Deciding between GARCH and stochastic volatility via strong decision rules

The GARCH and stochastic volatility (SV) models are two competing, well-known and often used models to explain the volatility of financial series. In this paper, we consider a closed form estimator for a stochastic volatility model and derive its asymptotic properties. We confirm our theoretical results by a simulation study. In addition, we propose a set of simple, strongly consistent decision rules to compare the ability of the GARCH and the SV model to fit the characteristic features observed in high frequency financial data such as high kurtosis and slowly decaying autocorrelation function of the squared observations. These rules are based on a number of moment conditions that is allowed to increase with sample size. We show that our selection procedure leads to choosing the model that fits best, or the simplest model under equivalence, with probability one as the sample size increases. The finite sample size behavior of our procedure is analyzed via simulations. Finally, we provide an application to stocks in the Dow Jones industrial average index.     

Common Drifting Volatility in Large Bayesian VARs

The general pattern of estimated volatilities of macroeconomic and financial variables is often broadly similar. We propose two models in which conditional volatilities feature comovement and study them using U.S. macroeconomic data. The first model specifies the conditional volatilities as driven by a single common unobserved factor, plus an idiosyncratic component. We label this model BVAR with general factor stochastic volatility (BVAR-GFSV) and we show that the loss in terms of marginal likelihood from assuming a common factor for volatility is moderate. The second model, which we label BVAR with common stochastic volatility (BVAR-CSV), is a special case of the BVAR-GFSV in which the idiosyncratic component is eliminated and the loadings to the factor are set to 1 for all the conditional volatilities. Such restrictions permit a convenient Kronecker structure for the posterior variance of the VAR coefficients, which in turn permits estimating the model even with large datasets. While perhaps misspecified, the BVAR-CSV model is strongly supported by the data when compared against standard homoscedastic BVARs, and it can produce relatively good point and density forecasts by taking advantage of the information contained in large datasets.     

Common-factor stochastic volatility modelling with observable proxy

Multi-asset modelling is of fundamental importance to financial applications such as risk management and portfolio selection. In this article, we propose a multivariate stochastic volatility modelling framework with a parsimonious and interpretable correlation structure. Building on well-established evidence of common volatility factors among individual assets, we consider a multivariate diffusion process with a common-factor structure in the volatility innovations. Upon substituting an observable market proxy for the common volatility factor, we markedly improve the estimation of several model parameters and latent volatilities. The model is applied to a portfolio of several important constituents of the S&P500 in the financial sector, with the VIX index as the common-factor proxy. We find that the prediction intervals for asset forecasts are comparable to those of more complex dependence models, but that option-pricing uncertainty can be greatly reduced by adopting a common-volatility structure. The Canadian Journal of Statistics 48: 36–61; 2020 © 2020 Statistical Society of Canada     

FORECASTING VOLATILITY IN THE PRESENCE OF MODEL INSTABILITY

Recent advances in financial econometrics have allowed for the construction of efficient ex post measures of daily volatility. This paper investigates the importance of instability in models of realised volatility and their corresponding forecasts. Testing for model instability is conducted with a subsampling method. We show that removing structurally unstable data of a short duration has a negligible impact on the accuracy of conditional mean forecasts of volatility. In contrast, it does provide a substantial improvement in a model's forecast density of volatility. In addition, the forecasting performance improves, often dramatically, when we evaluate models on structurally stable data.     

Tree-structured generalized autoregressive conditional heteroscedastic models

We propose a new generalized autoregressive conditional heteroscedastic (GARCH) model with tree-structured multiple thresholds for the estimation of volatility in financial time series. The approach relies on the idea of a binary tree where every terminal node parameterizes a (local) GARCH model for a partition cell of the predictor space. The fitting of such trees is constructed within the likelihood framework for non-Gaussian observations: it is very different from the well-known regression tree procedure which is based on residual sums of squares. Our strategy includes the classical GARCH model as a special case and allows us to increase model complexity in a systematic and flexible way. We derive a consistency result and conclude from simulation and real data analysis that the new method has better predictive potential than other approaches.     

基于高维波动率网络模型的股票市场风险特征研究

波动率是金融风险管理研究的重要内容之一。本文基于复杂网络理论和数据挖掘技术提出股票市场的高维波动率网络模型。首先运用互信息度量不同股票价格波动之间的相关关系，其次对股票市场不同周期下的波动情况建立度的中心势、平均距离、幂律分布等网络拓扑指标，再次根据这些指标利用Prim算法构建出高维波动率网络模型，最后运用Newman-Girvan算法对股票价格波动率的相关性进行分层研究。高维波动率网络模型突破了传统波动率模型关于变量维数的限制，能够在依赖少量假设的基础上，挖掘出多个金融市场主体间的相互关系，反映金融市场的风险特征及网络拓扑性质。实证结果发现：与常用的Pearson相关系数法相比，在互信息框架下，股价波动的非线性相关关系得到了更好的度量；股票市场的整体波动性与个股波动率相关性变化趋势相反，市场处在高波动时期资产组合分散化效果较好；网络中存在少量度数大的关键节点和中心节点，风险通过这些节点可以迅速传递到整个市场；股票市场的运行具有明显的行业聚集现象；网络分层研究进一步直观的展现了风险在层与层之间的传递规律和与之对应的行业特征。高维波动率网络模型为挖掘股票市场的风险特征与管理金融风险提供了一个新的工具。     

