基于价格极差的GARCH模型

本文利用资产价格的极差序列,基于常规GARCH模型的框架,构造了一类关于波动率的新模型,即GARCH-R模型以及能够表达波动率变化非对称性特性的AGARCH-R模型。利用上证综合指数日收益率及相应的高频数据,通过比较不同模型对波动率以及VAR的预测效果,揭示了这种包含了极差信息的新的模型比传统的GARCH类模型的预测效果具有显著的优势。     

Bayesian modeling and forecasting of Value‐at‐Risk via threshold realized volatility

This study proposes a threshold realized generalized autoregressive conditional heteroscedastic (GARCH) model that jointly models daily returns and realized volatility, thereby taking into account the bias and asymmetry of realized volatility. We incorporate this threshold realized GARCH model with skew Student‐t innovations as the observation equation, view this model as a sharp transition model, and treat the realized volatility as a proxy for volatility under this nonlinear structure. Through the Bayesian Markov chain Monte Carlo method, the model can jointly estimate the parameters in the return equation, the volatility equation, and the measurement equation. As an illustration, we conduct a simulation study and apply the proposed method to the US and Japan stock markets. Based on quantile forecasting and volatility estimation, we find that the threshold heteroskedastic framework with realized volatility successfully models the asymmetric dynamic structure. We also investigate the predictive ability of volatility by comparing the proposed model with the traditional GARCH model as well as some popular asymmetric GARCH and realized GARCH models. This threshold realized GARCH model with skew Student‐t innovations outperforms the competing risk models in out‐of‐sample volatility and Value‐at‐Risk forecasting.     

Model selection of a switching mechanism for financial time series

The threshold autoregressive model with generalized autoregressive conditionally heteroskedastic (GARCH) specification is a popular nonlinear model that captures the well‐known asymmetric phenomena in financial market data. The switching mechanisms of hysteretic autoregressive GARCH models are different from threshold autoregressive model with GARCH as regime switching may be delayed when the hysteresis variable lies in a hysteresis zone. This paper conducts a Bayesian model comparison among competing models by designing an adaptive Markov chain Monte Carlo sampling scheme. We illustrate the performance of three kinds of criteria by comparing models with fat‐tailed and/or skewed errors: deviance information criteria, Bayesian predictive information, and an asymptotic version of Bayesian predictive information. A simulation study highlights the properties of the three Bayesian criteria and the accuracy as well as their favorable performance as model selection tools. We demonstrate the proposed method in an empirical study of 12 international stock markets, providing evidence to strongly support for both models with skew fat‐tailed innovations. Copyright © 2016 John Wiley & Sons, Ltd.     

基于Expectile和Realized GARCH模型的波动率预测

Realized GARCH模型是预测波动率的经典模型之一,最小化非对称二次损失函数的Expectile对收益率尾部分布更加敏感,我们在Realized GARCH模型的基础上引入Expectile提出Expectile-Realized GARCH模型。以沪深300指数的高频收益率为例建模分析,对比不同模型下的波动率预测效果,发现Expectile-Realized GARCH模型较Realized GARCH模型对波动率预测能力更好。其中,当风险水平为95%时,对应的Expectile-Realized GARCH波动率预测能力最好。     

Improving density forecast by modeling asymmetric features: An application to S&P500 returns

A density forecast is an estimate of the probability distribution of the possible future values of a random variable. From the current literature, an economic time series may have three types of asymmetry: asymmetry in unconditional distribution, asymmetry in conditional distribution, volatility asymmetry. In this paper, we propose three density forecasting methods under two-piece normal assumption to capture these asymmetric features. A GARCH model with two-piece normal distribution is developed to capture asymmetries in the conditional distributions. In this approach, we first estimate parameters of a GARCH model by assuming normal innovations, and then fit a two-piece normal distribution to the empirical residuals. Block bootstrap procedure, and moving average method with two-piece normal distribution are presented for volatility asymmetry and asymmetry in the conditional distributions. Application of the developed methods to the weekly S&P500 returns illustrates that forecast quality can be significantly improved by modeling these asymmetric features.     

Simulated likelihood inference for stochastic volatility models using continuous particle filtering

Discrete-time stochastic volatility (SV) models have generated a considerable literature in financial econometrics. However, carrying out inference for these models is a difficult task and often relies on carefully customized Markov chain Monte Carlo techniques. Our contribution here is twofold. First, we propose a new SV model, namely SV–GARCH, which bridges the gap between SV and GARCH models: it has the attractive feature of inheriting unconditional properties similar to the standard GARCH model but being conditionally heavier tailed. Second, we propose a likelihood-based inference technique for a large class of SV models relying on the recently introduced continuous particle filter. The approach is robust and simple to implement. The technique is applied to daily returns data for S&P 500 and Dow Jones stock price indices for various spans.     

