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A New Class of Stochastic Volatility Models with Jumps: Theory and Estimation

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  • Mikhail Chernov
  • A. Ronald Gallant
  • Eric Ghysels

    ()

  • George Tauchen

Abstract

The purpose of this paper is to propose a new class of jump diffusions which feature both stochastic volatility and random intensity jumps. Previous studies have focussed primarily on pure jump processes with constant intensity and log-normal jumps or constant jump intensity combined with a one factor stochastic volatility model. We introduce several generalizations which can better accommodate several empirical features of returns data. In their most general form we introduce a class of processes which nests jump-diffusions previously considered in empirical work and includes the affine class of random intensity models studied by Bates (1998) and Duffie, Pan and Singleton (1998) but also allows for non-affine random intensity jump components. We attain the generality of our specification through a generic Lévy process characterization of the jump component. The processes we introduce share the desirable feature with the affine class that they yield analytically tractable and explicit option pricing formula. The non-affine class of processes we study include specifications where the random intensity jump component depends on the size of the previous jump which represent an alternative to affine random intensity jump processes which feature correlation between the stochastic volatility and jump component. We also allow for and experiment with different empirical specifications of the jump size distributions. We use two types of data sets. One involves the S&P500 and the other comprises of 100 years of daily Dow Jones index. The former is a return series often used in the literature and allows us to compare our results with previous studies. The latter has the advantage to provide a long time series and enhances the possibility of estimating the jump component more precisely. The non-affine random intensity jump processes are more parsimonious than the affine class and appear to fit the data much better. Nous présentons une nouvelle classe de processus à sauts avec volatilité stochastique. Cette nouvelle classe généralise les modèles affinés proposés par Duffie, Pan et Singleton (1998). La généralité se manifeste par une représentation générique des sauts par un processus de Lévy. La classe des processus que nous présentons nous fournit également des prix d'options. Une application empirique démontre la présence de sauts dans des séries financières telles le S&P500 et le Dow Jones. De plus, les processus n'ont pas une intensité constante. Nous analysons plusieurs spécifications empiriques.

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 99s-48.

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Date of creation: 01 Nov 1999
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Handle: RePEc:cir:cirwor:99s-48

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Keywords: Efficient method of moments; Poisson processes; jump processes; stochastic volatility models; filtering; Processus à sauts; mesures de Lévy; modèles à volatilité stochastique;

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References

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  1. Tauchen, George E. & Gallant, A. Ronald, 1995. "Which Moments to Match," Working Papers 95-20, Duke University, Department of Economics.
  2. Gallant, A Ronald & Rossi, Peter E & Tauchen, George, 1992. "Stock Prices and Volume," Review of Financial Studies, Society for Financial Studies, vol. 5(2), pages 199-242.
  3. Lo, Andrew W. (Andrew Wen-Chuan) & Wang, Jiang, 1959-, 1993. "Implementing option pricing models when asset returns are predictable," Working papers 3593-93., Massachusetts Institute of Technology (MIT), Sloan School of Management.
  4. Ho, Mun S & Perraudin, William R M & Sorensen, Bent E, 1996. "A Continuous-Time Arbitrage-Pricing Model with Stochastic Volatility and Jumps," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(1), pages 31-43, January.
  5. Gallant, A. Ronald & Hsieh, David & Tauchen, George, 1995. "Estimation of Stochastic Volatility Models with Diagnostics," Working Papers 95-36, Duke University, Department of Economics.
  6. GHYSELS, Eric & HARVEY, Andrew & RENAULT, Eric, 1995. "Stochastic Volatility," CORE Discussion Papers 1995069, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  7. Bates, David S., 2000. "Post-'87 crash fears in the S&P 500 futures option market," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 181-238.
  8. repec:cup:etheor:v:12:y:1996:i:4:p:657-81 is not listed on IDEAS
  9. Duffie, Darrell & Singleton, Kenneth J, 1993. "Simulated Moments Estimation of Markov Models of Asset Prices," Econometrica, Econometric Society, vol. 61(4), pages 929-52, July.
  10. A. Ronald Gallant & Chien-Te Hsu & George Tauchen, 1999. "Using Daily Range Data To Calibrate Volatility Diffusions And Extract The Forward Integrated Variance," The Review of Economics and Statistics, MIT Press, vol. 81(4), pages 617-631, November.
  11. Jarrow, Robert A & Rosenfeld, Eric R, 1984. "Jump Risks and the Intertemporal Capital Asset Pricing Model," The Journal of Business, University of Chicago Press, vol. 57(3), pages 337-51, July.
  12. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
  13. Chernov, Mikhail & Ghysels, Eric, 2000. "A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation," Journal of Financial Economics, Elsevier, vol. 56(3), pages 407-458, June.
  14. Ball, Clifford A. & Torous, Walter N., 1983. "A Simplified Jump Process for Common Stock Returns," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 18(01), pages 53-65, March.
  15. Philippe Jorion, 1988. "On Jump Processes in the Foreign Exchange and Stock Markets," Review of Financial Studies, Society for Financial Studies, vol. 1(4), pages 427-445.
  16. Gallant, A.R. & Tauchen, G., 1988. "Seminonparametric Estimation Of Conditionally Constrained Heterogeneous Processes: Asset Pricing Applications," Papers 88-59, Chicago - Graduate School of Business.
  17. Victor Fenton & Gallant, A. Ronald, 1996. "Qualitative and Asymptotic Performance of SNP Density Estimators," Working Papers 96-17, Duke University, Department of Economics.
  18. Ball, Clifford A & Torous, Walter N, 1985. " On Jumps in Common Stock Prices and Their Impact on Call Option Pricing," Journal of Finance, American Finance Association, vol. 40(1), pages 155-73, March.
  19. Chacko, George & Viceira, Luis M., 2003. "Spectral GMM estimation of continuous-time processes," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 259-292.
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Citations

