On asymmetric generalised t stochastic volatility models

In stochastic volatility (SV) models, asset returns conditional on the latent volatility are usually assumed to have a normal, Student-t or exponential power (EP) distribution. An earlier study uses a generalised t (GT) distribution for the conditional returns and the results indicate that the GT distribution provides a better model fit to the Australian Dollar/Japanese Yen daily exchange rate than the Student-t distribution. In fact, the GT family nests a number of well-known distributions including the commonly used normal, Student-t and EP distributions. This paper extends the SV model with a GT distribution by incorporating general volatility asymmetry. We compare the empirical performance of nested distributions of the GT distribution as well as different volatility asymmetry specifications. The new asymmetric GT SV models are estimated using the Bayesian Markov chain Monte Carlo (MCMC) method to obtain parameter and log-volatility estimates. By using daily returns from the Standard and Poors (S&P) 500 index, we investigate the effects of the specification of error distributions as well as volatility asymmetry on parameter and volatility estimates. Results show that the choice of error distributions has a major influence on volatility estimation only when volatility asymmetry is not accounted for.     

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.     

Leverage, heavy-tails and correlated jumps in stochastic volatility models

Efficient and fast Markov chain Monte Carlo estimation methods for the stochastic volatility model with leverage effects, heavy-tailed errors and jump components, and for the stochastic volatility model with correlated jumps are proposed. The methods are illustrated using simulated data and are applied to analyze daily stock returns data on S&P500 index and TOPIX. Model comparisons are conducted based on the marginal likelihood for various SV models including the superposition model.     

Testing volatility autocorrelation in the constant elasticity of variance stochastic volatility model

The sample autocovariance of the suitably scaled squared returns of a given stock is shown here to be a consistent and asymptotically normal estimator of the theoretical autocovariance of the mean variance, when the data is generated by the Constant Elasticity of Variance stochastic volatility (CEV SV) process. By computing explicitly the asymptotic variance of the estimator, confidence bands are obtained for the theoretical autocovariance. For each one of the stock indexes S&P500, CAC40, FTSE, DAX and SMI the estimated confidence bands are compared with the theoretical autocovariances computed for several values of the model parameters. The results suggest that the CEV SV model is able to capture the empirical autocovariance detected on the observed data. Analogous results are derived for the theoretical autocorrelation function.     

Stochastic volatility models with leverage and heavy-tailed distributions: A Bayesian approach using scale mixtures

This paper studies a heavy-tailed stochastic volatility (SV) model with leverage effect, where a bivariate Student-t distribution is used to model the error innovations of the return and volatility equations. Choy et al. (2008) studied this model by expressing the bivariate Student-t distribution as a scale mixture of bivariate normal distributions. We propose an alternative formulation by first deriving a conditional Student-t distribution for the return and a marginal Student-t distribution for the log-volatility and then express these two Student-t distributions as a scale mixture of normal (SMN) distributions. Our approach separates the sources of outliers and allows for distinguishing between outliers generated by the return process or by the volatility process, and hence is an improvement over the approach of Choy et al. (2008). In addition, it allows an efficient model implementation using the WinBUGS software. A simulation study is conducted to assess the performance of the proposed approach and its comparison with the approach by Choy et al. (2008). In the empirical study, daily exchange rate returns of the Australian dollar to various currencies and daily stock market index returns of various international stock markets are analysed. Model comparison relies on the Deviance Information Criterion and convergence diagnostic is monitored by Geweke’s convergence test.     

Forecasting stock returns based on information transmission across global markets using support vector machines

This paper provides evidence that forecasts based on global stock returns transmission yield better returns in day trading, for both developed and emerging stock markets. The study investigates the performance of global stock market price transmission information in forecasting stock prices using support vector regression for six global markets—USA (Dow Jones, S&P500), UK (FTSE-100), India (NSE), Singapore (SGX), Hong Kong (Hang Seng) and China (Shanghai Stock Exchange) over the period 1999–2011. The empirical analysis shows that models with other global market price information outperform forecast models based merely on auto-regressive past lags and technical indicators. Shanghai stock index movement was predicted best by Hang Seng Index opening price (57.69), Hang Seng Index by previous day’s S&P500 closing price (54.34), FTSE by previous day’s S&P500 closing price (57.94), Straits Times Index by previous day’s Dow Jones closing price (54.44), Nifty by HSI opening price (60), S&P500 by STI closing price (55.31) and DJIA by HSI opening price (55.22), and Nifty was found to be the most predictable stock index. Trading using global cues-based forecast model generates greater returns than other models in all the markets. The study provides evidence that stock markets across the globe are integrated and the information on price transmission across markets, including emerging markets, can induce better returns in day trading.     

