Maximum likelihood estimation of continuous time stochastic volatility models with partially observed GARCH
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Wei-Fang Niu
Abstract
This paper proposes a method for the maximum likelihood estimation of continuous time stochastic volatility models. The key step is to introduce approximating GARCH processes that have higher frequencies of construction but are observed at lower frequencies. The latency of the volatility process is retained by augmenting data points between price observations. The convergence of the likelihood function can be obtained with mild regularity conditions. Such an approach reconciles discrete and continuous time models, and it can be implemented easily under the context of the simulated maximum likelihood. As an extension to the commonly used modified Brownian bridge sampler, we propose generating paths with skewed density to match the dynamics of the volatilities.
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Richard and Zhang (2007) suggested an alternative approach to find efficient importance samplers. An attractive feature of their method is that the minimization procedure can be reduced to the ordinary least squares problem when the importance sampler belongs to the exponential family. Pastorello and Rossi (2010) applied this method to estimate univariate processes and suggested an algorithm that can be easily implemented. We also tried to implement the method. However, when the transition density does not explicitly depends on the log-price Xt, the solution tends to degenerate to that in which the first n–2 steps of φ are the same as q and then *Xt–Δ, the last simulated point between t and t+1, is chosen to get as close to Xt as possible. So the importance sampler will be very similar to that of Pedersen (1995), producing huge jumps, and thus could be less efficient.
References
Aït-Sahalia, Y. 2002. “Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approximation Approach.” Econometrica 70: 223–262.10.1111/1468-0262.00274Suche in Google Scholar
Aït-Sahalia, Y. 2008. “Closed-Form Likelihood Expansions for Multivariate Diffusions.” Annals of Statistics 36: 906–937.10.1214/009053607000000622Suche in Google Scholar
Anderson, T. G., and J. Lund. 1997. “Estimating Continuous-Time Stochastic Volatility Models of the Short-Term Interest Rate.” Journal of Econometrics 77: 343–377.10.1016/S0304-4076(96)01819-2Suche in Google Scholar
Azzalini, A. 1985. “A Class of Distributions which Includes the Normal Ones.” Scandinavian Journal of Statistics 12: 171–178.Suche in Google Scholar
Bandi, F., and P. Phillips. 2003. “Fully Nonparametric Estimation of Scalar Diffusion Models.” Econometrica 71: 241–283.10.1111/1468-0262.00395Suche in Google Scholar
Bollerslev, T. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31: 307–327.10.1016/0304-4076(86)90063-1Suche in Google Scholar
Brown, L. D., Y. Wang, and L. H. Zhao. 2003. “On the Statistical Equivalence at Suitable Frequencies of GARCH and Stochastic Volatility Models with the Corresponding Diffusion Models.” Statistica Sinica 13: 993–1013.Suche in Google Scholar
Corradi, V. 2000. “Reconsidering the Diffusion Limit of the GARCH(1,1) Process.” Journal of Econometrics 96: 145–153.10.1016/S0304-4076(99)00053-6Suche in Google Scholar
Duan, J. C. 1997. “Augmented GARCH(p,q) Process and its Diffusion Limit.” Journal of Econometrics 79: 97–127.10.1016/S0304-4076(97)00009-2Suche in Google Scholar
Duan, J. C., G. Gauthier, and J. G. Simonato. 1999. “An Analytical Approximation for the GARCH Option Pricing Model.” Journal of Computational Finance 2: 75–116.10.21314/JCF.1999.033Suche in Google Scholar
Duan, J. C., P. Ritchken, and Z. Sun. 2006. “Approximating GARCH-Jump Models, Jump-Diffusion Processes, and Option Pricing.” Mathematical Finance 16: 21–52.10.1111/j.1467-9965.2006.00259.xSuche in Google Scholar
Durham, G. B., and A. R.Gallant. 2002. “Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes.” Journal of Business & Economic Statistics 20: 297–316.10.1198/073500102288618397Suche in Google Scholar
Elerian, O., S. Chib, and N. Shephard. 2001. “Likelihood Inference for Discretely Observed Non-Linear Diffusions.” Econometrica 69: 959–993.10.1111/1468-0262.00226Suche in Google Scholar
