Fitting stochastic volatility models in the presence of irregular sampling via particle methods and the EM algorithm.

*(English)*Zbl 1199.62042The stochastic volatility model proposed by S.J. Taylor [Financial returns modelled by the product of two stochastic processes – a study of daily sugar prices, 1961-1979. O.D. Anderson (ed.), Time Ser. Anal.: Theory and Practice, Vol. 1, NYk: Elsevier, Amsterdam: North-Holland Publ., 203–226 (1982)] has become popular for explaining the behaviour of financial time series, such as stock prices and exchange rates caused by the existence of irregular sampling. Stretches of data may not be available for various reasons, such as unexpected political events or natural or man-made disasters. In addition, financial data often have heavier tails than can be captured by standard stochastic volatility models. In the standard stochastic volatility model framework volatility is modelled as an AR process \(h_t=\theta h_{t-1}+w_t\), and the returns \(r_t\) are given by \(r_t=\beta\exp(h_t/2)\epsilon_t\), where \(w_t\) are i.i.d. \(N(0,Q)\), \(h_0\) is \(N(\mu_0,\sigma_0^2)\), \(\epsilon_t\) are i.i.d. \(N(0,1)\), and \(w_t\) and \(\epsilon_t\) are independent processes. The last equation may be represented as \(y_t=\alpha+h_t+v_t\), where
\[
y_t = \log(r_y^2),\;\alpha =\log(\beta^2)+E(\log(\epsilon_t^2)),\;v_t=\log(\epsilon_t^2)-E(\log(\epsilon_t^2)).
\]
Note that \(\epsilon_t^2\sim\chi_1^2\), so that \(v_t\) has a centred \(\log\chi_1^2\) distribution. The first and the third equations represent the standard univariate stochastic volatility model and together they form a linear non-Gaussian state-space model. The authors consider a modification of the standard stochastic volatility model where it is assumed that the observational noise process, \(v_t\), is a mixture of two normals with unknown parameters. The model presented by R.H. Shumway and D.S. Stoffer [Time series analysis and its applicartions. With R examples. 2 nd ed., NY: Springer (2006; Zbl 1096.62088)] retains the state equation for the volatility but the observations are changed to \(y_t=\alpha +h_t+v_t\) with \(v_t=I_tz_{t1}+(1-I_t)z_{t0}-\mu\pi\), where \(z_{t0}\) are i.i.d \(N(0,R_0)\), \(z_{t1}\) are i.i.d. \(N(\mu,R_1)\), \(I_t\) are indicator variables i.i.d. \(B(\pi)\), where \(\pi\) is an unknown mixing probability. The observational noise is constructed using two normals: the \(N(0,R_0)\) term is used to account for most of the noise, whereas the \(N(\mu,R_1)\) term is used to account for the lower tail behaviour of the noise.

The main advantage of the mixture stochastic volatility model over the standard stochastic volatility model is its flexibility. In the modified model \(v_t\) has its own parameter set which must be estimated along with the state parameters \(\{\theta,Q\}\). Therefore this modification can give better parameter estimates for \(\{\theta,Q\}\). For model fitting in the presence of missing or irregular observations the authors combine the expectation-maximization (EM) algorithm with particle filters and smoothers to estimate the parameters of the model. The convergence of the method is discussed. To solve the irregular data problem an imputation technique based on properties of state-space models is used. To demonstrate the viability of the proposed methods several numerical exercises and data analyses are performed.

The main advantage of the mixture stochastic volatility model over the standard stochastic volatility model is its flexibility. In the modified model \(v_t\) has its own parameter set which must be estimated along with the state parameters \(\{\theta,Q\}\). Therefore this modification can give better parameter estimates for \(\{\theta,Q\}\). For model fitting in the presence of missing or irregular observations the authors combine the expectation-maximization (EM) algorithm with particle filters and smoothers to estimate the parameters of the model. The convergence of the method is discussed. To solve the irregular data problem an imputation technique based on properties of state-space models is used. To demonstrate the viability of the proposed methods several numerical exercises and data analyses are performed.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

##### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62M20 | Inference from stochastic processes and prediction |

91G70 | Statistical methods; risk measures |

65C60 | Computational problems in statistics (MSC2010) |

##### Keywords:

financial time series; missing data; mixtures of normals; particle filtering; particle smoothing; state-space model
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\textit{J. Kim} and \textit{D. S. Stoffer}, J. Time Ser. Anal. 29, No. 5, 811--833 (2008; Zbl 1199.62042)

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