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Cited by:
  1. Luca Benzoni & Pierre Collin-Dufresne & Robert S. Goldstein, 2005. "Can Standard Preferences Explain the Prices of out of the Money S&P 500 Put Options," NBER Working Papers 11861, National Bureau of Economic Research, Inc.
  2. René Garcia & Eric Ghysels & Éric Renault, 2004. "The Econometrics of Option Pricing," CIRANO Working Papers 2004s-04, CIRANO.
  3. Jing-zhi Huang & Liuren Wu, 2004. "Specification Analysis of Option Pricing Models Based on Time-Changed Levy Processes," Econometric Society 2004 North American Winter Meetings 405, Econometric Society.
  4. John M. Maheu & Thomas H. McCurdy, 2002. "Nonlinear Features of Realized FX Volatility," The Review of Economics and Statistics, MIT Press, vol. 84(4), pages 668-681, November.
  5. Sassan Alizadeh & Michael W. Brandt & Francis X. Diebold, 1999. "Range-Based Estimation of Stochastic Volatility Models or Exchange Rate Dynamics are More Interesting Than You Think," Center for Financial Institutions Working Papers 00-28, Wharton School Center for Financial Institutions, University of Pennsylvania.
  6. Stefano Galluccio & Yann Le Cam, 2005. "Implied Calibration of Stochastic Volatility Jump Diffusion Models," Finance 0510028, EconWPA.
  7. Nour Meddahi, 2001. "An Eigenfunction Approach for Volatility Modeling," CIRANO Working Papers 2001s-70, CIRANO.
  8. Das, Sanjiv Ranjan & Uppal, Raman, 2002. "Systemic Risk and International Portfolio Choice," CEPR Discussion Papers 3305, C.E.P.R. Discussion Papers.
  9. David Neto & Sylvain Sardy & Paul Tseng, 2009. "l1-Penalized Likelihood Smoothing of Volatility Processes allowing for Abrupt Changes," Research Papers by the Institute of Economics and Econometrics, Geneva School of Economics and Management, University of Geneva 2009.05, Institut d'Economie et Econométrie, Université de Genève.
  10. Pan, Jun, 2002. "The jump-risk premia implicit in options: evidence from an integrated time-series study," Journal of Financial Economics, Elsevier, vol. 63(1), pages 3-50, January.
  11. Daal, Elton & Naka, Atsuyuki & Yu, Jung-Suk, 2006. "Volatility Clustering, Leverage Effects, and Jump Dynamics in the US and Emerging Asian Equity Markets," Working Papers 2005-03, University of New Orleans, Department of Economics and Finance.
  12. Torben G. Andersen & Luca Benzoni & Jesper Lund, 2002. "An Empirical Investigation of Continuous-Time Equity Return Models," Journal of Finance, American Finance Association, vol. 57(3), pages 1239-1284, 06.
  13. Chib, Siddhartha & Nardari, Federico & Shephard, Neil, 2002. "Markov chain Monte Carlo methods for stochastic volatility models," Journal of Econometrics, Elsevier, vol. 108(2), pages 281-316, June.
  14. Tsyplakov, Alexander, 2010. "Revealing the arcane: an introduction to the art of stochastic volatility models," MPRA Paper 25511, University Library of Munich, Germany.

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