On leverage in a stochastic volatility model

This paper is concerned with the specification for modelling financial leverage effect in the context of stochastic volatility (SV) models. Two alternative specifications co-exist in the literature. One is the Euler approximation to the well-known continuous time SV model with leverage effect and the other is the discrete time SV model of Jacquier et al. (J. Econometrics 122 (2004) 185). Using a Gaussian nonlinear state space form with uncorrelated measurement and transition errors, I show that it is easy to interpret the leverage effect in the conventional model whereas it is not clear how to obtain and interpret the leverage effect in the model of Jacquier et al. Empirical comparisons of these two models via Bayesian Markov chain Monte Carlo (MCMC) methods further reveal that the specification of Jacquier et al. is inferior. Simulation experiments are conducted to study the sampling properties of Bayes MCMC for the conventional model.     

Stochastic volatility models: conditional normality versus heavy‐tailed distributions

Most of the empirical applications of the stochastic volatility (SV) model are based on the assumption that the conditional distribution of returns, given the latent volatility process, is normal. In this paper, the SV model based on a conditional normal distribution is compared with SV specifications using conditional heavy‐tailed distributions, especially Student's t‐distribution and the generalized error distribution. To estimate the SV specifications, a simulated maximum likelihood approach is applied. The results based on daily data on exchange rates and stock returns reveal that the SV model with a conditional normal distribution does not adequately account for the two following empirical facts simultaneously: the leptokurtic distribution of the returns and the low but slowly decaying autocorrelation functions of the squared returns. It is shown that these empirical facts are more adequately captured by an SV model with a conditional heavy‐tailed distribution. It also turns out that the choice of the conditional distribution has systematic effects on the parameter estimates of the volatility process. Copyright © 2000 John Wiley & Sons, Ltd.     

A semiparametric stochastic volatility model

In this paper the correlation structure in the classical leverage stochastic volatility (SV) model is generalized based on a linear spline. In the new model the correlation between the return and volatility innovations is time varying and depends nonparametrically on the type of news arrived to the market. Theoretical properties of the proposed model are examined. The model estimation and comparison are conducted by Bayesian methods. The performance of the estimates are examined in simulations. The new model is fitted to daily and weekly US data and compared with the classical SV and GARCH models in terms of their in-sample and out-of-sample performances. Empirical results suggest evidence in favor of the proposed model. In particular, the new model finds strong evidence of time varying leverage effect in individual stocks when the classical model fails to identify the leverage effect.     

An empirical application of stochastic volatility models

This paper studies the empirical performance of stochastic volatility models for twenty years of weekly exchange rate data for four major currencies. We concentrate on the effects of the distribution of the exchange rate innovations for both parameter estimates and for estimates of the latent volatility series. The density of the log of squared exchange rate innovations is modelled as a flexible mixture of normals. We use three different estimation techniques: quasi-maximum likelihood, simulated EM, and a Bayesian procedure. The estimated models are applied for pricing currency options. The major findings of the paper are that: (1) explicitly incorporating fat-tailed innovations increases the estimates of the persistence of volatility dynamics; (2) the estimation error of the volatility time series is very large; (3) this in turn causes standard errors on calculated option prices to be so large that these prices are rarely significantly different from a model with constant volatility. © 1998 John Wiley & Sons, Ltd.     

No-arbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects,jumps and i.i.d. noise: Theory and testable distributional implications

We develop a sequential procedure to test the adequacy of jump-diffusion models for return distributions. We rely on intraday data and nonparametric volatility measures, along with a new jump detection technique and appropriate conditional moment tests, for assessing the import of jumps and leverage effects. A novel robust-to-jumps approach is utilized to alleviate microstructure frictions for realized volatility estimation. Size and power of the procedure are explored through Monte Carlo methods. Our empirical findings support the jump-diffusive representation for S&P500 futures returns but reveal it is critical to account for leverage effects and jumps to maintain the underlying semi-martingale assumption.     

A non-linear filtering approach to stochastic volatility models with an application to daily stock returns

This paper develops a new model for the analysis of stochastic volatility (SV) models. Since volatility is a latent variable in SV models, it is difficult to evaluate the exact likelihood. In this paper, a non-linear filter which yields the exact likelihood of SV models is employed. Solving a series of integrals in this filter by piecewise linear approximations with randomly chosen nodes produces the likelihood, which is maximized to obtain estimates of the SV parameters. A smoothing algorithm for volatility estimation is also constructed. Monte Carlo experiments show that the method performs well with respect to both parameter estimates and volatility estimates. We illustrate our model by analysing daily stock returns on the Tokyo Stock Exchange. Since the method can be applied to more general models, the SV model is extended so that several characteristics of daily stock returns are allowed, and this more general model is also estimated. Copyright © 1999 John Wiley & Sons, Ltd.     