基于Markov区制转换模型的极值风险度量研究

将马尔科夫区制转换模型与极值理论相结合研究金融风险度量问题.首先用SWARCH-t模型捕捉收益率序列的剧烈波动和结构变换特征,然后将收益序列转化为标准残差序列,在此基础上通过SWARCH-t模型与极值理论相结合拟合标准残差的尾部分布,进而构建基于SWARCH- t- EVT的动态VaR模型,最后对模型的有效性进行检验.研究表明,SWARCH-t-EVT模型能够有效识别上证综指的波动区制特征,且能有效合理地测度上证综指收益风险,尤其在高的置信水平下表现更好.     

HEAVY-TAILED-DISTRIBUTED THRESHOLD STOCHASTIC VOLATILITY MODELS IN FINANCIAL TIME SERIES

To capture mean and variance asymmetries and time‐varying volatility in financial time series, we generalize the threshold stochastic volatility (THSV) model and incorporate a heavy‐tailed error distribution. Unlike existing stochastic volatility models, this model simultaneously accounts for uncertainty in the unobserved threshold value and in the time‐delay parameter. Self‐exciting and exogenous threshold variables are considered to investigate the impact of a number of market news variables on volatility changes. Adopting a Bayesian approach, we use Markov chain Monte Carlo methods to estimate all unknown parameters and latent variables. A simulation experiment demonstrates good estimation performance for reasonable sample sizes. In a study of two international financial market indices, we consider two variants of the generalized THSV model, with US market news as the threshold variable. Finally, we compare models using Bayesian forecasting in a value‐at‐risk (VaR) study. The results show that our proposed model can generate more accurate VaR forecasts than can standard models.     

New HEAVY Models for Fat-Tailed Realized Covariances and Returns

ABSTRACT

We develop a new score-driven model for the joint dynamics of fat-tailed realized covariance matrix observations and daily returns. The score dynamics for the unobserved true covariance matrix are robust to outliers and incidental large observations in both types of data by assuming a matrix-F distribution for the realized covariance measures and a multivariate Student's t distribution for the daily returns. The filter for the unknown covariance matrix has a computationally efficient matrix formulation, which proves beneficial for estimation and simulation purposes. We formulate parameter restrictions for stationarity and positive definiteness. Our simulation study shows that the new model is able to deal with high-dimensional settings (50 or more) and captures unobserved volatility dynamics even if the model is misspecified. We provide an empirical application to daily equity returns and realized covariance matrices up to 30 dimensions. The model statistically and economically outperforms competing multivariate volatility models out-of-sample. Supplementary materials for this article are available online.     

Analysis and modelling of wind speed in New York

In this paper we propose an ARMA time-series model for the wind speed at a single spatial location, and estimate it on in-sample data recorded in three different wind farm regions in New York state. The data have a three-hour granularity, but based on applications to financial wind derivatives contracts, we also consider daily average wind speeds. We demonstrate that there are large discrepancies in the behaviour of daily average and three-hourly wind speed records. The validation procedure based on out-of-sample observations reflects that the proposed model is reliable and can be used for various practical applications, like, for instance, weather prediction, pricing of financial wind contracts, wind generated power, etc. Furthermore, we discuss some striking resemblances with temperature dynamics.     

Bayesian Dynamic Factor Models and Portfolio Allocation

We discuss the development of dynamic factor models for multivariate financial time series, and the incorporation of stochastic volatility components for latent factor processes. Bayesian inference and computation is developed and explored in a study of the dynamic factor structure of daily spot exchange rates for a selection of international currencies. The models are direct generalizations of univariate stochastic volatility models and represent specific varieties of models recently discussed in the growing multivariate stochastic volatility literature. We discuss model fitting based on retrospective data and sequential analysis for forward filtering and short-term forecasting. Analyses are compared with results from the much simpler method of dynamic variance-matrix discounting that, for over a decade, has been a standard approach in applied financial econometrics. We study these models in analysis, forecasting, and sequential portfolio allocation for a selected set of international exchange-rate-return time series. Our goals are to understand a range of modeling questions arising in using these factor models and to explore empirical performance in portfolio construction relative to discount approaches. We report on our experiences and conclude with comments about the practical utility of structured factor models and on future potential model extensions.     

