BAYESIAN ANALYSIS OF GENERALIZED AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY AND STOCHASTIC VOLATILITY: MODELING LEVERAGE,JUMPS AND HEAVY‐TAILS FOR FINANCIAL TIME SERIES*

This paper develops a Bayesian model comparison of two broad major classes of varying volatility model, the generalized autoregressive conditional heteroskedasticity and stochastic volatility models, on financial time series. The leverage effect, jumps and heavy‐tailed errors are incorporated into the two models. For estimation, the efficient Markov chain Monte Carlo methods are developed and the model comparisons are examined based on the marginal likelihood. The empirical analyses are illustrated using the daily return data of US stock indices, individual securities and exchange rates of UK sterling and Japanese yen against the US dollar. The estimation results indicate that the stochastic volatility model with leverage and Student‐t errors yield the best performance among the competing models.     

Computing DSGE models with recursive preferences and stochastic volatility

This paper compares different solution methods for computing the equilibrium of dynamic stochastic general equilibrium (DSGE) models with recursive preferences such as those in Epstein and Zin, 1989, Epstein and Zin, 1991 and stochastic volatility. Models with these two features have recently become popular, but we know little about the best ways to implement them numerically. To fill this gap, we solve the stochastic neoclassical growth model with recursive preferences and stochastic volatility using four different approaches: second- and third-order perturbation, Chebyshev polynomials, and value function iteration. We document the performance of the methods in terms of computing time, implementation complexity, and accuracy. Our main finding is that perturbations are competitive in terms of accuracy with Chebyshev polynomials and value function iteration while being several orders of magnitude faster to run. Therefore, we conclude that perturbation methods are an attractive approach for computing this class of problems.     

Estimating general equilibrium models with stochastic volatility and changing parameters

This paper explores the importance of specification in estimated general equilibrium models with changing monetary policy parameters and stochastic volatility. Simulated data is used to estimate models with incorrectly specified exogenous shocks (time-varying vs. constant variance) and models misspecifying the way Taylor rule parameters change over time (constant vs. drifting vs. regime-switching). The model correctly identifies some changes in monetary policy parameters, even when misspecified. The inclusion of stochastic volatility greatly improves model fit even when the data is generated using constant variance exogenous shocks; this relationship is stronger in data generated from models with changing policy parameters.     

Empirical Evidence of the Leverage Effect in a Stochastic Volatility Model: A Realized Volatility Approach

Increasing attention has been focused on the analysis of the realized volatility, which can be treated as a proxy for the true volatility. In this paper, we study the potential use of the realized volatility as a proxy in a stochastic volatility model estimation. We estimate the leveraged stochastic volatility model using the realized volatility computed from five popular methods across six sampling-frequency transaction data (from 1-min to 60- min) based on the trust region method. Availability of the realized volatility allows us to estimate the model parameters via the MLE and thus avoids computational challenge in the high dimensional integration. Six stock indices are considered in the empirical investigation. We discover some consistent findings and interesting patterns from the empirical results. In general, the significant leverage effect is consistently detected at each sampling frequency and the volatility persistence becomes weaker at the lower sampling frequency.     

Modelling conditional heteroskedasticity: Application to the “IBEX-35” stock-return index

This paper compares alternative time-varying volatility models for daily stock-returns using data from Spanish equity index IBEX-35. Specifically, we estimate a parametric family of models of generalized autoregressive heteroskedasticity (which nests the most popular symmetric and asymmetric GARCH models), a semiparametric GARCH model, the generalized quadratic ARCH model, the stochastic volatility model, the Poisson Jump Diffusion model and, finally, a nonparametric model. Those models which use conditional standard deviation (specifically, TGARCH and AGARCH models) produce better fits than all other GARCH models. We also compare the within sample predictive power of all models using a standard efficiency test. Our results show that the asymmetric behaviour of responses is a statistically significant characteristic of these data. Moreover, we observe that specifications with a distribution which allows for fatter tails than a normal distribution do not necessarily outperform specifications with a normal distribution.     

Volatility transmission between Japan,UK and USA in daily stock returns

In the past, there are a lot of studies which conclude that the holiday, asymmetry and day-of-the-week effects influence stock price volatility. Most of the studies are based on a class of generalized auto-regressive conditional heteroskedasticity (GARCH) models. No one examines these effects simultaneously using stochastic volatility (SV) models. In this paper, using the SV model, we examine whether these effects play an important role in stock price volatilities. Furthermore, we consider spillover effects between Japan, UK and USA, where spillover effects in price level as well as volatility are taken into account. We are grateful to two anonymous referees for suggestions and comments. We also acknowledge Toshiaki Watanabe who gave us a lot of helpful suggestions and comments in the preliminary version of this paper. This research is partially supported from Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) #18530158, 2006–2009, and Grant-in-Aid for COE Research) and the Zengin Foundation (Grant-in-Aid for Studies on Economics and Finance), which are acknowledged by H. Tanizaki.     