GARCH模型族在沪深300中的比较研究

风险测量一直是金融研究领域的热门话题,而如何构建合适的模型来衡量风险自然而然成为众多学者研究的关注点.VaR方法是当今应用最广泛的衡量金融风险的方法之一,其核心又在构建良好的波动率估计模型.GARCH模型族能很好地描述股指波动率呈现的重尾、波动性聚集、杠杆效用等,是当前效果比较好的条件异方差性的模型.本文着重研究基于GARCH模型族(GARCH、EGARCH、PGARCH)在不同分布假定下(高斯分布、t分布、广义误差分布)的表现,从而计算出沪深300的在险价值( VaR),比较分析模型拟合效果,选出适合的模型,对规范国内沪深300的风险管理提供了理论依据.     

Leverage effect on stochastic volatility for option pricing in Hong Kong: A simulation and empirical study

This paper explores the importance of incorporating the financial leverage effect in the stochastic volatility models when pricing options. For the illustrative purpose, we first conduct the simulation experiment by using the Markov Chain Monte Carlo (MCMC) sampling method. We then make an empirical analysis by applying the volatility models to the real return data of the Hang Seng index during the period from January 1, 2013 to December 31, 2017. Our results highlight the accuracy of the stochastic volatility models with leverage in option pricing when leverage is high. In addition, the leverage effect becomes more significant as the maturity of options increases. Moreover, leverage affects the pricing of in-the-money options more than that of at-the-money and out-of-money options. Our study is therefore useful for both asset pricing and portfolio investment in the Hong Kong market where volatility is an inherent nature of the economy.     

Modelling Regime‐Specific Stock Price Volatility*

Single‐state generalized autoregressive conditional heteroscedasticity (GARCH) models identify only one mechanism governing the response of volatility to market shocks, and the conditional higher moments are constant, unless modelled explicitly. So they neither capture state‐dependent behaviour of volatility nor explain why the equity index skew persists into long‐dated options. Markov switching (MS) GARCH models specify several volatility states with endogenous conditional skewness and kurtosis; of these the simplest to estimate is normal mixture (NM) GARCH, which has constant state probabilities. We introduce a state‐dependent leverage effect to NM‐GARCH and thereby explain the observed characteristics of equity index returns and implied volatility skews, without resorting to time‐varying volatility risk premia. An empirical study on European equity indices identifies two‐state asymmetric NM‐GARCH as the best fit of the 15 models considered. During stable markets volatility behaviour is broadly similar across all indices, but the crash probability and the behaviour of returns and volatility during a crash depends on the index. The volatility mean‐reversion and leverage effects during crash markets are quite different from those in the stable regime.     

Global and regional integration of the Middle East and North African (MENA) stock markets

We investigate financial integration of MENA region to facilitate a more in-depth exploration of the structure of interdependence and transmission mechanism of stock returns and volatility between MENA and world stock markets. The EGARCH-M models with a generalized error distribution are employed to consider both leverage effect of negative shocks and leptokurtosis prevalent in the MENA stock markets. The estimation results of multivariate AR-GARCH models indicate that there are large and predominantly positive volatility spillovers and volatility persistence in conditional volatility between MENA and world stock markets. Own-volatility spillovers are generally higher than cross-volatility spillovers for all the markets.     

A flexible parametric approach for estimating switching regime models and treatment effect parameters

In this paper, we propose a flexible, parametric class of switching regime models allowing for both skewed and fat-tailed outcome and selection errors. Specifically, we model the joint distribution of each outcome error and the selection error via a newly constructed class of multivariate distributions which we call generalized normal mean–variance mixture distributions. We extend Heckman’s two-step estimation procedure for the Gaussian switching regime model to the new class of models. When the distributions of the outcome errors are asymmetric, we show that an additional correction term accounting for skewness in the outcome error distribution (besides the analogue of the well known inverse mill’s ratio) needs to be included in the second step regression. We use the two-step estimators of parameters in the model to construct simple estimators of average treatment effects and establish their asymptotic properties. Simulation results confirm the importance of accounting for skewness in the outcome errors in estimating both model parameters and the average treatment effect and the treatment effect for the treated.     

Inference with non-Gaussian Ornstein–Uhlenbeck processes for stochastic volatility

Continuous-time stochastic volatility models are becoming an increasingly popular way to describe moderate and high-frequency financial data. Barndorff-Nielsen and Shephard (2001a) proposed a class of models where the volatility behaves according to an Ornstein–Uhlenbeck (OU) process, driven by a positive Lévy process without Gaussian component. These models introduce discontinuities, or jumps, into the volatility process. They also consider superpositions of such processes and we extend that to the inclusion of a jump component in the returns. In addition, we allow for leverage effects and we introduce separate risk pricing for the volatility components. We design and implement practically relevant inference methods for such models, within the Bayesian paradigm. The algorithm is based on Markov chain Monte Carlo (MCMC) methods and we use a series representation of Lévy processes. MCMC methods for such models are complicated by the fact that parameter changes will often induce a change in the distribution of the representation of the process and the associated problem of overconditioning. We avoid this problem by dependent thinning methods. An application to stock price data shows the models perform very well, even in the face of data with rapid changes, especially if a superposition of processes with different risk premiums and a leverage effect is used.     

