Manabu Asai

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

We provide a detailed summary of the large and vibrant emerging literature that deals with the multivariate modeling of conditional volatility of financial time series within the framework of stochastic volatility. The developments and achievements in this area represent one of the great success stories of financial econometrics. Three broad classes of multivariate stochastic volatility models have emerged: one that is a direct extension of the univariate class of stochastic volatility model, another that is related to the factor models of multivariate analysis and a third that is based on the direct modeling of time-varying correlation matrices via matrix exponential transformations, Wishart processes and other means. We discuss each of the various model formulations, provide connections and differences and show how the models are estimated. Given the interest in this area, further significant developments can be expected, perhaps fostered by the overview and details delineated in this paper, especially in the fitting of high-dimensional models.

Dynamic Asymmetric Leverage in Stochastic Volatility Models

ABSTRACT In the class of stochastic volatility (SV) models, leverage effects are typically specified through the direct correlation between the innovations in both returns and volatility, resulting in the dynamic leverage (DL) model. Recently, two asymmetric SV models based on threshold effects have been proposed in the literature. As such models consider only the sign of the previous return and neglect its magnitude, this paper proposes a dynamic asymmetric leverage (DAL) model that accommodates the direct correlation as well as the sign and magnitude of the threshold effects. A special case of the DAL model with zero direct correlation between the innovations is the asymmetric leverage (AL) model. The dynamic asymmetric leverage models are estimated by the Monte Carlo likelihood (MCL) method. Monte Carlo experiments are presented to examine the finite sample properties of the estimator. For a sample size of T = 2000 with 500 replications, the sample means, standard deviations, and root mean squared errors of the MCL estimators indicate only a small finite sample bias. The empirical estimates for S&P 500 and TOPIX financial returns, and USD/AUD and YEN/USD exchange rates, indicate that the DAL class, including the DL and AL models, is generally superior to threshold SV models with respect to AIC and BIC, with AL typically providing the best fit to the data.

Asymmetric Multivariate Stochastic Volatility

This paper proposes and analyses two types of asymmetric multivariate stochastic volatility (SV) models, namely, (i) the SV with leverage (SV-L) model, which is based on the negative correlation between the innovations in the returns and volatility, and (ii) the SV with leverage and size effect (SV-LSE) model, which is based on the signs and magnitude of the returns. The paper derives the state space form for the logarithm of the squared returns, which follow the multivariate SV-L model, and develops estimation methods for the multivariate SV-L and SV-LSE models based on the Monte Carlo likelihood (MCL) approach. The empirical results show that the multivariate SV-LSE model fits the bivariate and trivariate returns of the S&P 500, the Nikkei 225, and the Hang Seng indexes with respect to AIC and BIC more accurately than does the multivariate SV-L model. Moreover, the empirical results suggest that the univariate models should be rejected in favor of their bivariate and trivariate counterparts.

Multivariate Stochastic Volatility, Leverage and News Impact Surfaces

Alternative multivariate stochastic volatility (MSV) models with leverage have been proposed in the literature. However, the existing MSV with leverage models are unclear about the definition of leverage, specifically the timing of the relationship between the innovations in financial returns and the associated shocks to volatility, as well as their connection to partial correlations. This paper proposes a new MSV with leverage (MSVL) model in which leverage is defined clearly in terms of the innovations in both financial returns and volatility, such that the leverage effect associated with one financial return is not related to the leverage effect of another. News impact surfaces are developed for MSV models with leverage based on both log-volatility and volatility and are compared with the special case of news impact functions for their univariate counterparts. In order to capture heavy tails in each return distribution, we incorporate an additional factor for the volatility of each return. An empirical example based on bivariate data for Standard and Poors 500 Composite Index and the Nikkei 225 Index is presented to illustrate the usefulness of the new MSVL model and the associated news impact surfaces. Likelihood ratio (LR) tests are considered for model selection. The LR tests show that the two-factor MSVL model is supported, indicating that the restrictions considered in the paper are empirically adequate under heavy-tailed return distributions. Copyright

Bayesian analysis of stochastic volatility models with mixture-of-normal distributions

Stochastic volatility (SV) models usually assume that the distribution of asset returns conditional on the latent volatility is normal. This article analyzes SV models with a mixture-of-normal distributions in order to compare with other heavy-tailed distributions such as the Student-t distribution and generalized error distribution (GED). A Bayesian method via Markov-chain Monte Carlo (MCMC) techniques is used to estimate parameters and Bayes factors are calculated to compare the fit of distributions. The method is illustrated by analyzing daily data from the Yen/Dollar exchange rate and the Tokyo stock price index (TOPIX). According to Bayes factors, we find that while the t distribution fits the TOPIX better than the normal, the GED and the normal mixture, the mixture-of-normal distributions give a better fit to the Yen/Dollar exchange rate than other models. The effects of the specification of error distributions on the Bayesian confidence intervals of future returns are also examined. Comparison of SV with GARCH models shows that there are cases that the SV model with the normal distribution is less effective to capture leptokurtosis than the GARCH with heavy-tailed distributions.

The relationship between stock return volatility and trading volume: the case of the Philippines

This article reconsiders the relationship between stock return volatility and trading volume. Based on the multi-factor stochastic volatility model for stock return, we suggest several specifications for the trading volume. This approach enables the unobservable information arrival to follow the ARMA process. We apply the model to the data of Philippine Stock Exchange Composite Index and find that two factors are adequate to describe the movements of stock return volatility and variance of trading volume. We also find that the weights for the factors of the return and volume models are different from each other. The empirical results show (i) a negative correlation between stock return volatility and variance of trading volume, and (ii) a lack of effect of information arrivals on the level of trading volume. These findings are contrary to the results for the equity markets of advanced countries.

General asymmetric stochastic volatility models using range data: estimation and empirical evidence from emerging equity markets

We extend the range-based approach of Alizadeh et al. (2002) in order to deal with leverage and size effects and nonnormal conditional distribution in Stochastic Volatility (SV) models. We employ the Efficient Importance Sampling (EIS) method to estimate the range-based asymmetric SV models. Empirical results for the stock market indices of the Association of Southeast Asian Nations (ASEAN5) countries show that the conditional distributions of stock returns are nonnormal and that the model considered captures the existence/absence of the leverage and size effects.

Leverage and feedback effects on multifactor Wishart stochastic volatility for option pricing

The paper proposes a general asymmetric multifactor Wishart stochastic volatility (AMWSV) di usion process which accommodates leverage, feedback e ects and mul- tifactor for the covariance process. The paper gives the closed-form solution for the conditional and unconditional Laplace transform of the AMWSV models. The paper also suggests estimating the AMWSV model by the generalized method of moments using information not only of stock prices but also of realized volatilities and co- volatilities. The empirical results for the bivariate data of the NASDAQ 100 and S&P 500 indices show that the general AMWSV model is preferred among several nested models.

A fractionally integrated Wishart stochastic volatility model

ABSTRACT 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.