Efficient Bayesian inference in generalized inverse gamma processes for stochastic volatility

This paper develops a novel and efficient algorithm for Bayesian inference in inverse Gamma stochastic volatility models. It is shown that by conditioning on auxiliary variables, it is possible to sample all the volatilities jointly directly from their posterior conditional density, using simple and easy to draw from distributions. Furthermore, this paper develops a generalized inverse gamma process with more flexible tails in the distribution of volatilities, which still allows for simple and efficient calculations. Using several macroeconomic and financial datasets, it is shown that the inverse gamma and generalized inverse gamma processes can greatly outperform the commonly used log normal volatility processes with Student’s t errors or jumps in the mean equation.     

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.     

Markov switch smooth transition HYGARCH model: Stability and estimation

Abstract

HYGARCH model is basically used to model long-range dependence in volatility. We propose Markov switch smooth-transition HYGARCH model, where the volatility in each state is a time-dependent convex combination of GARCH and FIGARCH. This model provides a flexible structure to capture different levels of volatilities and also short and long memory effects. The necessary and sufficient condition for the asymptotic stability is derived. Forecast of conditional variance is studied by using all past information through a parsimonious way. Bayesian estimations based on Gibbs sampling are provided. A simulation study has been given to evaluate the estimations and model stability. The competitive performance of the proposed model is shown by comparing it with the HYGARCH and smooth-transition HYGARCH models for some period of the S&P500 and Dow Jones industrial average indices based on volatility and value-at-risk forecasts.     

Robust bootstrap forecast densities for GARCH returns and volatilities

Bootstrap procedures are useful to obtain forecast densities for both returns and volatilities in the context of generalized autoregressive conditional heteroscedasticity models. In this paper, we analyse the effect of additive outliers on the finite sample properties of these bootstrap densities and show that, when obtained using maximum likelihood estimates of the parameters and standard filters for the volatilities, they are badly affected with dramatic consequences on the estimation of Value-at-Risk. We propose constructing bootstrap densities for returns and volatilities using a robust parameter estimator based on variance targeting implemented together with an adequate modification of the volatility filter. We show that the performance of the proposed procedure is adequate when compared with available robust alternatives. The results are illustrated with both simulated and real data.     

Estimating the Stock/Portfolio Volatility and the Volatility of Volatility: A New Simple Method

We devise a convenient way to estimate stochastic volatility and its volatility. Our method is applicable to both cross-sectional and time series data, and both high-frequency and low-frequency data. Moreover, this method, when applied to cross-sectional data (a collection of risky assets, portfolio), provides a great simplification in the sense that estimating the volatility of the portfolio does not require an estimation of a volatility matrix (the volatilities of the individual assets in the portfolio and their correlations). Furthermore, there is no need to generate volatility data.     

Factor Multivariate Stochastic Volatility via Wishart Processes

This paper proposes a high dimensional factor multivariate stochastic volatility (MSV) model in which factor covariance matrices are driven by Wishart random processes. The framework allows for unrestricted specification of intertemporal sensitivities, which can capture the persistence in volatilities, kurtosis in returns, and correlation breakdowns and contagion effects in volatilities. The factor structure allows addressing high dimensional setups used in portfolio analysis and risk management, as well as modeling conditional means and conditional variances within the model framework. Owing to the complexity of the model, we perform inference using Markov chain Monte Carlo simulation from the posterior distribution. A simulation study is carried out to demonstrate the efficiency of the estimation algorithm. We illustrate our model on a data set that includes 88 individual equity returns and the two Fama–French size and value factors. With this application, we demonstrate the ability of the model to address high dimensional applications suitable for asset allocation, risk management, and asset pricing.     

Factor Multivariate Stochastic Volatility via Wishart Processes

This paper proposes a high dimensional factor multivariate stochastic volatility (MSV) model in which factor covariance matrices are driven by Wishart random processes. The framework allows for unrestricted specification of intertemporal sensitivities, which can capture the persistence in volatilities, kurtosis in returns, and correlation breakdowns and contagion effects in volatilities. The factor structure allows addressing high dimensional setups used in portfolio analysis and risk management, as well as modeling conditional means and conditional variances within the model framework. Owing to the complexity of the model, we perform inference using Markov chain Monte Carlo simulation from the posterior distribution. A simulation study is carried out to demonstrate the efficiency of the estimation algorithm. We illustrate our model on a data set that includes 88 individual equity returns and the two Fama-French size and value factors. With this application, we demonstrate the ability of the model to address high dimensional applications suitable for asset allocation, risk management, and asset pricing.     