Non-linear properties of conditional returns under scale mixtures

This article provides a framework for analyzing multifactor financial returns that violate the Gaussian distributional assumption. Analytical expressions are provided for the non-linear regression equation and its prediction error (heteroscedasticity) by modeling the returns of financial assets as scale mixtures of the multivariate normal distribution. The expressions involve conditional moments of the mixing variable. These conditional moments are explicitly derived when the mixing variable belongs to the generalized inverse Gaussian family, of which gamma, inverse gamma and the inverse Gaussian distributions are distinguished members. The derived expressions are non-linear in the parameters and involve the modified Bessel function of the third kind. The effects of the non-linear model, in terms of both the regression equation and heteroscedasticity against the corresponding values for the standard linear regression model, are captured through simulations for the gamma, inverse gamma and inverse Gaussian distributions. The proposed scale mixture models extend the well-known arbitrage pricing theory (APT) in financial modeling to non-Gaussian cases. The methodology is applied to analyze the intra-day log returns quarterly data for DELL and COKE regressed against S&P 500 for the years 1998-2000.     

Modeling and simulation of the market fluctuations by the finite range contact systems

In this paper, we model a new random stock price model for the stock markets based on the finite range contact process, which is a model for epidemic spreading that mimics the interplay of local infections and recovery of individuals, it is a member of a class of stochastic processes known as interacting particle systems. Then, we analyze the statistical behaviors of Shanghai Stock Exchange (SSE) Composite Index, Shenzhen Stock Exchange (SZSE) Composite Index, Dow Jones Industrial Average Index (DJIA), Nasdaq Composite Index (IXIC), the standard and Poor’s 500 Index (S&P500) and the simulative data derived from the finite range contact model by comparison. And six individual Chinese stocks from large-cap, mid-cap and small-cap categories are discussed. Furthermore, we investigate the long range correlations of the returns for these indices and the corresponding simulative data by applying the detrended fluctuation analysis. At last, the positive part of the probability distributions of the logarithmic returns for the actual data and the simulative data are studied by the q-Gaussian dynamic systems. The main objective of this work is to discuss the impact on the returns with the different range financial models.     

Quantile autoregression neural network model with applications to evaluating value at risk

We develop a new quantile autoregression neural network (QARNN) model based on an artificial neural network architecture. The proposed QARNN model is flexible and can be used to explore potential nonlinear relationships among quantiles in time series data. By optimizing an approximate error function and standard gradient based optimization algorithms, QARNN outputs conditional quantile functions recursively. The utility of our new model is illustrated by Monte Carlo simulation studies and empirical analyses of three real stock indices from the Hong Kong Hang Seng Index (HSI), the US S&P500 Index (S&P500) and the Financial Times Stock Exchange 100 Index (FTSE100).     

Realized stochastic volatility with leverage and long memory

The daily return and the realized volatility are simultaneously modeled in the stochastic volatility model with leverage and long memory. The dependent variable in the stochastic volatility model is the logarithm of the squared return, and its error distribution is approximated by a mixture of normals. In addition, the logarithm of the realized volatility is incorporated into the measurement equation, assuming that the latent log volatility follows an Autoregressive Fractionally Integrated Moving Average (ARFIMA) process to describe its long memory property. The efficient Bayesian estimation method using Markov chain Monte Carlo method (MCMC) was proposed and implemented in the state space representation. Model comparisons are performed based on the marginal likelihood, and the volatility forecasting performances are investigated using S&P500 stock index returns.     