On the effects of rare disasters and uncertainty shocks for risk premia in non-linear DSGE models

This paper studies how rare disasters and uncertainty shocks affect risk premia in DSGE models approximated to second and third order. Based on an extension of the results in Schmitt-Grohé and Uribe (2004) to third order, we derive propositions for how rare disasters, stochastic volatility, and GARCH affect any type of risk premia in a wide class of DSGE models. To quantify the effects, we set up a standard New Keynesian DSGE model where total factor productivity includes rare disasters, stochastic volatility, and GARCH. We find that rare disasters increase the level of the 10-year nominal term premium, whereas a key effect of uncertainty shocks, i.e. stochastic volatility and GARCH, is an increase in the variability of this premium.     

Foreign equity option pricing under stochastic volatility model with double jumps

This paper considers the challenging problem advocated by Huang and Hung (2005), that is to incorporate the stochastic volatility into the foreign equity option pricing. Foreign equity options (quanto options) are contingent claims where the payoff is determined by an equity in one currency but the actual payoff is done in another currency. Huang and Hung (2005) priced foreign equity options under the Lévy processes. In Huang and Hung's paper, they considered jumps in the foreign asset prices and exchange rates and assumed the volatility as constant. However, many studies showed that constant volatility and jumps in returns are incapable of fully capturing the empirical features of equity returns or option prices. In this paper, the stochastic volatility with simultaneous jumps in prices and volatility is proposed to model foreign asset prices and exchange rates. The foreign equity option pricing formula is given by using the Fourier inverse transformation. The numerical results show that the use of stochastic volatility with simultaneous jumps in prices and volatility proposed to model foreign asset prices and exchange rates is necessary and this approach can help us to capture more accurately the foreign equity option prices.     

VOLATILITY MODELS*

Models for estimating the volatility of financial assets are reviewed in this paper. The volatility can be estimated by the univariate GARCH family of models, or stochastic volatility models. These univariate models are developed intomultivariate models. Finally, the search for an adequate framework for the estimation has led to the analysis of high frequency intraday data. The variance over a fixed interval can be estimated accurately as the sum of squared realizations, provided the data are available at sufficiently high sampling frequencies. The future of this new area is wide open for theoretical developments and for applied studies.     

An improved framework for approximating option prices with application to option portfolio hedging

As the price of the underlying asset changes over time, delta of the option changes and a gamma hedge is required along with delta hedge to reduce risk. This paper develops an improved framework to compute delta and gamma values with the average of a range of underlying prices rather than at the conventional fixed ‘one point’. We find that models with time-varying volatility price options satisfactorily, and perform remarkably well in combination with the delta and delta-gamma approximations. Significant improvements are achieved for the GARCH model followed by stochastic volatility models. The new approach can ensure significant improvement in modelling option prices leading to better risk-management decision-making.     

Business cycle fluctuations in Taiwan — A Bayesian DSGE analysis

This paper examines the Taiwanese economy in a small open economy DSGE model using Bayesian estimation. The model consists of two countries and 12 exogenous shocks with stochastic volatility to capture the fluctuations in the business cycle. The main results are: (1) shock innovations with stochastic volatility increase the model fit, (2) shocks originated from outside the country are important sources of fluctuations in the Taiwanese business cycle.     

Semi-Parametric Comparison of Stochastic Volatility Models using Realized Measures

This paper proposes a procedure to test for the correct specification of the functional form of the volatility process within the class of eigenfunction stochastic volatility models. The procedure is based on the comparison of the moments of realized volatility measures with the corresponding ones of integrated volatility implied by the model under the null hypothesis.
We first provide primitive conditions on the measurement error associated with the realized measure, which allow to construct asymptotically valid specification tests.
Then we establish regularity conditions under which the considered realized measures, namely, realized volatility, bipower variation, and modified subsampled realized volatility, satisfy the given primitive assumptions.
Finally, we provide an empirical illustration based on three stocks from the Dow Jones Industrial Average.     

Asset prices and economic fluctuations: The implications of stochastic volatility

This paper investigates whether the multi-factor stochastic volatility of stock returns is related to economic fluctuations and affects asset prices. We address these issues in a dynamic Fama-French three-factor volatility model framework. Consistent with the ICAPM with stochastic volatility (Campbell et al., 2017), we find that the conditional volatility of the size and value factors is significantly related to economic uncertainty. These volatilities are also significant pricing factors. The out-of-sample forecasting analysis further reveals that the conditional volatility can predict stock returns and deliver economic gain in asset allocation. Our analysis sharpens the understanding on the link between the stock market and economic fundamentals.     