The stochastic volatility in mean model: empirical evidence from international stock markets

In this paper we present an exact maximum likelihood treatment for the estimation of a Stochastic Volatility in Mean (SVM) model based on Monte Carlo simulation methods. The SVM model incorporates the unobserved volatility as an explanatory variable in the mean equation. The same extension is developed elsewhere for Autoregressive Conditional Heteroscedastic (ARCH) models, known as the ARCH in Mean (ARCH‐M) model. The estimation of ARCH models is relatively easy compared with that of the Stochastic Volatility (SV) model. However, efficient Monte Carlo simulation methods for SV models have been developed to overcome some of these problems. The details of modifications required for estimating the volatility‐in‐mean effect are presented in this paper together with a Monte Carlo study to investigate the finite sample properties of the SVM estimators. Taking these developments of estimation methods into account, we regard SV and SVM models as practical alternatives to their ARCH counterparts and therefore it is of interest to study and compare the two classes of volatility models. We present an empirical study of the intertemporal relationship between stock index returns and their volatility for the United Kingdom, the United States and Japan. This phenomenon has been discussed in the financial economic literature but has proved hard to find empirically. We provide evidence of a negative but weak relationship between returns and contemporaneous volatility which is indirect evidence of a positive relation between the expected components of the return and the volatility process. Copyright © 2002 John Wiley & Sons, Ltd.     

Temporal aggregation of volatility models

In this paper, we consider temporal aggregation of volatility models. We introduce semiparametric volatility models, termed square-root stochastic autoregressive volatility (SR-SARV), which are characterized by autoregressive dynamics of the stochastic variance. Our class encompasses the usual GARCH models and various asymmetric GARCH models. Moreover, our stochastic volatility models are characterized by multiperiod conditional moment restrictions in terms of observables. The SR-SARV class is a natural extension of the class of weak GARCH models. This extension has four advantages: (i) we do not assume that fourth moments are finite; (ii) we allow for asymmetries (skewness, leverage effect) that are excluded from weak GARCH models; (iii) we derive conditional moment restrictions and (iv) our framework allows us to study temporal aggregation of IGARCH models.     

Kuhn-Tucker,likelihood ratio and Wald tests for nonlinear models with inequality constraints on the parameters

This paper considers the problem of testing statistical hypothesis in nonlinear regression models with inequality constraints on the parameters. First, the Kuhn-Tucker test procedure is defined. Next, it is shown that the distribution of the Kuhn-Tucker, the likelihood ratio and the Wald test statistics converges to the same mixture of chi-square distributions under the null hypothesis. To illustrate these results two examples are considered: (1) the problem of testing that individual effects are missing in an error component model, and (2) the problem of testing equilibrium for a model of markets in disequilibrium.     

A class of adaptive importance sampling weighted EM algorithms for efficient and robust posterior and predictive simulation

A class of adaptive sampling methods is introduced for efficient posterior and predictive simulation. The proposed methods are robust in the sense that they can handle target distributions that exhibit non-elliptical shapes such as multimodality and skewness. The basic method makes use of sequences of importance weighted Expectation Maximization steps in order to efficiently construct a mixture of Student-t$t$ densities that approximates accurately the target distribution–typically a posterior distribution, of which we only require a kernel–in the sense that the Kullback–Leibler divergence between target and mixture is minimized. We label this approach Mixture of  t$t$by Importance Sampling weighted Expectation Maximization (MitISEM). The constructed mixture is used as a candidate density for quick and reliable application of either Importance Sampling (IS) or the Metropolis–Hastings (MH) method. We also introduce three extensions of the basic MitISEM approach. First, we propose a method for applying MitISEM in a sequential manner, so that the candidate distribution for posterior simulation is cleverly updated when new data become available. Our results show that the computational effort reduces enormously, while the quality of the approximation remains almost unchanged. This sequential approach can be combined with a tempering approach, which facilitates the simulation from densities with multiple modes that are far apart. Second, we introduce a permutation-augmented MitISEM approach. This is useful for importance or Metropolis–Hastings sampling from posterior distributions in mixture models without the requirement of imposing identification restrictions on the model’s mixture regimes’ parameters. Third, we propose a partial MitISEM approach, which aims at approximating the joint distribution by estimating a product of marginal and conditional distributions. This division can substantially reduce the dimension of the approximation problem, which facilitates the application of adaptive importance sampling for posterior simulation in more complex models with larger numbers of parameters. Our results indicate that the proposed methods can substantially reduce the computational burden in econometric models like DCC or mixture GARCH models and a mixture instrumental variables model.     

