Modelling stochastic volatility with leverage and jumps: a simulated maximum likelihood approach via particle filtering
Malik, Sheheryar and Pitt, Michael K. (2009) Modelling stochastic volatility with leverage and jumps: a simulated maximum likelihood approach via particle filtering. Working Paper. Coventry: University of Warwick, Department of Economics. (Warwick economic research papers).
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In this paper we provide a unified methodology in order to conduct likelihood-based inference on the unknown parameters of a general class of discrete-time stochastic volatility models, characterized by both a leverage effect and jumps in returns. Given the non-linear/non-Gaussian state-space form, approximating the likelihood for the parameters is conducted with output generated by the particle filter. Methods are employed to ensure that the approximating likelihood is continuous as a function of the unknown parameters thus enabling the use of Newton-Raphson type maximization algorithms. Our approach is robust and efficient relative to alternative Markov Chain Monte Carlo schemes employed in such contexts. In addition it provides a feasible basis for undertaking the non-trivial task of model comparison. The technique is applied to daily returns data for various stock price indices. We find strong evidence in favour of a leverage effect in all cases. Jumps are an important component in two out of the four series we consider.
|Item Type:||Working or Discussion Paper (Working Paper)|
|Subjects:||H Social Sciences > HB Economic Theory|
|Divisions:||Faculty of Social Sciences > Economics|
|Library of Congress Subject Headings (LCSH):||Financial leverage, Jump processes, Stochastic processes, Markov processes, Monte Carlo method, Stock price indexes|
|Series Name:||Warwick economic research papers|
|Publisher:||University of Warwick, Department of Economics|
|Place of Publication:||Coventry|
|Date:||3 April 2009|
|Number of Pages:||29|
|Status:||Not Peer Reviewed|
|Access rights to Published version:||Open Access|
|References:||Bates, D. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark Options. Review of Financial Studies, 9, 69-107. Bates, D. (2000). Post-'87 crash fears in S&P 500 futures options. Journal of Econometrics, 94, 181-238. Bakshi, G., C. Cao and Z. Chen (1997). Empirical performance of alternative options pricing models. Journal of Finance, 52, 2003-2049. Black, F. (1976). Studies of stock market volatility changes. Proceedings of the American Statistical Association, Business and Economic Statistics Section 177-181. Carpenter, J. R., P. Cli®ord, and P. Fearnhead (1999). An improved particle ¯lter for non- linear problems. IEE Proceedings on Radar, Sonar and Navigation 146, 2-7. Chib, S., F. Nardari, and N. Shephard (2006). Analysis of high dimensional multivariate stochastic volatility models. Journal of Econometrics 134, 341-371. Christie, A. A. (1982).The stochastic behaviour of common stock variances. Journal of Financial Economics 10, 407-432. Christo®ersen, P., Jocobs, K., and Mimouni, K. (2007). Models for S&P dynamics: Evi- dence from realized volatility, daily returns, and options prices.(McGill University, unpublished working paper). Del-Moral, P. (2004). Feynman-Kac Formulae: Genealogical and Interacting Particle Sys- tems with Applications. New York. Springer. Durham, G. B. (2006). SV mixture models with application to S&P 500 index returns. Journal of Financial Economics (forthcoming). Du±e, D., K. Singleton and J. Pan (2000). Transform analysis and asset pricing for a±ne jump-di®usions. Econometrica, 68, 1343-1376. Doucet, A., J.F.G. De Freitas and N. Gordon (2000). Sequential Monte Carlo Methods in Practice. Cambridge University Press, Cambridge. Eraker, B., M. Johannes, and N. Polson (2003). Journal of Finance, 58(3), 1269-3000. Engle, R., and Ng, V. (1993). Measuring and testing the impact of news in volatility. Journal of Finance, 43, 1749-1778 Gallant, A. R. and G. Tauchen (1998). Reprojection partially observed systems with appli- cations to interest rate di®usions. Journal of the American Statistical Association, 93, 10-24. Glosten, L.R., Jagannanthan, R. and