Quantile forecasts for financial volatilities based on parametric and asymmetric models

For financial volatilities such as realized volatility and volatility index, a new parametric quantile forecast strategy is proposed, focusing on forecast interval and value at risk (VaR) forecast. This fully addresses asymmetries in 3 parts: mean, volatility and distribution. The asymmetries are addressed by the LHAR (leverage heterogeneous autoregressive) model of McAleer and Medeiros (2008) and Corsi and Reno (2009) for the mean part, by the EGARCH model for the volatility part, and by the skew-t distribution for the error distribution part. The method is applied to the realized volatilities and the volatility indexes of the US S&P 500 index, the US NASDAQ index, the Korea KOSPI index in which significant asymmetries are identified. Considerable out-of-sample forecast improvements of the forecast interval and VaR forecast are demonstrated for the volatilities: forecast intervals of volatilities have better coverages with shorter lengths and VaR forecasts of volatility indexes have better violations if asymmetries are properly addressed rather than ignored. The proposed parametric method reveals considerably better out-of-sample performance than the recently proposed semiparametric quantile regression approach of Zikes and Barunik (2016).     

Splines for financial volatility

Summary.  We propose a flexible generalized auto-regressive conditional heteroscedasticity type of model for the prediction of volatility in financial time series. The approach relies on the idea of using multivariate B -splines of lagged observations and volatilities. Estimation of such a B -spline basis expansion is constructed within the likelihood framework for non-Gaussian observations. As the dimension of the B -spline basis is large, i.e. many parameters, we use regularized and sparse model fitting with a boosting algorithm. Our method is computationally attractive and feasible for large dimensions. We demonstrate its strong predictive potential for financial volatility on simulated and real data, and also in comparison with other approaches, and we present some supporting asymptotic arguments.     

BL-GARCH models and asymmetries in volatility

In this paper the class of Bilinear GARCH (BL-GARCH) models is proposed. BL-GARCH models allow to capture asymmetries in the conditional variance of financial and economic time series by means of interactions between past shocks and volatilities. The availability of likelihood based inference is an attractive feature of BL-GARCH models. Under the assumption of conditional normality, the log-likelihood function can be maximized by means of an EM type algorithm. The main reason for using the EM algorithm is that it allows to obtain parameter estimates which naturally guarantee the positive definiteness of the conditional variance with no need for additional parameter constraints. We also derive a robust LM test statistic which can be used for model identification. Finally, the effectiveness of BL-GARCH models in capturing asymmetric volatility patterns in financial time series is assessed by means of an application to a time series of daily returns on the NASDAQ Composite stock market index.     

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     

Analytical and numerical studies on the second-order asymptotic expansion method for European option pricing under two-factor stochastic volatilities

The celebrated Black–Scholes model made the assumption of constant volatility but empirical studies on implied volatility and asset dynamics motivated the use of stochastic volatilities. Christoffersen in 2009 showed that multi-factor stochastic volatilities models capture the asset dynamics more realistically. Fouque in 2012 used it to price European options. In 2013, Chiarella and Ziveyi considered Christoffersen’s ideas and introduced an asset dynamics where the two volatilities of the Heston type act separately and independently on the asset price, and using Fourier transform for the asset price process and double Laplace transform for the two volatilities processes, solved a pricing problem for American options. This paper considers the Chiarella and Ziveyi model and parameterizes it so that the volatilities revert to the long-run-mean with reversion rates that mimic fast (for example daily) and slow (for example seasonal) random effects. Applying asymptotic expansion method presented by Fouque in 2012, we make an extensive and detailed derivation of the approximation prices for European options. We also present numerical studies on the behavior and accuracy of our first- and second-order asymptotic expansion formulas.     

Conditional Euro Area Sovereign Default Risk

We propose an empirical framework to assess the likelihood of joint and conditional sovereign default from observed CDS prices. Our model is based on a dynamic skewed-t distribution that captures all salient features of the data, including skewed and heavy-tailed changes in the price of CDS protection against sovereign default, as well as dynamic volatilities and correlations that ensure that uncertainty and risk dependence can increase in times of stress. We apply the framework to euro area sovereign CDS spreads during the euro area debt crisis. Our results reveal significant time-variation in distress dependence and spill-over effects for sovereign default risk. We investigate market perceptions of joint and conditional sovereign risk around announcements of Eurosystem asset purchases programs, and document a strong impact on joint risk.     

