×

Estimating the Wishart affine stochastic correlation model using the empirical characteristic function. (English) Zbl 1329.91148

Summary: This paper provides the first estimation strategy for the Wishart Affine Stochastic Correlation (WASC) model. We provide elements showing that the use of empirical characteristic function-based estimates is advisable as this function is exponential affine in the WASC case. We use a GMM estimation strategy with a continuum of moment conditions based on the characteristic function. We present the estimation results obtained using a dataset of equity indexes. The WASC model captures most of the known stylized facts associated with financial markets, including leverage and asymmetric correlation effects.

MSC:

91G70 Statistical methods; risk measures
62F10 Point estimation
62H20 Measures of association (correlation, canonical correlation, etc.)
60E10 Characteristic functions; other transforms
62P05 Applications of statistics to actuarial sciences and financial mathematics

Software:

HRMSYM; PATSYM; QUADPACK
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aït-Sahalia, Y., J. Cacho-Diaz, and R J. Laeven. 2010. “Modeling Financial Contagion Using Mutually Exciting Jump Processes.” Working Paper NBER No 15850.;
[2] Amisano, G., and R. Giacomini. 2007. “Comparing Density Forecasts via Weighted Likelihood Ratio Tests.” Journal of Business and Economic Statistics 25 (02): 177-190.;
[3] Ang, A., and J. Chen. 2002. “Asymmetric Correlations of Equity Portfolios.” Journal of Financial Economics (63): 443-494.;
[4] Asai, M., M. McAleer, and J. Yu. 2006. “Multivariate Stochastic Volatility: A Review.” Econometric Review 25 (2-3): 145-175.; · Zbl 1107.62108
[5] Bates, D. S. 2006. “Maximum Likelihood Estimation of Latent Affine Processes.” The Review of Financial Studies 19 (3): 909-965.;
[6] Bathia, R. 2005. Matrix Analysis. Graduate Texts in Mathematics. Berlin, Heidelberg, New York: Springer.;
[7] Bauer, G. H., and K. Vorkink. 2011. “Forecasting Multivariate Realized Stock Market Volatility.” Journal of Econometrics, 160 (1): 93-101.; · Zbl 1441.62601
[8] Bauwens, L., S. Laurent, and J. Rombouts. 2006. “Multivariate Garch Models: A Survey.” Journal of Applied Econometrics 21 (1): 79-109.;
[9] Belisle, C. J. P. 1992. “Convergence Theorems for a Class of Simulated Annealing Algorithms.” J Applied Probability 29: 885-895.; · Zbl 0765.65059
[10] Bertholon, H., A. Monfort, and F. Pegoraro. 2008. “Econometric Asset Pricing Modelling.” Journal of Financial Econometrics, 6 (4): 407-458.;
[11] Black, F. 1976.“Studies of Stock Prices Volatility Changes.” Proceeding from the American Statistical Association, Business and Economics Statistics Section, pp. 177-181.;
[12] Bru, M. F. 1991. “Wishart Processes.” Journal of Theoretical Probability 4: 725-751.; · Zbl 0737.60067
[13] Buraschi, A., P. Porchia, and F. Trojani. 2006. “Correlation Risk and Optimal Portfolio Choice.” Working paper, SSRN-908664.;
[14] Cappiello, L., R. F. Engle, and K. Sheppard. 2006. “Asymmetric Dynamics in the Correlations of Global Equity and Bond Returns.” Journal of Financial Econometrics 4 (4): 537-572.;
[15] Carrasco, M., and J. Florens. 2000. “Generalization of GMM to a Continuum of Moment Conditions.” Econometric Theory (16): 797-834.; · Zbl 0968.62028
[16] Carrasco, M., and R. Kotchoni. 2010. “Efficient Estimation using the Characteristic Function.” University of Montreal Working Paper.; · Zbl 1442.62732
