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.     

Bayesian changepoint and time-varying parameter learning in regime switching volatility models

This paper proposes a combined state and piecewise time-varying parameter learning technique in regime switching volatility models using multiple changepoint detection. This approach is a Sequential Monte Carlo method for estimating GARCH & EGARCH based volatility models with an unknown number of changepoints. Modern auxiliary particle filtering techniques are used to calculate the posterior densities and online forecasts. This approach also automatically deals with the common ancestral path dependence problem faced in these type volatility models. The model is tested on Borsa Istanbul (BIST) formerly known as Istanbul Stock Exchange (ISE) market data using daily log returns. A full structural changepoint specification is defined in which all parameters of the conditional variance of the volatility models are dynamic. Finally, it is shown with simulation experiments that the proposed approach partitions the series into several regimes and learns the parameters of each regime's volatility model in parallel with the multiple changepoint detection process.     

Financial volatility trading using recurrent neural networks

We simulate daily trading of straddles on financial indexes. The straddles are traded based on predictions of daily volatility differences in the indexes. The main predictive models studied are recurrent neural nets (RNN). Such applications have often been studied in isolation. However, due to the special character of daily financial time-series, it is difficult to make full use of RNN representational power. Recurrent networks either tend to overestimate noisy data, or behave like finite-memory sources with shallow memory; they hardly beat classical fixed-order Markov models. To overcome data nonstationarity, we use a special technique that combines sophisticated models fitted on a larger data set, with a fixed set of simple-minded symbolic predictors using only recent inputs. Finally, we compare our predictors with the GARCH family of econometric models designed to capture time-dependent volatility structure in financial returns. GARCH models have been used to trade volatility. Experimental results show that while GARCH models cannot generate any significantly positive profit, by careful use of recurrent networks or Markov models, the market makers can generate a statistically significant excess profit, but then there is no reason to prefer RNN over much more simple and straightforward Markov models. We argue that any report containing RNN results on financial tasks should be accompanied by results achieved by simple finite-memory sources combined with simple techniques to fight nonstationarity in the data.     

A double-threshold GARCH model of stock market and currency shocks on stock returns

International integration of financial markets provides a channel for currency movements to affect stock prices. This paper applies a four-regime double-threshold GARCH (DTGARCH) model of stock market returns to investigate empirically the effects of daily currency movements on five stock market returns, namely in Taiwan, Singapore, South Korea, Japan and the USA. The asymmetric reactions of the mean and volatility stock returns in five markets to stock market and foreign exchange news are investigated using linear and nonlinear models. We discuss a four-regime DTGARCH model, which allows for asymmetry in both the conditional mean and conditional variance simultaneously by using two threshold variables to analyze stock market reactions to different types of information (that is, positive and negative news) that are generated from stock and foreign exchange markets. By applying the four-regime DTGARCH model, this paper finds that the interactions between the information of stock and foreign exchange markets lead to asymmetric reactions of stock returns and their associated variability. The empirical results show that international fund managers who invest in newly emerging stock markets need to evaluate the value and stability of domestic currencies as part of their stock market investment decisions.     

Generalised long-memory GARCH models for intra-daily volatility

The class of fractionally integrated generalised autoregressive conditional heteroskedastic (FIGARCH) models is extended for modelling the periodic long-range dependence typically shown by volatility of most intra-daily financial returns. The proposed class of models introduces generalised periodic long-memory filters, based on Gegenbauer polynomials, into the equation describing the time-varying volatility of standard GARCH models. A fitting procedure is illustrated and its performance is evaluated by means of Monte Carlo simulations. The effectiveness of these models in describing periodic long-memory volatility patterns is shown through an empirical application to the Euro-Dollar intra-daily exchange rate.     

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.     

Fat tails and asymmetry in financial volatility models

Although the generalised autoregressive conditional heteroskedasticity (GARCH) model has been quite successful in capturing important empirical aspects of financial data, particularly for the symmetric effects of volatility, it has had far less success in capturing the effects of extreme observations, outliers and skewness in returns. This paper examines the GARCH model under various non-normal error distributions in order to evaluate skewness and leptokurtosis. The empirical results show that GARCH models estimated using asymmetric leptokurtic distributions are superior to their counterparts estimated under normality, in terms of: (i) capturing skewness and leptokurtosis; (ii) the maximized log-likelihood values; and (iii) isolating the ARCH and GARCH parameter estimates from the adverse effects of outliers. Overall, the flexible asymmetric Student’s t-distribution performs best in capturing the non-normal aspects of the data.     

