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.     

Simplified specifications of a multivariate generalized autoregressive conditional heteroscedasticity model

Recent developments in multivariate volatility modeling suggest that the conditional correlation matrix can be described by a time series recursion, where the total number of parameters grows by the power-of-two of the dimension of financial returns. The power of two computational requirement makes high-dimensional multivariate volatility modeling very time consuming. In this paper, we propose two simplified specifications in a multivariate autoregressive conditional heteroscedasticity model. The first specification computes an unconditional correlation matrix from standardized residuals of the model. The second specification restricts the sum of the weights in a time-varying conditional correlation equation to be one. Applying a Bayesian sampling scheme allows the number of parameters to be reduced from the power of two of the dimension to the linear order of the dimension only and simultaneously provides us a framework for model comparison. We test our simplified specifications using simulated and real data from three sectoral indices in Hong Kong, three market indices and four exchange rates. The results suggest that our simplified specifications are more effective than the original formulation.     

Volatility forecast using hybrid Neural Network models

In this research the testing of a hybrid Neural Networks-GARCH model for volatility forecast is performed in three Latin-American stock exchange indexes from Brazil, Chile and Mexico. A detail of the methodology and application of the volatility forecast of financial series using a hybrid artificial Neural Network model are presented.The results demonstrate that the ANN models can improve the forecasting performance of the GARCH models when studied in the three Latin-American markets and it is shown that the results are robust and consistent for different ANN specifications and different volatility measures.     

Inductive transfer for learning Bayesian networks

In several domains it is common to have data from different, but closely related problems. For instance, in manufacturing, many products follow the same industrial process but with different conditions; or in industrial diagnosis, where there is equipment with similar specifications. In these cases it is common to have plenty of data for some scenarios but very little for others. In order to learn accurate models for rare cases, it is desirable to use data and knowledge from similar cases; a technique known as transfer learning. In this paper we propose an inductive transfer learning method for Bayesian networks, that considers both structure and parameter learning. For structure learning we use conditional independence tests, by combining measures from the target task with those obtained from one or more auxiliary tasks, using a novel weighted sum of the conditional independence measures. For parameter learning, we propose two variants of the linear pool for probability aggregation, combining the probability estimates from the target task with those from the auxiliary tasks. To validate our approach, we used three Bayesian networks models that are commonly used for evaluating learning techniques, and generated variants of each model by changing the structure as well as the parameters. We then learned one of the variants with a small dataset and combined it with information from the other variants. The experimental results show a significant improvement in terms of structure and parameters when we transfer knowledge from similar tasks. We also evaluated the method with real-world data from a manufacturing process considering several products, obtaining an improvement in terms of log-likelihood between the data and the model when we do transfer learning from related products.     

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.     

Comparison of nonnested asymmetric heteroskedastic models

The GJR-GARCH model is a popular choice among nonlinear models of the well-known asymmetric volatility phenomenon in financial market data. However, recent work employs double threshold nonlinear models to capture both mean and volatility asymmetry. A Bayesian model comparison procedure is proposed to compare the GJR-GARCH with various double threshold GARCH specifications, by designing a reversible jump Markov chain Monte Carlo algorithm. A simulation experiment illustrates good performance in estimation and model selection over reasonable sample sizes. In a study of seven markets strong evidence is found that the DTGARCH, with US market news as threshold variable, outperforms the GJR-GARCH and traditional self-exciting DTGARCH models. This result was consistent across six markets, excluding Canada.     

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.     

Robust Bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions

A Bayesian analysis of stochastic volatility (SV) models using the class of symmetric scale mixtures of normal (SMN) distributions is considered. In the face of non-normality, this provides an appealing robust alternative to the routine use of the normal distribution. Specific distributions examined include the normal, student-t, slash and the variance gamma distributions. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo (MCMC) algorithm is introduced for parameter estimation. Moreover, the mixing parameters obtained as a by-product of the scale mixture representation can be used to identify outliers. The methods developed are applied to analyze daily stock returns data on S&P500 index. Bayesian model selection criteria as well as out-of-sample forecasting results reveal that the SV models based on heavy-tailed SMN distributions provide significant improvement in model fit as well as prediction to the S&P500 index data over the usual normal model.     