[17] Carrasco, M., M. Chernov, J.-P. Florens, and E. Ghysels. 2007. “Efficient Estimation of Jump Diffusions and General Dynamic Models with a Continuum of Moment Conditions.” Journal of Econometrics (140): 529-573.; · Zbl 1247.91116
[18] Chacko, G., and L. M. Viceira. 2003. “Spectral GMM Estimation of Continuous-Time Processes.” Journal of Econometrics 116 (1-2): 259-292.; · Zbl 1026.62085
[19] Da Fonseca, J., M. Grasselli, and C. Tebaldi. 2007. “Option Pricing when Correlations are Stochastic: an Analytical Framework.” Review of Derivatives Research 10 (2): 151-180.; · Zbl 1174.91006
[20] Da Fonseca, J., M. Grasselli, and C. Tebaldi. 2008. “A Multifactor Volatility Heston Model.” Quantitative Finance, 8 (6): 591-604. An earlier version of this paper circulated in 2005 as “Wishart multi-dimensional stochastic volatility”, RR31, ESILV.; · Zbl 1152.91500
[21] Daleckii, J. 1974. “Differentiation of Non-Hermitian Matrix Functions Depending on a Parameter.” AMS Translations 47 (2): 73-87.;
[22] Daleckii, J. and S. Krein. 1974. “Integration and Differentiation of Functions of Hermitian Operators and Applications to the Theory of Perturbations.” AMS Translations 47 (2): 1-30.;
[23] Donoghue, W. J. 1974. Monotone Matrix Functions and Analytic Continuation. Berlin, Heidelberg, New York: Springer.; · Zbl 0278.30004
[24] Duffie, D., and K. Singleton. 1993. “Simulated Moments Estimation of Markov Models of Asset Prices.” Econometrica 61: 929-952.; · Zbl 0783.62099
[25] Engle, R. 2002. “Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models.” Journal of Business & Economic Statistics 20 (3): 339-350.;
[26] Eraker, B., M. Johannes, and N. Polson. 2003. “The Impact of Jumps in Volatility and Returns.” The Journal of Finance 58 (3): 1269-1300.;
[27] Faraut, J. 2006. Analyse sur les groupes de Lie. Montrouge, France: Calvage & Mounet.; · Zbl 1096.22001
[28] Gallant, R. A., and G. Tauchen. 1996. “Which moments to match?” Econometric Theory 12 (04): 657-681.;
[29] Garcia, R., A. Lewis, S. Pastorello, and E. Renault. 2011. “Estimation of the Objective and Risk-Neutral Distributions based on Moments of Integrated Volatility.” Journal of Econometrics 160 (1): 22-32.; · Zbl 1441.62698
[30] Genz, A., and B. Keister. 1996. “Fully Symmetric Interpolatory Rules for Multiple Integrals Over Infinite Regions with Gaussian Weight.” Journal of Computational and Applied Mathematics 71: 299-309.; · Zbl 0856.65011
[31] Gouriéroux, C. 2006. “Continuous Time Wishart Process for Stochastic Risk.” Econometric Review 25 (2-3): 177-217.; · Zbl 1105.62104
[32] Gouriéroux, C., and J. Jasiak. 2001. Financial Econometrics. Princeton, New Jersey: Princeton University Press.; · Zbl 1028.62083
[33] Gouriéroux, C., and A. Monfort. 2007. “Estimating the Historical Mean Reverting Parameter in the CIR model.” CREST Working Paper.;
[34] Gouriéroux, C., and R. Sufana. 2010. “Derivative Pricing with Multivariate Stochastic Volatility.” Journal of Business and Economic Statistics 28 (3): 438-451.; · Zbl 1209.91179
[35] Gouriéroux, C., A. Monfort, and E. Renault. 1993. “Indirect Inference.” Journal of Applied Econometrics 8: 85-118.; · Zbl 1448.62202
[36] Gouriéroux, C., E. Renault, and N. Touzi. 2000. “Calibration by Simulation for Small Sample Bias Correction.” In Simulation Based Inference in Econometrics,edited by Mariano, R., Schuerman, T. and M. Week, pp. 328-358, Cambridge, UK: Cambridge University Press.; · Zbl 1184.62048