A hybrid modeling approach for forecasting the volatility of S&P 500 index return

Forecasting volatility is an essential step in many financial decision makings. GARCH family of models has been extensively used in finance and economics, particularly for estimating volatility. The motivation of this study is to enhance the ability of GARCH models in forecasting the return volatility. We propose two hybrid models based on EGARCH and Artificial Neural Networks to forecast the volatility of S&P 500 index. The estimates of volatility obtained by an EGARCH model are fed forward to a Neural Network. The input to the first hybrid model is complemented by historical values of other explanatory variables. The second hybrid model takes as inputs both series of the simulated data and explanatory variables. The forecasts obtained by each of those hybrid models have been compared with those of EGARCH model in terms of closeness to the realized volatility. The computational results demonstrate that the second hybrid model provides better volatility forecasts.     

On estimation of a heteroscedastic measurement error model under heavy-tailed distributions

It is common in epidemiology and other fields that the analyzing data is collected with error-prone observations and the variances of the measurement errors change across observations. Heteroscedastic measurement error (HME) models have been developed for such data. This paper extends the structural HME model to situations in which the observations jointly follow scale mixtures of normal (SMN) distribution. We develop the EM algorithm to compute the maximum likelihood estimates for the model with and without equation error respectively, and derive closed forms of asymptotic variances. We also conduct simulations to verify the effective of the EM estimates and confirm their robust behaviors based on heavy-tailed SMN distributions. A practical application is reported for the data from the WHO MONICA Project on cardiovascular disease.     

Measuring actual daily volatility from high frequency intraday returns of the S&P futures and index observations

In this study 10 min frequency realized variance series are used to forecast the volatility of S&P 500 index (SPX) daily returns. The logarithm-transformed realized variances are modeled directly in the AR(FI)MA model specification in which the structure of the model is optimized using the AICc criterion. As reported in previous literature, the approximately normal structure of distribution of the logarithm-transformed realized variance series can be modeled directly in structure of the AR(FI)MA process. However, in this study, it is recognized the statistically significant non-normal property of the logarithm-transformed realized variances. Hence, to forecast volatility the non-normality is exploited to improve efficiency of volatility forecasts. It is also observed that in the context of the AR(FI)MA model specification the futures and index based deseasonalized returns for the realized variance estimates improve the forecast performance. Considering the seasonality effect and the distributional properties of the estimated realized variance series, it is evident that the information content of the futures (ES) high frequency observations produces the most accurate forecasts.     

Bayesian nonlinear regression models with scale mixtures of skew-normal distributions: Estimation and case influence diagnostics

The purpose of this paper is to develop a Bayesian analysis for nonlinear regression models under scale mixtures of skew-normal distributions. This novel class of models provides a useful generalization of the symmetrical nonlinear regression models since the error distributions cover both skewness and heavy-tailed distributions such as the skew-t, skew-slash and the skew-contaminated normal distributions. The main advantage of these class of distributions is that they have a nice hierarchical representation that allows the implementation of Markov chain Monte Carlo (MCMC) methods to simulate samples from the joint posterior distribution. In order to examine the robust aspects of this flexible class, against outlying and influential observations, we present a Bayesian case deletion influence diagnostics based on the Kullback-Leibler divergence. Further, some discussions on the model selection criteria are given. The newly developed procedures are illustrated considering two simulations study, and a real data previously analyzed under normal and skew-normal nonlinear regression models.     

Bayesian Analysis of Student t Linear Regression with Unknown Change-Point and Application to Stock Data Analysis

This article devotes to studying the variance change-points problem in student t linear regression models. By exploiting the equivalence of the student t distribution and an appropriate scale mixture of normal distributions, a Bayesian approach combined with Gibbs sampling is developed to detect the single and multiple change points. Some simulation studies are performed to display the process of the detection and investigate the effects of the developed approach. Finally, for illustration, the Dow Jones index closed data of U.S. market are analyzed and three variance change-points are detected.     