Estimating stochastic volatility models using daily returns and realized volatility simultaneously

Realized volatility, which is the sum of squared intraday returns over a certain interval such as a day, has recently attracted the attention of financial economists and econometricians as an accurate measure of the true volatility. In the real market, however, the presence of non-trading hours and market microstructure noise in transaction prices may cause bias in the realized volatility. On the other hand, daily returns are less subject to noise and therefore may provide additional information on the true volatility. From this point of view, modeling realized volatility and daily returns simultaneously based on the well-known stochastic volatility model is proposed. Empirical studies using intraday data of Tokyo stock price index show that this model can estimate realized volatility biases and parameters simultaneously. The Bayesian approach is taken and an efficient sampling algorithm is proposed to implement the Markov chain Monte Carlo method for our simultaneous model. The result of the model comparison between the simultaneous models using both naive and scaled realized volatilities indicates that the effect of non-trading hours is more essential than that of microstructure noise and that asymmetry is crucial in stochastic volatility models. The proposed Bayesian approach provides an estimate of the entire conditional predictive distribution of returns under consideration of the uncertainty in the estimation of both biases and parameters. Hence common risk measures, such as value-at-risk and expected shortfall, can be easily estimated.     

基于GARCH模型的网络新闻与舆情的波动性分析

网络新闻产生的舆情波动一般具有异方差特征,难以用普通模型拟合。由诺贝尔经济学奖获得者恩格尔教授提出的条件异方差(GARCH)模型在分析证券价格波动性方面获得极大成功。本文利用GARCH模型分析网络新闻与舆情的波动性,通过典型事件的舆情采集,分析数据的特征。研究表明,网络新闻与舆情的波动性符合GARCH模型的特征,通过参数的调整和检验,可以实现模型与数据的良好拟合。     

Testing for expected return and market price of risk in Chinese A and B share markets: A geometric Brownian motion and multivariate GARCH model approach

There exist dual listed stocks which are issued by the same company in some stock markets. Although these stocks bare the same firm-specific risks and enjoy identical dividends and voting policies, they are priced differently. Some previous studies show this seeming deviation from the law of one price can be solved by allowing different expected returns and market prices of risk for investors holding heterogeneous beliefs. This paper provides empirical evidence for that argument by testing the expected return and market price of risk between Chinese A and B shares listed in Shanghai and Shenzhen stock markets. Models with dynamic of Geometric Brownian Motion are adopted. Multivariate GARCH models are also introduced to capture the feature of time-varying volatility in stock returns. The results suggest that the different pricing can be explained by the difference in expected returns between A and B shares. However, the difference between market price of risk is insignificant for both markets if GARCH models are adopted.     

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.     

Heavy-tailed mixture GARCH volatility modeling and Value-at-Risk estimation

This paper presents a heavy-tailed mixture model for describing time-varying conditional distributions in time series of returns on prices. Student-t component distributions are taken to capture the heavy tails typically encountered in such financial data. We design a mixture MT(m)-GARCH(p, q) volatility model for returns, and develop an EM algorithm for maximum likelihood estimation of its parameters. This includes formulation of proper temporal derivatives for the volatility parameters. The experiments with a low order MT(2)-GARCH(1, 1) show that it yields results with improved statistical characteristics and economic performance compared to linear and nonlinear heavy-tail GARCH, as well as normal mixture GARCH. We demonstrate that our model leads to reliable Value-at-Risk performance in short and long trading positions across different confidence levels.     

Robust M-estimation of multivariate GARCH models

The Gaussian quasi-maximum likelihood estimator of Multivariate GARCH models is shown to be very sensitive to outliers in the data. A class of robust M-estimators for MGARCH models is developed. To increase the robustness of the estimators, the use of volatility models with the property of bounded innovation propagation is recommended. The Monte Carlo study and an empirical application to stock returns document the good robustness properties of the M-estimator with a fat-tailed Student t loss function.     