V-uniform ergodicity of a continuous time asymmetric power GARCH(1,1) model

A continuous time asymmetric power GARCH(1,1) model is presented and the V-uniform ergodicity and β-mixing property of the process with exponential decay rate are proved. The V-uniform ergodicity of the COGARCH(1,1) model is obtained as a special case.     

A semiparametric Bayesian approach to the analysis of financial time series with applications to value at risk estimation

GARCH models are commonly used for describing, estimating and predicting the dynamics of financial returns. Here, we relax the usual parametric distributional assumptions of GARCH models and develop a Bayesian semiparametric approach based on modeling the innovations using the class of scale mixtures of Gaussian distributions with a Dirichlet process prior on the mixing distribution. The proposed specification allows for greater flexibility in capturing the usual patterns observed in financial returns. It is also shown how to undertake Bayesian prediction of the Value at Risk (VaR). The performance of the proposed semiparametric method is illustrated using simulated and real data from the Hang Seng Index (HSI) and Bombay Stock Exchange index (BSE30).     

Restricted normal mixture QMLE for non-stationary TGARCH(1, 1) models

The threshold GARCH(TGARCH)models have been very useful for analyzing asymmetric volatilities arising from financial time series.Most research on TGARCH has been directed to the stationary case.This paper studies the estimation of non-stationary first order TGARCH models.Restricted normal mixture quasi-maximum likelihood estimation(NM-QMLE)for non-stationary TGARCH models is proposed in the sense that we estimate the other parameters with any fixed location parameter.We show that the proposed estimators(except location parameter)are consistent and asymptotically normal under mild regular conditions.The impact of relative leptokursis and skewness of the innovations’distribution and quasi-likelihood distributions on the asymptotic efficiency has been discussed.Numerical results lend further support to our theoretical results.Finally,an illustrated real example is presented.     

Inference for 2-D GARCH models

The purpose of this paper is, in the first step, to consider a class of GMM estimators with interesting asymptotic properties and a reasonable number of computations for two dimensionally indexed Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. In the second step, we use the central limit theorem of Huang (1992) for spatial martingale differences to establish the LAN property for general two-dimensional discrete models on a regular grid with Gaussian errors. We then apply this result to the spatial GARCH model and derive the limit distribution of the maximum likelihood estimators of the parameters. Results of numerical simulations are presented.     

GARCH option pricing: A semiparametric approach

Option pricing based on GARCH models is typically obtained under the assumption that the random innovations are standard normal (normal GARCH models). However, these models fail to capture the skewness and the leptokurtosis in financial data. We propose a new method to compute option prices using a nonparametric density estimator for the distribution of the driving noise. We investigate the pricing performances of this approach using two different risk neutral measures: the Esscher transform pioneered by Gerber and Shiu [Gerber, H.U., Shiu, E.S.W., 1994a. Option pricing by Esscher transforms (with discussions). Trans. Soc. Actuar. 46, 99–91], and the extended Girsanov principle introduced by Elliot and Madan [Elliot, R.J., Madan, D.G., 1998. A discrete time equivalent martingale 9 measure. Math. Finance 8, 127–152]. Both measures are justified by economic arguments and are consistent with Duan’s [Duan, J.-C., 1995. The GARCH option pricing model. Math. Finance 5, 13–32] local risk neutral valuation relationship (LRNVR) for normal GARCH models. The main advantage of the two measures is that one can price derivatives using skewed or heavier tailed innovations distributions to model the returns. An empirical study regarding the European Call option valuation on S&P500 Index shows: (i) under both risk neutral measures our semiparametric algorithm performs better than the existing normal GARCH models if we allow for a leverage effect and (ii) the pricing errors when using the Esscher transform are quite small even though our estimation procedure is based only on historical return data.     

On linear processes with dependent innovations

We consider asymptotic behavior of partial sums and sample covariances for linear processes whose innovations are dependent. Central limit theorems and invariance principles are established under fairly mild conditions. Our results go beyond earlier ones by allowing a quite wide class of innovations which includes many important nonlinear time series models. Applications to linear processes with GARCH innovations and other nonlinear time series models are discussed.     

Bayesian tail‐risk forecasting using realized GARCH

A realized generalized autoregressive conditional heteroskedastic (GARCH) model is developed within a Bayesian framework for the purpose of forecasting value at risk and conditional value at risk. Student‐t and skewed‐t return distributions are combined with Gaussian and student‐t distributions in the measurement equation to forecast tail risk in eight international equity index markets over a 4‐year period. Three realized measures are considered within this framework. A Bayesian estimator is developed that compares favourably, in simulations, with maximum likelihood, both in estimation and forecasting. The realized GARCH models show a marked improvement compared with ordinary GARCH for both value‐at‐risk and conditional value‐at‐risk forecasting. This improvement is consistent across a variety of data and choice of distributions. Realized GARCH models incorporating a skewed student‐t distribution for returns are favoured overall, with the choice of measurement equation error distribution and realized measure being of lesser importance. Copyright © 2017 John Wiley & Sons, Ltd.     