Modelling financial time series based on heavy-tailed market microstructure models with scale mixtures of normal distributions

This paper presents a type of heavy-tailed market microstructure models with the scale mixtures of normal distributions (MM-SMN), which include two specific sub-classes, viz. the slash and the Student-t distributions. Under a Bayesian perspective, the Markov Chain Monte Carlo (MCMC) method is constructed to estimate all the parameters and latent variables in the proposed MM-SMN models. Two evaluating indices, namely the deviance information criterion (DIC) and the test of white noise hypothesis on the standardised residual, are used to compare the MM-SMN models with the classic normal market microstructure (MM-N) model and the stochastic volatility models with the scale mixtures of normal distributions (SV-SMN). Empirical studies on daily stock return data show that the MM-SMN models can accommodate possible outliers in the observed returns by use of the mixing latent variable. These results also indicate that the heavy-tailed MM-SMN models have better model fitting than the MM-N model, and the market microstructure model with slash distribution (MM-s) has the best model fitting. Finally, the two evaluating indices indicate that the market microstructure models with three different distributions are superior to the corresponding stochastic volatility models.     

Intra-daily information of range-based volatility for MEM-GARCH

Conventional GARCH modeling formulates an additive-error mean equation for daily return and an autoregressive moving-average specification for its conditional variance, without much consideration on the effects of intra-daily data. Using Engle’s multiplicative-error model (MEM) formulation, range-based volatility is proposed as an intraday proxy for several GARCH frameworks. The performances of these different approaches for two 8-year market data sets: the S&P 500 and the NASDAQ composite index, are studied and compared. The impact of significant changes in intraday data has been found to reflect in the MEM-GARCH volatility. For some frameworks it is also possible to use lagged values of range-based volatility to delay the intraday effects in the conditional variance estimation.     

The effectiveness of the combined use of VIX and Support Vector Machines on the prediction of S&P 500

The aim of this research is to analyse the effectiveness of the Chicago Board Options Exchange Market Volatility Index (VIX) when used with Support Vector Machines (SVMs) in order to forecast the weekly change in the S&P 500 index. The data provided cover the period between 3 January 2000 and 30 December 2011. A trading simulation is implemented so that statistical efficiency is complemented by measures of economic performance. The inputs retained are traditional technical trading rules commonly used in the analysis of equity markets such as Relative Strength Index, Moving Average Convergence Divergence, VIX and the daily return of the S&P 500. The SVM identifies the best situations in which to buy or sell in the market. The two outputs of the SVM are the movement of the market and the degree of set membership. The obtained results show that SVM using VIX produce better results than the Buy and Hold strategy or SVM without VIX. The influence of VIX in the trading system is particularly significant when bearish periods appear. Moreover, the SVM allows the reduction in the Maximum Drawdown and the annualised standard deviation.     

Nonlinearity in Forecasting of High-Frequency Stock Returns

Using high-frequency S&P 500 data, we examined intraday efficiency by comparing the ability of several nonlinear models to forecast returns for horizons of 5, 10, 30 and 60?min. Taking into account fat tails and volatility dynamics, we compared the forecasting performance of simple random walk and autoregressive models with Markov switching, artificial neural network and support vector machine regression models in terms of both statistical and economic criteria. Our empirical results for out-of-sample forecasts for high and low volatility samples at different time periods provide weak evidence of intraday predictability in terms of statistical criteria, but corroborate the superiority of nonlinear model predictability using economic criteria such as trading rule profitability and value-at-risk calculations.     

Generation of cubic power-law for high frequency intra-day returns: Maximum Tsallis entropy framework

A theoretical framework based on the maximum Tsallis entropy is proposed to explain the tail behavior of the intra-day stock returns, providing a rationale for the cubic law behavior for high frequency data. The specification of first two time-dependent moment constraints yields a q-Gaussian distribution for the intra-day stock returns. The value of the parameter q is estimated by minimizing appropriately modified Jensen–Shannon (JS) divergence in Tsallis entropy framework between q-Gaussian distribution and empirical NASDAQ 100 data. The estimated value of q yields the well-known empirically observed cubic law tail behavior of the intra-day stock returns which has been observed for high frequency data sets. To validate the cubic law stylized fact, five more data sets from high frequency NASDAQ 100, S&P 500 and NYSE index have been examined and it is found that the cubic law operates.