A common factor of stochastic volatilities between oil and commodity prices

This article analyses the multivariate stochastic volatilities (SVs) with a common factor influencing volatilities in the prices of crude oil and agricultural commodities, used for both biofuel and nonbiofuel purposes. Modelling the volatility is crucial because the volatility is an important variable for asset allocation, risk management and derivative pricing. We develop a SV model comprising a latent common volatility factor with two asymptotic regimes with a smooth transition between them. In contrast to conventional volatility models, SVs are generated by the logistic transformation of latent factors, which comprise two components: the common volatility factor and an idiosyncratic component. We present a SV model with a common factor for oil, corn and wheat from 8 August 2005 to 10 October 2014, using a Markov chain Monte Carlo method to estimate the SVs and extract the common volatility factor. We find that the volatilities of oil and grain markets are persistent. According to the estimated common volatility factor, high volatility periods match the 2007–2009 recession and the 2007–2008 financial crisis quite well. Finally, the extracted common volatility factor exhibits a distinct pattern.     

FORECASTING GLOBAL EQUITY INDICES USING LARGE BAYESIAN VARS

This paper proposes a large Bayesian Vector Autoregressive (BVAR) model with common stochastic volatility to forecast global equity indices. Using a monthly dataset on global stock indices, the BVAR model controls for co‐movement commonly observed in global stock markets. Moreover, the time‐varying specification of the covariance structure accounts for sudden shifts in the level of volatility. In an out‐of‐sample forecasting application we show that the BVAR model with stochastic volatility significantly outperforms the random walk both in terms of point as well as density predictions. The BVAR model without stochastic volatility, on the other hand, shows some merits relative to the random walk for forecast horizons greater than six months ahead. In a portfolio allocation exercise we moreover provide evidence that it is possible to use the forecasts obtained from our model with common stochastic volatility to set up simple investment strategies. Our results indicate that these simple investment schemes outperform a naive buy‐and‐hold strategy.     

Can GARCH-class models capture long memory in WTI crude oil markets?

This paper investigates the issue whether GARCH-type models can well capture the long memory widely existed in the volatility of WTI crude oil returns. In this frame, we model the volatility of spot and futures returns employing several GARCH-class models. Then, using two non-parametric methods, detrended fluctuation analysis (DFA) and rescaled range analysis (R/S), we compare the long memory properties of conditional volatility series obtained from GARCH-class models to that of actual volatility series. Our results show that GARCH-class models can well capture the long memory properties for the time scale larger than a year. However, for the time scale smaller than a year, the GARCH-class models are misspecified.     

Uncertainty and crude oil market volatility: new evidence

ABSTRACT

The main goal of this paper is to investigate the predictability of five economic uncertainty indices for oil price volatility in a changing world. We employ the standard predictive regression framework, several model combination approaches, as well as two prevailing model shrinkage methods to evaluate the performances of the uncertainty indices. The empirical results based on simple autoregression models including only one index suggest that global economic policy uncertainty (GEPU) and US equity market volatility (EMV) indices have significant predictive power for crude oil market volatility. In addition, the model combination approaches adopted in this paper can improve slightly the performances of individual autoregressive models. Lastly, the two model shrinkage methods, namely Elastin net and Lasso, outperform other individual AR-type model and combination models in most forecasting cases. Other empirical results based on alternative forecasting methods, estimation window sizes, high/low volatility and economic expansion/recession time periods further make sure the robustness of our major conclusions. The findings in this paper also have several important economic implications for oil investors.     

Nonparametric estimation of stochastic volatility models

This letter introduces nonparametric estimators of the drift and diffusion coefficient of stochastic volatility models which exploit techniques for estimating integrated volatility with high-frequency data. The performance of the proposed estimators is assessed on simulations of two popular stochastic volatility models.     

Estimation of Asymmetric Stochastic Volatility Models for Stock-Exchange Index Returns

The objective of this paper is to put forward a new autoregressive asymmetric stochastic volatility model for modeling volatility and to compare results obtained for this model with an autoregressive stochastic model and another asymmetric volatility model, such as, asymmetric generalized autoregressive conditional heteroskedasticity model. The results obtained from the estimation by maximum likelihood have shown the volatility behavior is asymmetric in the majority of cases. This fact is better shown by the ARSVA model, than the rest of alternative models. Moreover, the ARSVA model is able to reproduce other stylized facts of such series, such as high kurtosis, no autocorrelation of returns, slow decreasing of the autocorrelation function of the squared returns and high persistence.

Asymmetry, realised volatility and stock return risk estimates

In this paper we estimate minimum capital risk requirements for short and long positions with three investment horizons, using the traditional GARCH model and two other GARCH-type models that incorporate the possibility of asymmetric responses of volatility to price changes. We also address the problem of the extremely high estimated persistence of the GARCH model to generate observed volatility patterns by including realised volatility as an explanatory variable into the model??s variance equation. The results suggest that the inclusion of realised volatility improves the GARCH forecastability as well as its ability to calculate accurate minimum capital risk requirements and makes it quite competitive when compared with asymmetric conditional heteroscedastic models such as the GJR and the EGARCH.