[37] Gouriéroux, C., J. Jasiak, and R. Sufana. 2009. “The Wishart Autoregressive Process of Multivariate Stochastic Volatility.” Journal of Econometrics 150 (2): 167-181.; · Zbl 1429.62397
[38] Hall, B. C. 2003. Lie Groups, Lie Algebras, and Representations: An Elementary introduction. Graduate Texts in Mathematics 222, Berlin, Heidelberg, New York: Springer.; · Zbl 1026.22001
[39] Harvey, A., E. Ruiz, and N. Shephard. 1994. “Multivariate stochastic variance models.” Review of Economic Studies 61 (2): 247-264.; · Zbl 0805.90026
[40] Heston, S. 1993. “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” The Review of Financial Studies 6 (2): 327-343.; · Zbl 1384.35131
[41] Jackwerth, J, C. 2000. “Recovering Risk Aversion from Option Prices and Realized Returns.” Review of Financial Studies 13 (2): 433-451.;
[42] Jiang, G. J., and J. L. Knight. 2002. “Estimation of Continuous-Time Processes via the Empirical Characteristic Function.” Journal of Business & Economic Statistics, 20 (2): 198-212.;
[43] Lo, A. W. 1988. “Maximum Likelihood Estimation of Generalized Ito Processes with Discretely Sampled Data.” Econometric Theory 4: 231-247.;
[44] Meddahi, N., and E. Renault. 2004. “Temporal Aggregation of Volatility Models.” Journal of Econometrics 119: 355-379.; · Zbl 1282.91239
[45] Nelson, D. B. 1990. “ARCH Models as Diffusion Approximations.” Journal of Econometrics 45: 7-38.; · Zbl 0719.60089
[46] Nelson, D. B., and D. P. Foster. 1994. “Asymptotic Filtering Theory for Univariate Arch Models.” Econometrica (1): 1-41.; · Zbl 0804.62085
[47] Newey, W. K., and K. D. West. 1994. “Automatic lag Selection in Covariance Matrix Estimation.” Review of Economic Studies 61 (4): 631-653.; · Zbl 0815.62063
[48] Pastorello, S., E. Renault, and N. Touzi. 2000. “Statistical Inference for Random-Variance Option Pricing.” Journal of Business and Economic Statistics 18 (03): 358-367.;
[49] Piessens, R., E. de Doncker-Kapenga, C. Ueberhuber, and D. Kahaner. 1983. QUADPACK: A subroutine package for automatic integration. Berlin, Heidelberg, New York: Springer.; · Zbl 0508.65005
[50] Rivers, D., and Q. Vuong. 2002. “Model Selection Tests for Nonlinear Dynamic Models.” Econometrics Journal 5 (1): 1-39.; · Zbl 1010.62110
[51] Rockinger, M., and M. Semenova. 2005. “Estimation of Jump-Diffusion Process via Empirical Characteristic Function.” FAME Research Paper Series rp150, International Center for Financial Asset Management and Engineering.;
[52] Roll, R. 1988. “The International Crash of October, 1987.” Financial Analysts Journal (September-October): 19-35.;
[53] Singleton, K. 2001. “Estimation of Affine Pricing Models Using the Empirical Characteristic Function.” Journal of Econometrics (102): 111-141.; · Zbl 0973.62096
[54] Singleton, K. J. 2006. Empirical Dynamic Asset Pricing: Model Specification and Econometric Assessment. Princeton, New Jersey: Princeton University Press.; · Zbl 1094.91030
[55] Vuong, Q. 1989. “Likelihood Ratio Tests for Model Selection and non-nested Hypotheses,” Econometrica 57 (2): 307-333.; · Zbl 0701.62106
[56] Yu, J. 2004. “Empirical Characteristic Function Estimation and its Applications,” Econometric Reviews 23 (2): 93-123.; · Zbl 1123.62030
[57] Yu, J. 2005. “On Leverage in a Stochastic Volatility Model.” Journal of Econometrics 127 (2): 165-178.; · Zbl 1335.91116
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.