AN APPLICATION OF THE ECF METHOD AND NUMERICAL INTEGRATION IN ESTIMATION OF THE STOCHASTIC VOLATILITY MODELS

Gradimir Milovanovic, Biljana Popovic, Vladica Stojanovic

DOI Number
-
First page
295
Last page
311

Abstract


In this paper, the Empirical Characteristic Function (ECF) method is described in parameter estimations of the stochastic volatility (SV) models,  as well as the original thresholds modification (and a generalization) of these models, named the Split-SV model. The estimation procedure is based on minimization of the objective function which represents the double integral with respect to the some weight function $g:\mathbb R^2\rightarrow\mathbb R$. Some typical, exponential classes of the weight functions $g(u_1,u_2)$ are considered, as well as the different types of cubature formulas. Estimation procedures are realized by the original authors' codes written in statistical programming language ``R'', and the performances of the ECF method are examined, by statistical and numerical aspects. The numerical simulation of the obtained estimates is given, also. Finally, the standard SV model, and the Split-SV model as its alternative, are applied for fitting  the empirical data: the daily returns of the exchange rates of GBP and USD per euro, and the efficiency of their fitting is compared.

Keywords


SV models, ECF estimation, numerical integration

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References


bibitem{OrtPol} {sc A.S. Cvetkovi'c, G.V. Milovanovi'c}: textit{The Mathematica Package

``OrthogonalPolynomials''}. Facta Univ. Ser. Math. Inform. {bf 19} (2004), 17--36.

bibitem{Dan} {sc J. Dan'ielsson}: textit{Stochastic volatility in asset prices: estimation with simulated maximum likelihood}. J. Econometrics textbf{61} (1994), 375--400.

bibitem{Durhan}{sc G.,B. Durhan}: textit{SV mixture models with application to S&P 500 index}. Journal of Financial Economics textbf{85 (3)}

(2007), 822--856.

bibitem{Haeg_Pies} {sc A. Haegemans, R. Piessens}: textit{Construction of cubature formulas of degree eleven for symmetric planar regions, using orthogonal polynomials}. Numer. Math. {bf 25} (1976), 139--148.

bibitem{Harvey} {sc A. Harvey, E. Ruiz, N. Shephard}: textit{Multivariate

stochastic variance model}. The Review of Economic Studies {bf 61} (1994), 247--264.

bibitem{Kastner1} {sc G. Kastner}: textit{Stochvol: Efficient Bayesian Inference for Stochastic Volatility (SV)

Models}. R package version 0.6-0, http://CRAN.R-project.org/package=stochvol, 2013.

bibitem{Kastner} {sc G. Kastner, S. Fr"uhwirth-Schnatter}: textit{Ancillarity-sufficiency interweaving strategy (ASIS)

for boosting MCMC estimation of stochastic volatility models}. Computational Statistics and Data

Analysis {bf 76} (2014), 408--423.

bibitem{Knight&Yu} {sc J.,L. Knight, J. Yu}: textit{Empirical characteristic

function in time series estimation}. Econometric Theory {bf 18} (2002), 691--721.

bibitem{Mas_Mil2008} {sc G. Mastroianni, G.V. Milovanovi'c}: textit{Interpolation Processes -- Basic Theory and Applications}. Springer Monographs in Mathematics, Springer -- Verlag, Berlin -- Heidelberg, 2008.

bibitem{OrtPol2} {sc G.,V. Milovanovi'c, A.,S. Cvetkovi'c}: textit{Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type}. Math. Balkanica {bf 26} (2012), 169--184.

bibitem{Mor_Pat} {sc C.,R. Morrow, T.,N.,L. Patterson}: textit{Construction of algebraic cubature rules using polynomial ideal theory}. SIAM J. Numer. Anal. {bf15} (1978), 953--976.

bibitem{Split-ARCH1} {sc B.,v C. Popovi' c, V.,S. Stojanovi' c}: textit{Split-ARCH}. Pliska Stud. Math. Bulgar. textbf{17}

(2005), 201--220.

bibitem{Split-SV1} {sc B.,v C. Popovi' c, V.,S. Stojanovi' c}:

textit{The Distribution of Split-SV(1) Model}. Proceeding of the 2th International Conference MIT, 335--340, 2011.

bibitem{Rab} {sc Ph. Rabinowitz, N. Richter}:

textit{Perfectly symmetric two-dimensional integration formulas with minimal number of points}. Math. Comp. textbf{23} (1969), 765--779.

bibitem{Ruiz} {sc E. Ruiz}: textit{Quasi-maximum likelihood estimation of stochastic

volatility models}. J. Econometrics textbf{63(1)} (1994), 289--306.

bibitem{So&Li} {sc M. So, W. Li}: textit{A threshold stochastic volatility model}. J. Forecast. textbf{21} (2002), 473--500.

bibitem{Split-ARCH2} {sc V.,S. Stojanovi' c, B.,v C. Popovi' c}:

textit{Estimation in real data set by Split-ARCH model}. Filomat {bf 21(2)} (2007), 133--152.

bibitem{Split-SV2} textsc{V.S. Stojanovi' c, B.v C. Popovi' c, G.V. Milovanovi'c}: textit{The Split-SV Model}. Comput. Statist. Data Anal. (2014) doi:

1016/j.csda.2014.08.010 (to appear).

bibitem{Split-BREAK} {sc V. Stojanovi' c, B.,v C Popovi' c, P.,M. Popovi' c}: textit{The Split-BREAK model}. Braz. J. Probab. Stat. {bf 25(1)} (2011), 44--63.

bibitem{GSB} {sc V. Stojanovi' c, B.,v C. Popovi' c, P.,M. Popovi' c}:

textit{Stochastic analysis of GSB process}. Publ. Inst. Math. (Beograd) (N.S.) {bf 95(109)} (2014), 149--159.

bibitem{Str_Sec} {sc A.,H. Stroud, D. Secrest}: textit{Approximate integration formulas for certain spherically symmetric regions}. Math. Comp. {bf17} (1963), 105--135.

bibitem{Taylor} {sc S.,J. Taylor}: textit{Modelling Financial Time Series}. John Wiley & Sons, Chichester, 1986.

bibitem{Tsiotas} {sc G. Tsiotas}: textit{On generalised asymmetric stochastic volatility models}. Comput. Statist. Data Anal. textbf{56}

(2012), 151--172.

bibitem{Yu-PhD} {sc J. Yu}: textit{Empirical Characteristic Function in Time Series

Estimation and a Test Statistic in Financial

Modelling}. PhD Thesis, The University of Western Ontario, 1998.

bibitem{Yu} {sc J. Yu}: textit{Empirical characteristic function estimation

and its applications}. Econometric Rev. {bf 23(2)} (2004), 93--123.


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