The Gaussian Mixture Dynamic Conditional Correlation Model: Parameter Estimation,Value at Risk Calculation,and Portfolio Selection

A multivariate generalized autoregressive conditional heteroscedasticity model with dynamic conditional correlations is proposed, in which the individual conditional volatilities follow exponential generalized autoregressive conditional heteroscedasticity models and the standardized innovations follow a mixture of Gaussian distributions. Inference on the model parameters and prediction of future volatilities are addressed by both maximum likelihood and Bayesian estimation methods. Estimation of the Value at Risk of a given portfolio and selection of optimal portfolios under the proposed specification are addressed. The good performance of the proposed methodology is illustrated via Monte Carlo experiments and the analysis of the daily closing prices of the Dow Jones and NASDAQ indexes.     

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.     

Inference for a Class of Stochastic Volatility Models Using Option and Spot Prices: Application of a Bivariate Kalman Filter

In this paper Bayesian methods are applied to a stochastic volatility model using both the prices of the asset and the prices of options written on the asset. Posterior densities for all model parameters, latent volatilities and the market price of volatility risk are produced via a Markov Chain Monte Carlo (MCMC) sampling algorithm. Candidate draws for the unobserved volatilities are obtained in blocks by applying the Kalman filter and simulation smoother to a linearization of a nonlinear state space representation of the model. Crucially, information from both the spot and option prices affects the draws via the specification of a bivariate measurement equation, with implied Black–Scholes volatilities used to proxy observed option prices in the candidate model. Alternative models nested within the Heston (1993) framework are ranked via posterior odds ratios, as well as via fit, predictive and hedging performance. The method is illustrated using Australian News Corporation spot and option price data.     

Forecasting energy futures volatility with threshold augmented heterogeneous autoregressive jump models

Abstract

This study forecasts the volatility of two energy futures markets (oil and gas), using high-frequency data. We, first, disentangle volatility into continuous volatility and jumps. Second, we apply wavelet analysis to study the relationship between volume and the volatility measures for different horizons. Third, we augment the heterogeneous autoregressive (HAR) model by nonlinearly including both jumps and volume. We then propose different empirical extensions of the HAR model. Our study shows that oil and gas volatilities nonlinearly depend on public information (jumps), private information (continuous volatility), and trading volume. Moreover, our threshold augmented HAR model with heterogeneous jumps and continuous volatility outperforms HAR model in forecasting volatility.     

Capturing the Spillover Effect With Multiplicative Error Models

Recent statistical models for the analysis of volatility in financial markets serve the purpose of incorporating the effect of other markets in their structure, in order to study the spillover or the contagion phenomena. Extending the Multiplicative Error Model we are able to capture these characteristics, under the assumption that the conditional mean of the volatility can be decomposed into the sum of one component representing the proper volatility of the time series analyzed, and other components, each representing the volatility transmitted from one other market. Each component follows a proper dynamics with elements that can be usefully interpreted. This particular decomposition allows to establish, each time, the contribution brought by each individual market to the global volatility of the market object of the analysis. We experiment this model with four stock indices.     

Nonparametric Estimation and Forecasting for Time-Varying Coefficient Realized Volatility Models

This article introduces a new specification for the heterogenous autoregressive (HAR) model for the realized volatility of S&P 500 index returns. In this modeling framework, the coefficients of the HAR are allowed to be time-varying with unspecified functional forms. The local linear method with the cross-validation (CV) bandwidth selection is applied to estimate the time-varying coefficient HAR (TVC-HAR) model, and a bootstrap method is used to construct the point-wise confidence bands for the coefficient functions. Furthermore, the asymptotic distribution of the proposed local linear estimators of the TVC-HAR model is established under some mild conditions. The results of the simulation study show that the local linear estimator with CV bandwidth selection has favorable finite sample properties. The outcomes of the conditional predictive ability test indicate that the proposed nonparametric TVC-HAR model outperforms the parametric HAR and its extension to HAR with jumps and/or GARCH in terms of multi-step out-of-sample forecasting, in particular in the post-2003 crisis and 2007 global financial crisis (GFC) periods, during which financial market volatilities were unduly high.     

Inference for a Class of Stochastic Volatility Models Using Option and Spot Prices: Application of a Bivariate Kalman Filter

In this paper Bayesian methods are applied to a stochastic volatility model using both the prices of the asset and the prices of options written on the asset. Posterior densities for all model parameters, latent volatilities and the market price of volatility risk are produced via a Markov Chain Monte Carlo (MCMC) sampling algorithm. Candidate draws for the unobserved volatilities are obtained in blocks by applying the Kalman filter and simulation smoother to a linearization of a nonlinear state space representation of the model. Crucially, information from both the spot and option prices affects the draws via the specification of a bivariate measurement equation, with implied Black-Scholes volatilities used to proxy observed option prices in the candidate model. Alternative models nested within the Heston (1993) framework are ranked via posterior odds ratios, as well as via fit, predictive and hedging performance. The method is illustrated using Australian News Corporation spot and option price data.