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The hierarchical-likelihood approach to autoregressive stochastic volatility models. (English) Zbl 1247.91141
Summary: Many volatility models used in financial research belong to a class of hierarchical generalized linear models with random effects in the dispersion. Therefore, the hierarchical-likelihood (h-likelihood) approach can be used. However, the dimension of the Hessian matrix is often large, so techniques of sparse matrix computation are useful to speed up the procedure of computing the inverse matrix. Using numerical studies we show that the h-likelihood approach gives better long-term prediction for volatility than the existing MCMC method, while the MCMC method gives better short-term prediction. We show that the h-likelihood approach gives comparable estimations of fixed parameters to those of existing methods.

91B82 Statistical methods; economic indices and measures
62P10 Applications of statistics to biology and medical sciences; meta analysis
Full Text: DOI
[1] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economy, 81, 637-659, (1973) · Zbl 1092.91524
[2] Castillo, J.; Lee, Y., GLM-methods for volatility models, Statistical modelling, 8, 263-283, (2008)
[3] Durbin, J.; Koopman, S.J., Time series analysis of non-gaussian observations based on state space models from both classical and Bayesian perspectives (with discussion), Journal of royal statistical society series B, 62, 3-56, (2000) · Zbl 0945.62084
[4] Fridman, M.; Harris, L., A maximum likelihood approach for the non-Gaussian stochastic volatility models, Journal of business and economic statistics, 16, 284-291, (1998)
[5] Golub, G.; Loan, C.V., Matrix computation, (1996), The Johns Hopkins University Press London
[6] Harvey, A.; Ruiz, E.; Shephard, N., Multivariate stochastic variance models, Review of economic studies, 61, 247-264, (1994) · Zbl 0805.90026
[7] Harvey, A.; Shephard, N., Estimation of an asymmetric stochastic volatility model for asset returns, Journal of business & economic statistics, 96, 429-434, (1996)
[8] Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, The review of financial studies, 6, 327-343, (1993) · Zbl 1384.35131
[9] Hull, J.; White, A., The pricing of options on assets with stochastic volatilities, The journal of finance, 42, 281-300, (1987)
[10] Jacquier, E.; Polson, N.; Rossi, P., Bayesian analysis of stochastic volatility models (with discussion), Journal of business and economic statistics, 12, 371-417, (1994)
[11] Joe, H., Ng, T., Qu, J., Lee, Y., 2009. Composite likelihood approach to stochastic volatility models (submitted for publication).
[12] Kim, S.; Shephard, N.; Chib, S., Stochastic volatility: likelihood inference and comparison with ARCH models, Review of economic studies, 65, 361-393, (1998) · Zbl 0910.90067
[13] Lee, Y., Ha, I., 2010. Orthodox BLUP versus h-likelihood methods for inferences about random effects in Tweedie mixed models, Statistics and Computing (in press).
[14] Lee, Y.; Nelder, J.A., Hierarchical generalized linear models (with discussion), Journal of the royal statistical society series B, 58, 619-678, (1996) · Zbl 0880.62076
[15] Lee, Y.; Nelder, J.A., Conditional and marginal models: another view (with discussion), Statistical science, 19, 219-238, (2004) · Zbl 1100.62591
[16] Lee, Y.; Nelder, J.A., Double hierarchical generalized linear models (with discussion), Applied statistics, 55, 139-185, (2006) · Zbl 05188732
[17] Lee, Y.; Nelder, J.; Pawitan, Y., Generalized linear models with random effects: unified analysis via H-likelihood, (2006), Chapman and Hall London · Zbl 1110.62092
[18] Liesenfeld, R.; Richard, J.F., Univariate and multivariate stochastic volatility models: estimation and diagnostics, Journal of empirical finance, 10, 505-531, (2003)
[19] Meyer, R.; Fournier, D.; Berg, A., Stochastic volatility: Bayesian computation using automatic differentiation and extended Kalman filter, Econometrics journal, 6, 408-420, (2003) · Zbl 1065.91533
[20] Meyer, R.; Yu, J., BUGS for a Bayesian analysis of stochastic volatility models, Econometrics journal, 3, 198-215, (2000) · Zbl 0970.91060
[21] Pollock, D.S.G., Recursive estimation in econometrics, Computational statistics & data analysis, 44, 37-75, (2003) · Zbl 1429.62692
[22] Ruiz, E., Quasi-maximum likelihood estimation of stochastic volatility models, Journal of econometrics, 63, 284-306, (1994) · Zbl 0825.62949
[23] Sandmann, G.; Koopman, S., Estimation of stochastic volatility models via Monte Carlo maximum likelihood, Journal of econometrics, 87, 271-301, (1998) · Zbl 0937.62110
[24] Scott, L., Option pricing when the variance changes randomly: theory, estimation, and an application, The journal of financial and quantitative analysis, 22, 419-438, (1987)
[25] Skaug, H.J.; Fournier, D.A., Automatic approximation of the marginal likelihood in non-Gaussian hierarchical models, Computational statistics & data analysis, 51, 699-709, (2006) · Zbl 1157.65317
[26] Stein, E.; Stein, J., Stock price distributions with stochastic volatility: an analytic approach, The review of financial studies, 4, 727-752, (1991) · Zbl 06857133
[27] Shimada, J.; Tsukuda, Y., Estimation of stochastic volatility models: an approximation to the nonlinear state space representation, Communications in statistical simulation and computation, 34, 429-450, (2005) · Zbl 1066.62105
[28] Shun, Z.; McCullagh, P., Laplace approximation of high dimensional integrals, Journal of royal statistical society, ser. B, 57, 749-760, (1995) · Zbl 0826.41026
[29] Taylor, S., Asset price dynamics, volatility and prediction, (2005), Princeton University Press
[30] Wei, W.S., Time series analysis: univariate and multivariate methods, (2006), Addison Wesley · Zbl 1170.62362
[31] Wiggins, J., Option values under stochastic volatility: theory and empirical estimates, Journal of financial economics, 19, 351-372, (1987)
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