Heterogeneous Traders,the Leverage Effect and Volatility of the Chinese P2P Market

The new financial industry represented by peer-to-peer lending has gradually become a new source of volatility due to the increasing complexity of the Chinese financial market. This volatility leads to greater risk to P2P investors and has become the focus of the regulatory authorities in China. Based on the background data of the P2P platform, Honglingchuangtou, we use the factor analysis method to construct a platform volatility (PV) index and we construct an HAR model to study the heterogeneous traders and leverage effect in the Chinese P2P market. The empirical results show that there are both short-term and long-term heterogeneous traders in the Chinese P2P market and that long-term traders have the greatest impact on market volatility. Similar to traditional financial markets, the volatility of the P2P market also shows a leverage effect, which means that the negative volatility of trader actions should have a negative impact on market fluctuations. With regard to the leverage effect, the LHAR-PV model is superior because of a higher goodness of fit and a lower prediction error.     

Bayesian modeling of joint and conditional distributions

In this paper, we study a Bayesian approach to flexible modeling of conditional distributions. The approach uses a flexible model for the joint distribution of the dependent and independent variables and then extracts the conditional distributions of interest from the estimated joint distribution. We use a finite mixture of multivariate normals (FMMN) to estimate the joint distribution. The conditional distributions can then be assessed analytically or through simulations. The discrete variables are handled through the use of latent variables. The estimation procedure employs an MCMC algorithm. We provide a characterization of the Kullback–Leibler closure of FMMN and show that the joint and conditional predictive densities implied by the FMMN model are consistent estimators for a large class of data generating processes with continuous and discrete observables. The method can be used as a robust regression model with discrete and continuous dependent and independent variables and as a Bayesian alternative to semi- and non-parametric models such as quantile and kernel regression. In experiments, the method compares favorably with classical nonparametric and alternative Bayesian methods.     

Volatility forecasting in the Bitcoin market: A new proposed measure based on the VS-ACARR approach

This paper proposes a new volatility-spillover-asymmetric conditional autoregressive range (VS-ACARR) approach that takes into account the intraday information, the volatility spillover from crude oil as well as the volatility asymmetry (leverage effect) to model/forecast Bitcoin volatility (price range). An empirical application to Bitcoin and crude oil (WTI) price ranges shows the existence of strong volatility spillover from crude oil to the Bitcoin market and a weak leverage effect in the Bitcoin market. The VS-ACARR model yields higher forecasting accuracy than the GARCH, CARR, and VS-CARR models regarding out-of-sample forecast performance, suggesting that accounting for the volatility spillover and asymmetry can significantly improve the forecasting accuracy of Bitcoin volatility. The superior forecast performance of the VS-ACARR model is robust to alternative out-of-sample forecast windows. Our findings highlight the importance of accommodating intraday information, spillover from crude oil, and volatility asymmetry in forecasting Bitcoin volatility.     

Exact and asymptotic tests for possibly non-regular hypotheses on stochastic volatility models

We study the problem of testing hypotheses on the parameters of one- and two-factor stochastic volatility models (SV), allowing for the possible presence of non-regularities such as singular moment conditions and unidentified parameters, which can lead to non-standard asymptotic distributions. We focus on the development of simulation-based exact procedures–whose level can be controlled in finite samples–as well as on large-sample procedures which remain valid under non-regular conditions. We consider Wald-type, score-type and likelihood-ratio-type tests based on a simple moment estimator, which can be easily simulated. We also propose a C(α)-type test which is very easy to implement and exhibits relatively good size and power properties. Besides usual linear restrictions on the SV model coefficients, the problems studied include testing homoskedasticity against a SV alternative (which involves singular moment conditions under the null hypothesis) and testing the null hypothesis of one factor driving the dynamics of the volatility process against two factors (which raises identification difficulties). Three ways of implementing the tests based on alternative statistics are compared: asymptotic critical values (when available), a local Monte Carlo (or parametric bootstrap) test procedure, and a maximized Monte Carlo (MMC) procedure. The size and power properties of the proposed tests are examined in a simulation experiment. The results indicate that the C(α)-based tests (built upon the simple moment estimator available in closed form) have good size and power properties for regular hypotheses, while Monte Carlo tests are much more reliable than those based on asymptotic critical values. Further, in cases where the parametric bootstrap appears to fail (for example, in the presence of identification problems), the MMC procedure easily controls the level of the tests. Moreover, MMC-based tests exhibit relatively good power performance despite the conservative feature of the procedure. Finally, we present an application to a time series of returns on the Standard and Poor’s Composite Price Index.