ARCH族模型及其对深圳股票市场的实证分析

ARCH族模型是动态非线性的股票定价模型,它在金融和经济领域具有广阔的应用前景.在短短二十年时间里取得了迅速的发展,先后提出了GARCH、ARCH-M、TARCH、EGRACH等模型丰富了ARCH族模型.本文从ARCH族模型出发,简要介绍了其主要形式和特点,然后将ARCH族模型运用于我国深圳股票市场的实证分析,从实证结果中总结深圳股市的总体特征.     

Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero

The asymptotic distribution of the quasi-maximum likelihood (QML) estimator is established for generalized autoregressive conditional heteroskedastic (GARCH) processes, when the true parameter may have zero coefficients. This asymptotic distribution is the projection of a normal vector distribution onto a convex cone. The results are derived under mild conditions. For an important subclass of models, no moment condition is imposed on the GARCH process. The main practical implication of these results concerns the estimation of overidentified GARCH models.     

基于EGB2分布族的GAS-EGARCH模型与VaR预测

GARCH族模型是刻画资产收益率的常用工具,在风险度量领域具有广泛应用。为了更有效地描述收益率的偏斜厚尾等特征,越来越多学者对GARCH族模型的条件分布形式进行了研究。但是仅对GARCH模型条件分布进行修正是不够的,还需要对模型本身的函数形式进行修正。基于得分函数的时变参数建模思想近年来受到广泛关注,本文借助这一思想对EGARCH模型中对数标准差进行时变波动建模,并利用EGB2分布族作为模型的条件分布,进而建立GAS-EGARCH-EGB2模型。以我国10只中证行业指数为研究对象考察GAS-EGARCH-EGB2模型的风险预测效果,GAS-EGARCH-EGB2模型样本外VaR预测表现普遍优于ACM-EGARCH-EGB2模型。     

COGARCH as a continuous-time limit of GARCH(1,1)

COGARCH is an extension of the GARCH time series concept to continuous time, which has been suggested by Klüppelberg, Lindner and Maller [C. Klüppelberg, A. Lindner, R. Maller, A continuous-time GARCH process driven by a Lévy process: Stationarity and second order behaviour, Journal of Applied Probability 41 (2004) 601–622]. We show that any COGARCH process can be represented as the limit in law of a sequence of GARCH(1,1) processes. As a by-product we derive the infinitesimal generator of the bivariate Markov process representation of COGARCH. Moreover, we argue heuristically that COGARCH and the classical bivariate diffusion limit of Nelson [D. Nelson, ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7–38] are probably the only continuous-time limits of GARCH.     

Asymptotic theory for multivariate GARCH processes

We provide in this paper asymptotic theory for the multivariate GARCH(p,q) process. Strong consistency of the quasi-maximum likelihood estimator (MLE) is established by appealing to conditions given by Jeantheau (Econometric Theory14 (1998), 70) in conjunction with a result given by Boussama (Ergodicity, mixing and estimation in GARCH models, Ph.D. Dissertation, University of Paris 7, 1998) concerning the existence of a stationary and ergodic solution to the multivariate GARCH(p,q) process. We prove asymptotic normality of the quasi-MLE when the initial state is either stationary or fixed.     

成交量信息有助于预测中国股票市场的波动吗?

就成交量信息是否有助于预测股票市场的波动率这一问题,目前学术界有两种截然相反的观点存在。本文以中国股票市场代表性指数的代表性波动周期为例,对上述问题进行了实证研究。通过采用较以往研究更为严谨和稳健的样本外滚动时间窗预测法和高级预测能力检验法(Superiorpredictive ability,SPA),本文得到的分析结论包括:(1)成交量信息对中国股票市场的波动过程有显著影响;(2)将成交量纳入GARCH族模型会导致条件方差方程中的波动持续性出现明显下降;(3)引入成交量作为附加解释变量的GARCH族模型并未表现出比一般GARCH族模型更优的波动率预测能力。最后对实证结果给出了理论解释。     

Multivariate Generalized Autoregressive Conditional Heteroscedastic Model

1 IlltroductionThe concept of ARCH, which stands for autoregressit,e conditional heteroscedasticity wasfrist introduced by EngelI1J to handIe time series with a changing conditional tariance.Bollersle.I2] extended the ARCH model into the sChcalled generalized autoregressive con-ditional heteroscedastic model(GARCH). This class of models has important applitalions,particularly in finance and economics(see, e.g., [3], [4]). Lingl5] found some simple sufficientconditions fOr the strict st…