Engle, R. 1982. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of UK Inflation.” Econometrica 50: 987–1008.10.2307/1912773Suche in Google Scholar
Eraker, B. 2001. “MCMC Analysis of Diffusion Models with Application to Finance.” Journal Business & Economic Statistics 19: 177–191.10.1198/073500101316970403Suche in Google Scholar
Eraker, B. 2004. “Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices.” Journal of Finance 59: 1367–1403.10.1111/j.1540-6261.2004.00666.xSuche in Google Scholar
Fiorentini, G., A. León, and G. Rubio. 2002. “Estimation and Empirical Performance of Heston’s Stochastic Volatility Model: The Case of A Thinly Traded Market.” Journal of Empirical Finance 9: 225–255.10.1016/S0927-5398(01)00052-4Suche in Google Scholar
Florens-Zmirou, D. 1993. “On estimating the Diffusion Coefficient from Discrete Observations.” Journal of Applied Probability 30: 790–804.10.2307/3214513Suche in Google Scholar
Fornari, F., and A. Mele. 2006. “Approximating Volatility Diffusions with CEV-ARCH Models.” Journal of Economic Dynamics and Control 30: 931–966.10.1016/j.jedc.2005.03.008Suche in Google Scholar
Gallant, R., and G. E. Tauchen. 1996. “Which Moment to Match?” Econometric Theory 12: 657–681.10.1017/S0266466600006976Suche in Google Scholar
Genon-Catalot, V., T. Jeantheau, and C. Laredo. 1999. “Parameter Estimation for Discretely Observed Stochastic Volatility Models.” Bernoulli 5: 855–872.10.2307/3318447Suche in Google Scholar
Gourieroux, C., A. Monfort, and E. Renault. 1993. “Indirect inference.” Journal of Applied Econometrics 8S: S85–S118.10.1002/jae.3950080507Suche in Google Scholar
Heston, S. 1993. “A Closed-Form Solution for Options with Stochastic Volatility with Application to Bond and Currency Options.” Review of Financial Studies 6: 327–343.10.1093/rfs/6.2.327Suche in Google Scholar
Hoffmann, M. 1999. “Lp Estimation of the Diffusion Coefficient.” Bernoulli 5: 447–481.10.2307/3318712Suche in Google Scholar
Kleppe, T. S., J. Yu, and H. J. Skaug. 2010. “Simulated Maximum Likelihood Estimation of Continuous Time Stochastic Volatility Models.” Advances in Econometrics 26: 137–161.10.1108/S0731-9053(2010)0000026009Suche in Google Scholar
Lo, A. 1988. “Maximum Likelihood Estimation of Generalized Itô Processes with Discretely-Sampled Data.” Econometric Theory 4: 231–247.10.1017/S0266466600012044Suche in Google Scholar
Nelson, D. 1990. “ARCH Models as Diffusion Approximation.” Journal of Econometrics 45: 7–38.10.1016/0304-4076(90)90092-8Suche in Google Scholar
O’Hagan, A., and T. Leonhard. 1976. “Bayes Estimation Subject to Uncertainty About Parameter Constraints.” Biometrika 63: 201–202.10.1093/biomet/63.1.201Suche in Google Scholar
Pastorello, S., and E. Rossi. 2010. “Efficient Importance Sampling Maximum Likelihood Estimation of Stochastic Differential Equations.” Computational Statistics and Data Analysis 54: 2753–2762.10.1016/j.csda.2010.02.001Suche in Google Scholar
Pedersen, A. R. 1995. “A New Approach to Maximum Likelihood Estimation for Stochastic Differential Equations Based on Discrete Observations.” Scandinavian Journal of Statistics 22: 55–71.Suche in Google Scholar
Phillips, P. C. B. and J. Yu. 2009. “A Two-Stage Realized Volatility Approach to Estimation of Diffusion Processes with Discrete Data.” Journal of Econometrics 150: 139–150.10.1016/j.jeconom.2008.12.006Suche in Google Scholar
Renò, R. 2008. “Nonparametric Estimation of the Diffusion Coefficient of Stochastic Volatility Models.” Econometric Theory 24: 1174–1206.10.1017/S026646660808047XSuche in Google Scholar
Richard, J.-F., and W. Zhang. 2007. “Efficient High-Dimensional Importance Sampling.” Journal of Econometrics 141: 1385–1411.10.1016/j.jeconom.2007.02.007Suche in Google Scholar
Sørensen, M. 2000. “Prediction-Based Estimating Functions.” Econometrics Journal 3: 123–147.10.1111/1368-423X.00042Suche in Google Scholar
Trifi, A. 2006. “Issues of Aggregation Over Time of Conditional Heteroscedastic Volatility Models: What Kind of Diffusion do we Recover?” Studies in Nonlinear Dynamics & Econometrics 10:4.10.2202/1558-3708.1314Suche in Google Scholar
Wang, Y. 2002. “Asymptotic Nonequivalence of GARCH Models and Diffusions.” Annals of Statistics 30: 754–783.10.1214/aos/1028674841Suche in Google Scholar
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