 1343–1376

    Article  Google Scholar 

  • Eraker B., Johannes M., Polson N. (2003) The impact of jumps in volatility and returns. J Finance 58(3): 1269–1300

    Article  Google Scholar 

  • Garcia, R., Ghysels, E., Renault, E.: The econometrics of option pricing, Working Paper, Université de Montréal - CIREQ / Département de sciences économiques (2004)

  • Hamilton J.D. (1994) Time Series Analysis. Princeton, Princeton University Press

    Google Scholar 

  • Honoré P. (1998) Pitfalls in estimating jump diffusion models. Working Paper, University of Aarhus

  • Huang, J., Huangm M.: How much of the corporate-treasury yield spread is due to credit risk?, Working Paper, Graduate School of Business, Stanford University (2003)

  • Huang J., Wu L. (2004) Specification analysis of option pricing models based on time-changed Lévy processes. J Finance 59(3): 1405–1439

    Article  Google Scholar 

  • Jackwerth J., Rubinstein M. (1996) Recovering probability distributions from option prices. J Finance 51, 1611–1631

    Article  Google Scholar 

  • Keppo, J., Meng, X., Shive, S., Sullivan, M.: Modelling and hedging options under stochastic pricing parameters, Working Paper, Industrial and Operations Engineering, University of Michigan at Ann Arbor (2003)

  • Kiefer N.M. (1978) Discrete parameter variation: efficient estimation of switching regression model. Econometrica 46(2): 427–34

    Article  Google Scholar 

  • Kou S. (2002) A jump diffusion model for option pricing. Manage Sci 48(8): 1086–1101

    Article  Google Scholar 

  • Kou S., Wang H. (2004) Option pricing under a double exponential jump diffusion model. Manage Sci 50(9): 1178–1192

    Article  Google Scholar 

  • Lee, R.W.: Implied volatility: Statics, dynamics, and probabilistic interpretation. Recent advances in applied probability (2006), (inpress)

  • Lewis, A.: A simple option formula for general jump-diffusion and other exponential Lévy processes. Envision financial system and optioncitynet (2001)

  • Maheu J.M., McCurdy T. (2004) News arrival, jump dynamics and volatility components for individual stock returns. J Finance 59(2): 755–768

    Article  Google Scholar 

  • Merton RC. (1976a) Option pricing when underlying stock returns are discontinuous. J Financ Econ 3, 224–244

    Article  Google Scholar 

  • Merton R.C. (1976b) The impact on option pricing of specification error in the underlying stock price returns. J Finance 31(2): 333–350

    Article  Google Scholar 

  • Milgrom P.R. (1981) Good news and bad news: representation theorems and applications. Bell J Econ 12, 380–391

    Article  Google Scholar 

  • Naik V. (1993) Option valuation and hedging strategies with jumps in the volatility of asset returns. J Finance 48(5): 1969–1984

    Article  Google Scholar 

  • Patel J., Kapadia C.H., Owen D.B. (1976) Handbook of Statistical Distributions. New York, Marcel Dekker Inc.

    Google Scholar 

  • Press S.J. (1967) A compound events model for security prices. J Busi 40, 317–335

    Article  Google Scholar 

  • Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P. (1992) Numerical Recipes in FORTRAN. London, Cambridge University Press

    Google Scholar 

  • Protter P. (1991) Stochastic Integration and Differential Equations. Berlin Heidelberg New York, Springer

    Google Scholar 

  • Ramezani, C., Zeng, Y.:Maximum likelihood estimation of asymmetric jump-diffusion processes: Application to security prices, Working Paper, Department of Mathematics and Statistics, University of Missouri, Kansas City, see http://papers.ssrn.com/sol3/papers.cfm?abstract_id=606361 (1998)

  • Ramezani, C., Zeng, Y.: An empirical assessment of the double exponential jump-diffusion process, Working Paper, Department of Mathematics and Statistics, University of Missouri, Kansas City, see http://papers.ssrn.com/sol3/papers.cfm?abstract_id=606101 (2005)

  • Schwarz G. (1978) Estimating the dimension of a model. Ann Stat 6, 461–464

    Google Scholar 

  • Sepp A. (2004) Analytical pricing of double-barrier options under a double-exponential jump diffusion process: Applications of laplace transform. Int J Theor Appl Finance 7(2): 151–175

    Article  Google Scholar 

  • Sorensen M. (1991) Likelihood methods for diffusions with jumps. In: Prabhu N.V., Basawa I.V. (eds) Statistical Inference in Stochastic Processes. New York, Marcel Dekker, Inc.

    Google Scholar 

  • Tsay R.S. (2002) Analysis of Financial Time Series. Wiley Series in Probability and Statistics. Wiley Inter-Science

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Authors and Affiliations

  1. Finance area, Orfalea college of business, California Polytechnic State University, San Luis Obispo, CA, 93407, USA

    Cyrus A. Ramezani

  2. Department of Mathematics and Statistics, University of Missouri, Kansas City, MO, 64110, USA

    Yong Zeng

Authors
  1. Cyrus A. Ramezani
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  2. Yong Zeng
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Correspondence to Cyrus A. Ramezani.

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Ramezani, C.A., Zeng, Y. Maximum likelihood estimation of the double exponential jump-diffusion process. Annals of Finance 3, 487–507 (2007). https://doi.org/10.1007/s10436-006-0062-y

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