Multivariate Markov switching dynamic conditional correlation GARCH representations for contagion analysis

This paper provides an extension of the Dynamic Conditional Correlation model of Engle (2002) by allowing both the unconditional correlation and the parameters to be driven by an unobservable Markov chain. We provide the estimation algorithm and perform an empirical analysis of the contagion phenomenon in which our model is compared to the traditional CCC and DCC representations. We acknowledge financial support from the Italian national research project on "The Euro and European financial market volatility: contagion, interdependence and volatility transmission" financed by the Italian Ministry of University and Research. Furthermore, we thank William De Pieri for research assistance and are grateful to Loriana Pelizzon, Claudio Pizzi, Domenico Sartore and the participants at the Forecasting Financial Markets 2004 conference and at the XLII Annual Meeting of the Italian Statistical Society for helpful comments. Usual disclaimer applies. Correspondence to: Monica Bilio     

Markov switching component GARCH model: Stability and forecasting

ABSTRACT

This paper introduces an extension of the Markov switching GARCH model where the volatility in each state is a convex combination of two different GARCH components with time varying weights. This model has the dynamic behavior to capture the variants of shocks. The asymptotic behavior of the second moment is investigated and an appropriate upper bound for it is evaluated. Using the Bayesian method via Gibbs sampling algorithm, a dynamic method for the estimation of the parameters is proposed. Finally, we illustrate the efficiency of the model by simulation and also by considering two different set of empirical financial data. We show that this model provides much better forecasts of the volatility than the Markov switching GARCH model.     

On double hysteretic heteroskedastic model

ABSTRACT

This paper proposes a hysteretic autoregressive model with GARCH specification and a skew Student's t-error distribution for financial time series. With an integrated hysteresis zone, this model allows both the conditional mean and conditional volatility switching in a regime to be delayed when the hysteresis variable lies in a hysteresis zone. We perform Bayesian estimation via an adaptive Markov Chain Monte Carlo sampling scheme. The proposed Bayesian method allows simultaneous inferences for all unknown parameters, including threshold values and a delay parameter. To implement model selection, we propose a numerical approximation of the marginal likelihoods to posterior odds. The proposed methodology is illustrated using simulation studies and two major Asia stock basis series. We conduct a model comparison for variant hysteresis and threshold GARCH models based on the posterior odds ratios, finding strong evidence of the hysteretic effect and some asymmetric heavy-tailness. Versus multi-regime threshold GARCH models, this new collection of models is more suitable to describe real data sets. Finally, we employ Bayesian forecasting methods in a Value-at-Risk study of the return series.     

Econometric analysis of realized volatility and its use in estimating stochastic volatility models

Summary. The availability of intraday data on the prices of speculative assets means that we can use quadratic variation-like measures of activity in financial markets, called realized volatility, to study the stochastic properties of returns. Here, under the assumption of a rather general stochastic volatility model, we derive the moments and the asymptotic distribution of the realized volatility error—the difference between realized volatility and the discretized integrated volatility (which we call actual volatility). These properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation-intensive methods.     

Derivation of Kurtosis and Option Pricing Formulas for Popular Volatility Models with Applications in Finance

This article discusses some topics relevant to financial modeling. The kurtosis of a distribution plays an important role in controlling tail-behavior and is used in edgeworth expansion of the call prices. We present derivations of the kurtosis for a number of popular volatility models useful in financial applications, including the class of random coefficient GARCH models. Option pricing formulas for various classes of volatility models are also derived and a simple proof of the option pricing formula under the Black–Scholes model is given.     

混合贝塔分布随机波动模型及其贝叶斯分析

为了更准确地揭示金融资产收益率数据的真实数据生成过程,提出了基于混合贝塔分布的随机波动模型,讨论了混合贝塔分布随机波动模型的贝叶斯估计方法,并给出了一种Gibbs抽样算法。以上证A股综指简单收益率为例,分别建立了基于正态分布和混合贝塔分布的随机波动模型,研究表明,基于混合贝塔分布的随机波动模型更准确地描述了样本数据的真实数据生成过程,而正态分布的随机波动模型将高峰厚尾等现象归结为波动冲击,从而低估了收益率的平均波动水平,高估了波动的持续性和波动的冲击扰动。     

Bayesian analysis of stochastic volatility models with flexible tails

An alternative distributional assumption is proposed for the stochastic volatility model. This results in extremely flexible tail behaviour of the sampling distribution for the observables, as well as in the availability of a simple Markov Chain Monte Carlo strategy for posterior analysis. By allowing the tail behaviour to be determined by a separate parameter, we reserve the parameters of the volatility process to dictate the degree of volatility clustering. Treatment of a mean function is formally integrated in the analysis.

Some empirical examples on both stock prices and exchange rates clearly indicate the presence of fat tails, in combination with high levels of volatility clustering. In addition, predictive distributions indicate a good fit with these typical financial data sets.     

Bayesian analysis of stochastic volatility models with flexible tails

An alternative distributional assumption is proposed for the stochastic volatility model. This results in extremely flexible tail behaviour of the sampling distribution for the observables, as well as in the availability of a simple Markov Chain Monte Carlo strategy for posterior analysis. By allowing the tail behaviour to be determined by a separate parameter, we reserve the parameters of the volatility process to dictate the degree of volatility clustering. Treatment of a mean function is formally integrated in the analysis.

Some empirical examples on both stock prices and exchange rates clearly indicate the presence of fat tails, in combination with high levels of volatility clustering. In addition, predictive distributions indicate a good fit with these typical financial data sets.     

Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics

Non-Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory.