Continuous Time Wishart Process for Stochastic Risk

Risks are usually represented and measured by volatility-covolatility matrices. Wishart processes are models for a dynamic analysis of multivariate risk and describe the evolution of stochastic volatility-covolatility matrices, constrained to be symmetric positive definite. The autoregressive Wishart process (WAR) is the multivariate extension of the Cox, Ingersoll, Ross (CIR) process introduced for scalar stochastic volatility. As a CIR process it allows for closed-form solutions for a number of financial problems, such as term structure of T-bonds and corporate bonds, derivative pricing in a multivariate stochastic volatility model, and the structural model for credit risk. Moreover, the Wishart dynamics are very flexible and are serious competitors for less structural multivariate ARCH models.     

Modeling and forecasting realized covariance matrices with accounting for leverage

The existing dynamic models for realized covariance matrices do not account for an asymmetry with respect to price directions. We modify the recently proposed conditional autoregressive Wishart (CAW) model to allow for the leverage effect. In the conditional threshold autoregressive Wishart (CTAW) model and its variations the parameters governing each asset's volatility and covolatility dynamics are subject to switches that depend on signs of previous asset returns or previous market returns. We evaluate the predictive ability of the CTAW model and its restricted and extended specifications from both statistical and economic points of view. We find strong evidence that many CTAW specifications have a better in-sample fit and tend to have a better out-of-sample predictive ability than the original CAW model and its modifications.     

Derivative Pricing With Wishart Multivariate Stochastic Volatility

This paper deals with the pricing of derivatives written on several underlying assets or factors satisfying a multivariate model with Wishart stochastic volatility matrix. This multivariate stochastic volatility model leads to a closed-form solution for the conditional Laplace transform, and quasi-explicit solutions for derivative prices written on more than one asset or underlying factor. Two examples are presented: (i) a multiasset extension of the stochastic volatility model introduced by Heston (1993), and (ii) a model for credit risk analysis that extends the model of Merton (1974) to a framework with stochastic firm liability, stochastic volatility, and several firms. A bivariate version of the stochastic volatility model is estimated using stock prices and moment conditions derived from the joint unconditional Laplace transform of the stock returns.     

A fractionally integrated Wishart stochastic volatility model

There has recently been growing interest in modeling and estimating alternative continuous time multivariate stochastic volatility models. We propose a continuous time fractionally integrated Wishart stochastic volatility (FIWSV) process, and derive the conditional Laplace transform of the FIWSV model in order to obtain a closed form expression of moments. A two-step procedure is used, namely estimating the parameter of fractional integration via the local Whittle estimator in the first step, and estimating the remaining parameters via the generalized method of moments in the second step. Monte Carlo results for the procedure show a reasonable performance in finite samples. The empirical results for the S&P 500 and FTSE 100 indexes show that the data favor the new FIWSV process rather than the one-factor and two-factor models of the Wishart autoregressive process for the covariance structure.     

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.     

Multivariate Stochastic Volatility: A Review

The literature on multivariate stochastic volatility (MSV) models has developed significantly over the last few years. This paper reviews the substantial literature on specification, estimation, and evaluation of MSV models. A wide range of MSV models is presented according to various categories, namely, (i) asymmetric models, (ii) factor models, (iii) time-varying correlation models, and (iv) alternative MSV specifications, including models based on the matrix exponential transformation, the Cholesky decomposition, and the Wishart autoregressive process. Alternative methods of estimation, including quasi-maximum likelihood, simulated maximum likelihood, and Markov chain Monte Carlo methods, are discussed and compared. Various methods of diagnostic checking and model comparison are also reviewed.     

Multivariate Stochastic Volatility: A Review

The literature on multivariate stochastic volatility (MSV) models has developed significantly over the last few years. This paper reviews the substantial literature on specification, estimation, and evaluation of MSV models. A wide range of MSV models is presented according to various categories, namely, (i) asymmetric models, (ii) factor models, (iii) time-varying correlation models, and (iv) alternative MSV specifications, including models based on the matrix exponential transformation, the Cholesky decomposition, and the Wishart autoregressive process. Alternative methods of estimation, including quasi-maximum likelihood, simulated maximum likelihood, and Markov chain Monte Carlo methods, are discussed and compared. Various methods of diagnostic checking and model comparison are also reviewed.     

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.     

Multivariate stochastic volatility with Bayesian dynamic linear models

This paper develops a Bayesian procedure for estimation and forecasting of the volatility of multivariate time series. The foundation of this work is the matrix-variate dynamic linear model, for the volatility of which we adopt a multiplicative stochastic evolution, using Wishart and singular multivariate beta distributions. A diagonal matrix of discount factors is employed in order to discount the variances element by element and therefore allowing a flexible and pragmatic variance modelling approach. Diagnostic tests and sequential model monitoring are discussed in some detail. The proposed estimation theory is applied to a four-dimensional time series, comprising spot prices of aluminium, copper, lead and zinc of the London metal exchange. The empirical findings suggest that the proposed Bayesian procedure can be effectively applied to financial data, overcoming many of the disadvantages of existing volatility models.     

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     

Multivariate Stochastic Volatility via Wishart Processes: A Comment

This comment refers to an error in the methodology for estimating the parameters of the model developed by Philipov and Glickman for modeling multivariate stochastic volatility via Wishart processes. For estimation they used Bayesian techniques. The derived expressions for the full conditionals of the model parameters as well as the expression for the acceptance ratio of the covariance matrix are erroneous. In this erratum all necessary formulae are given to guarantee an appropriate implementation and application of the model.     

Goodness-of-fit tests for centralized Wishart processes

Abstract

In this paper we present several goodness-of-fit tests for the centralized Wishart process, a popular matrix-variate time series model used to capture the stochastic properties of realized covariance matrices. The new test procedures are based on the extended Bartlett decomposition derived from the properties of the Wishart distribution and allows to obtain sets of independently and standard normally distributed random variables under the null hypothesis. Several tests for normality and independence are then applied to these variables in order to support or to reject the underlying assumption of a centralized Wishart process. In order to investigate the influence of estimated parameters on the suggested testing procedures in the finite-sample case, a simulation study is conducted. Finally, the new test methods are applied to real data consisting of realized covariance matrices computed for the returns on six assets traded on the New York Stock Exchange.     

Discussion of “Nonparametric Bayesian Inference in Applications”: Bayesian nonparametric methods in econometrics

The use of Bayesian nonparametrics models has increased rapidly over the last few decades driven by increasing computational power and the development of efficient Markov chain Monte Carlo algorithms. We review some applications of these models in economic applications including: volatility modelling (using both stochastic volatility models and GARCH-type models) with Dirichlet process mixture models, uses in portfolio allocation problems, long memory models with flexible forms of time-dependence, flexible extension of the dynamic Nelson-Siegel model for interest rate yields and multivariate time series models used in macroeconometrics.     

Monte Carlo Likelihood Estimation for Three Multivariate Stochastic Volatility Models

Estimating parameters in a stochastic volatility (SV) model is a challenging task. Among other estimation methods and approaches, efficient simulation methods based on importance sampling have been developed for the Monte Carlo maximum likelihood estimation of univariate SV models. This paper shows that importance sampling methods can be used in a general multivariate SV setting. The sampling methods are computationally efficient. To illustrate the versatility of this approach, three different multivariate stochastic volatility models are estimated for a standard data set. The empirical results are compared to those from earlier studies in the literature. Monte Carlo simulation experiments, based on parameter estimates from the standard data set, are used to show the effectiveness of the importance sampling methods.     

Multivariate Stochastic Volatility Model With Realized Volatilities and Pairwise Realized Correlations

Abstract

Although stochastic volatility and GARCH (generalized autoregressive conditional heteroscedasticity) models have successfully described the volatility dynamics of univariate asset returns, extending them to the multivariate models with dynamic correlations has been difficult due to several major problems. First, there are too many parameters to estimate if available data are only daily returns, which results in unstable estimates. One solution to this problem is to incorporate additional observations based on intraday asset returns, such as realized covariances. Second, since multivariate asset returns are not synchronously traded, we have to use the largest time intervals such that all asset returns are observed to compute the realized covariance matrices. However, in this study, we fail to make full use of the available intraday informations when there are less frequently traded assets. Third, it is not straightforward to guarantee that the estimated (and the realized) covariance matrices are positive definite.