D. Runkle (1993). Relationship between the expected value and the volatility of the excess return on stocks. Journal of Finance, 48, 1779-1802. Gordon, N. J., D. J. Salmond, and A. F. Smith (1993). A novel approach to non-linear and non-Gaussian Bayesian state estimation. IEE-Proceedings F 140, 107-13. Gourieroux, C. and Monfort, A. (1996). Simulation Based Econometric Methods. Oxford University Press, Oxford, pp 41-53. Gweke, J. (1992). Evaluating the accuracy of sampling-based approaches to calculation of moments (with discussion). In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (Eds.), Bayesian Statistics,Vol. 4. Oxford University Press, Oxford, pp. 169-193. Harvey, A. C. and N. Shephard (1996). The estimation of an asymmetric stochastic volatility model for asset returns. Journal of Business and Economic Statistics 14, 429-434. Harvey, A. C. , E. Ruiz, and N. Shephard (1994). Multivariate stochastic variance models. Review of Economic Studies 61, 247-246. Isard, M. and A. Blake (1996). Contour tracking by stochastic propagation of conditional density. Proceedings of the European Conference on Computer Vision, Cambridge 1, 343-356. Jacquier, E., N.G. Polson and P.E. Rossi (1994). Bayesian analysis of stochastic volatility models. Journal of Business and Economic Statistics 12, 371-389. Jacquier, E., N. G. Polson and P. E. Rossi (2004). Bayesian analysis of stochastic volatility models with fat-tails and correlated errors. Journal of Econometrics, 122(1), 185-212. Johannas, M., Polson, N. G., and Stroud, J. R (2009). Optimal Filtering of Jump-Di®usions: Extracting Latent States from Asset Prices. Forthcoming Review of Financial Studies. Sauer, R. and Keane M. P. (2009). A Computationally Practical Simulation Estimation Algorithm for Dynamic Panel Data Models with Unobserved Endogenous State Variables, In- ternational Economic Review, forthcoming. Kim, S., N. Shephard, and S. Chib (1998). Stochastic volatility: likelihood inference and comparison with ARCH models. Review of Economic Studies 65, 361-393. Kitagawa, G. (1996). Monte Carlo ¯lter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics, 5, 1-25. Liu, J. and R. Chen (1998). Sequential Monte Carlo methods for dynamic systems. Journal of American Statistical Association, 93, 1032-1044. Liu, J. and West, M. (2001). Combined parameters and state estimation in simulation-based ¯ltering. In: Sequential Monte Carlo Methods in Practice (by Doucet, A., J.F.G. De Freitas and N. Gordon) 97-233. Springer-Verlag, New York. Merton, R. C. (1976). Option pricing when underlying stock returns and discontinuous. Journal of Financial Economics 3, 125-144. Meyer, R. and J. Yu (2000). BUGS for Bayesian analysis of stochastic volatility model models. Econometrics Journal, 3, 198-215. Nelson, D. (1991). Conditional heteroskedasticity in asset pricing: A new approach. Econo- metrica, 59, 347-370. Omori, Y., S. Chib, N. Shephard and J. Nakajima (2007). Stochastic volatility with leverage: fast likelihood inference. Journal of Econometrics, 140, 425-449. Pakes, A. and Pollard, D. (1989). Simulation and Asymptotics of Optimization Estimators. Econometrica, 57, No. 5, 1027-1057. Lee, L. (1992). On E±ciency of Methods of Simulated Moments and Maximum Likelihood Estimation of Discrete Response Models. Econometric Theory, 8, 515-552. Pitt, M. K. (2003). Smooth particle ¯lters for likelihood evaluation and maximization. Unpublished working paper, University of Warwick. Pitt, M. K. with Shephard, N.(1999). Filtering via simulation: auxiliary particle ¯lter Journal of the American Statistical Association , 94, 590-9. Polson, N. G., Stroud, J. S. and Muller, P. (2008). Particle Filtering with Sequential Pa- rameter Learning, Journal of Royal Statistical Society, B, 70, 413-428. Sandmann, G. and S.J. Koopman (1998). Estimation of Stochastic Volatility Models via Monte Carlo Maximum Likelihood, Journal of Econometrics, 87, No.2, 271-301. Shephard, N. and M. K. Pitt (1997). Likelihood analysis of non-Gaussian measurement time series. Biometrika 84, 653-67. Yu, J. (2005). On leverage in a stochastic volatility model. Journal of Econometrics,127, 165-178.|
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