Forecasting volatility for the stock market: a new hybrid model

This study presents a new hybrid model that combines the grey forecasting model with the GARCH to improve the variance forecasting ability in variance as compared to the traditional GARCH. A range-based measure of ex post volatility is employed as a proxy for the unobservable volatility process in evaluating the forecasting ability due to true underlying volatility process not being observed. Overall, the results show that the new hybrid model can enhance the volatility forecasting ability of the traditional GARCH.     

Modelling conditional correlations in the volatility of Asian rubber spot and futures returns

Asia is presently the most important market for the production and consumption of natural rubber. World prices of rubber are subject to not only to changes in demand, but also speculation regarding future markets. Japan and Singapore are the major future markets for rubber, while Thailand is one of the world's largest producers of rubber. As rubber prices are influenced by external markets, it is important to analyse the relationship between the relevant markets in Thailand, Japan and Singapore. The analysis is conducted using several alternative multivariate GARCH models. The empirical results indicate that the constant conditional correlations arising from the CCC model lie in the low to medium range. The results from the VARMA-GARCH model and the VARMA-AGARCH model suggest the presence of volatility spillovers and asymmetric effects of positive and negative return shocks on conditional volatility. Finally, the DCC model suggests that the conditional correlations can vary dramatically over time. In general, the dynamic conditional correlations in rubber spot and futures returns shocks can be independent or interdependent.     

Forecasting large scale conditional volatility and covariance using neural network on GPU

Forecasting volatility is an important issue in financial econometric analysis. This paper aims to seek a computationally feasible approach for predicting large scale conditional volatility and covariance of financial time series. In the case of multi-variant time series, the volatility is represented by a Conditional Covariance Matrix (CCM). Traditional models for predicting CCM such as GARCH models are incapable of dealing with high-dimensional cases as there are O(N ²) parameters to be estimated in the case of N-variant asset return, and it is difficult to accelerate the computation of estimating these parameters by utilizing modern multi-core architecture. These GARCH models also have difficulties in modeling non-linear properties. The widely used Restricted Boltzmann Machine (RBM) is an energy-based stochastic recurrent neural network and its extended model, Conditional RBM (CRBM), has shown its capability in modeling high-dimensional time series. In this paper, we first propose a CRBM-based approach to forecast CCM and show how to capture the long memory properties in volatility, and then we implement the proposed model on GPU by using CUDA and CUBLAS. Experiment results indicate that the proposed CRBM-based model obtains better forecasting accuracy for low-dimensional volatility and it also shows great potential in modeling for large-scale cases compared with traditional GARCH models.     

Adaptive signal processing of asset price dynamics with predictability analysis

In this paper we illustrate the optimal filtering of log returns of commodity prices in which both the mean and volatility are modulated by a hidden Markov chain with finite state space. The optimal estimate of the Markov chain and the parameters of the price model are given in terms of discrete-time recursive filters. We provide an application on a set of high frequency gold price data for the period 1973-2006 and analyse the h-step ahead price predictions against the Diebold-Kilian metric. Within the modelling framework where the mean and volatility are switching regimes, our findings suggest that a two-state hidden Markov model is sufficient to describe the dynamics of the data and the gold price is predictable up to a certain extent in the short term but almost impossible to predict in the long term. The proposed model is also benchmarked with ARCH and GARCH models with respect to price predictability and forecasting errors.     

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.     

An algorithm for nonparametric GARCH modelling

A simple iterative algorithm for nonparametric first-order GARCH modelling is proposed. This method offers an alternative to fitting one of the many different parametric GARCH specifications that have been proposed in the literature. A theoretical justification for the algorithm is provided and examples of its application to simulated data from various stationary processes showing stochastic volatility, as well as empirical financial return data, are given. The nonparametric procedure is found to often give better estimates of the unobserved latent volatility process than parametric modelling with the standard GARCH(1,1) model, particularly in the presence of asymmetry and other departures from the standard GARCH specification. Extensions of the basic iterative idea to more complex time series models combining ARMA or GARCH features of possibly higher order are suggested.     

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.