Our contributions are the following: (1) we obtain the stable parameter estimates for the dynamic correlation models using the realized measures, (2) we make full use of intraday informations by using pairwise realized correlations, (3) the covariance matrices are guaranteed to be positive definite, (4) we avoid the arbitrariness of the ordering of asset returns, (5) we propose the flexible correlation structure model (e.g., such as setting some correlations to be zero if necessary), and (6) the parsimonious specification for the leverage effect is proposed. Our proposed models are applied to the daily returns of nine U.S. stocks with their realized volatilities and pairwise realized correlations and are shown to outperform the existing models with respect to portfolio performances.     

Monte Carlo Likelihood Estimation for Three Multivariate Stochastic Volatility Models

Estimating parameters in a stochastic volatility (SV) model is a challenging task. Among other estimation methods and approaches, efficient simulation methods based on importance sampling have been developed for the Monte Carlo maximum likelihood estimation of univariate SV models. This paper shows that importance sampling methods can be used in a general multivariate SV setting. The sampling methods are computationally efficient. To illustrate the versatility of this approach, three different multivariate stochastic volatility models are estimated for a standard data set. The empirical results are compared to those from earlier studies in the literature. Monte Carlo simulation experiments, based on parameter estimates from the standard data set, are used to show the effectiveness of the importance sampling methods.     

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.     

Stochastic Volatility Models Based on OU-Gamma Time Change: Theory and Estimation

We consider stochastic volatility models that are defined by an Ornstein–Uhlenbeck (OU)-Gamma time change. These models are most suitable for modeling financial time series and follow the general framework of the popular non-Gaussian OU models of Barndorff-Nielsen and Shephard. One current problem of these otherwise attractive nontrivial models is, in general, the unavailability of a tractable likelihood-based statistical analysis for the returns of financial assets, which requires the ability to sample from a nontrivial joint distribution. We show that an OU process driven by an infinite activity Gamma process, which is an OU-Gamma process, exhibits unique features, which allows one to explicitly describe and exactly sample from relevant joint distributions. This is a consequence of the OU structure and the calculus of Gamma and Dirichlet processes. We develop a particle marginal Metropolis–Hastings algorithm for this type of continuous-time stochastic volatility models and check its performance using simulated data. For illustration we finally fit the model to S&P500 index data.     

Monte Carlo Maximum Likelihood Estimation for Generalized Long-Memory Time Series Models

An exact maximum likelihood method is developed for the estimation of parameters in a non-Gaussian nonlinear density function that depends on a latent Gaussian dynamic process with long-memory properties. Our method relies on the method of importance sampling and on a linear Gaussian approximating model from which the latent process can be simulated. Given the presence of a latent long-memory process, we require a modification of the importance sampling technique. In particular, the long-memory process needs to be approximated by a finite dynamic linear process. Two possible approximations are discussed and are compared with each other. We show that an autoregression obtained from minimizing mean squared prediction errors leads to an effective and feasible method. In our empirical study, we analyze ten daily log-return series from the S&P 500 stock index by univariate and multivariate long-memory stochastic volatility models. We compare the in-sample and out-of-sample performance of a number of models within the class of long-memory stochastic volatility models.     

Books on probability ideas are reviewed

In this paper, semiparametric methods are applied to estimate multivariate volatility functions, using a residual approach as in [J. Fan and Q. Yao, Efficient estimation of conditional variance functions in stochastic regression, Biometrika 85 (1998), pp. 645–660; F.A. Ziegelmann, Nonparametric estimation of volatility functions: The local exponential estimator, Econometric Theory 18 (2002), pp. 985–991; F.A. Ziegelmann, A local linear least-absolute-deviations estimator of volatility, Comm. Statist. Simulation Comput. 37 (2008), pp. 1543–1564], among others. Our main goal here is two-fold: (1) describe and implement a number of semiparametric models, such as additive, single-index and (adaptive) functional-coefficient, in volatility estimation, all motivated as alternatives to deal with the curse of dimensionality present in fully nonparametric models; and (2) propose the use of a variation of the traditional cross-validation method to deal with model choice in the class of adaptive functional-coefficient models, choosing simultaneously the bandwidth, the number of covariates in the model and also the single-index smoothing variable. The modified cross-validation algorithm is able to tackle the computational burden caused by the model complexity, providing an important tool in semiparametric volatility estimation. We briefly discuss model identifiability when estimating volatility as well as nonnegativity of the resulting estimators. Furthermore, Monte Carlo simulations for several underlying generating models are implemented and applications to real data are provided.