GARCH dependence in extreme value models with Bayesian inference

Extreme value methods are widely used in financial applications such as risk analysis, forecasting and pricing models. One of the challenges with their application in finance is accounting for the temporal dependence between the observations, for example the stylised fact that financial time series exhibit volatility clustering. Various approaches have been proposed to capture the dependence. Commonly a two-stage approach is taken, where the volatility dependence is removed using a volatility model like a GARCH (or one of its many incarnations) followed by application of standard extreme value models to the assumed independent residual innovations.This study examines an alternative one stage approach, which makes parameter estimation and accounting for the associated uncertainties more straightforward than the two-stage approach. The location and scale parameters of the extreme value distribution are defined to follow a conditional autoregressive heteroscedasticity process. Essentially, the model implements GARCH volatility via the extreme value model parameters. Bayesian inference is used and implemented via Markov chain Monte Carlo, to permit all sources of uncertainty to be accounted for. The model is applied to both simulated and empirical data to demonstrate performance in extrapolating the extreme quantiles and quantifying the associated uncertainty.     

Forecasting conditional correlations in stock,bond and foreign exchange markets

The paper forecasts conditional correlations between three classes of international financial assets, namely stock, bond and foreign exchange. Two countries are considered, namely Australia and New Zealand. Forecasting will be conducted using three multivariate GARCH models, namely the CCC model [T. Bollerslev, Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model, Rev. Econ. Stat. 72 (1990) 498–505], VARMA-GARCH model [S. Ling, M. McAleer, Asymptotic theory for a vector ARMA-GARCH model, Econometric Theory 19 (2003) 280–310], and VARMA-AGARCH model [M. McAleer, S. Hoti, F. Chan, Structure and asymptotic theory for multivariate asymmetric volatility, Econometric Rev., in press]. A rolling window technique is used to forecast 1-day ahead conditional correlations. To evaluate the impact of model specification on conditional correlations forecasts, this paper calculates and compares the correlations between conditional correlations forecasts resulted from the three models. The paper finds the evidence of volatility spillovers and asymmetric effect of negative and positive shock on the conditional variance in most pairs of series. However, it suggests that incorporating volatility spillovers and asymmetric do not contribute to better conditional correlations forecasts.     

Modelling Australian interest rate swap spreads by mixture autoregressive conditional heteroscedastic processes

The observed difference between the swap rate and the government bond yield of corresponding maturity is known as the swap spread. The swap spread reflects the risk premium that is involved in a swap transaction instead of holding risk-free government bonds. It is primarily composed of the liquidity risk premium and the credit risk premium. In recent years there has been growing interest in modelling swap spreads because the swap spread is the key pricing variable for the swap rate. The Australian interest rate swap market is the most important over-the-counter (OTC) derivative market in Australia. In this paper we apply the class of mixture autoregressive conditional heteroscedastic (MARCH) models to three (3-year, 5-year and 10-year) swap spread series in Australia. The MARCH model is able to capture both of the stylised characteristics of the observed changes of the swap spread series: volatility persistence and the dependence of volatility on the level of the data. The proposed MARCH model also allows for regime switches in the swap spreads.     

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.     

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.     

Degrees of conditional (in)dependence: A framework for approximate Bayesian networks and examples related to the rough set-based feature selection

Bayesian networks provide the means for representing probabilistic conditional independence. Conditional independence is widely considered also beyond the theory of probability, with linkages to, e.g. the database multi-valued dependencies, and at a higher abstraction level of semi-graphoid models. The rough set framework for data analysis is related to the topics of conditional independence via the notion of a decision reduct, to be considered within a wider domain of the feature selection. Given probabilistic version of decision reducts equivalent to the data-based Markov boundaries, the studies were also conducted for other criteria of the rough-set-based feature selection, e.g. those corresponding to the multi-valued dependencies. In this paper, we investigate the degrees of approximate conditional dependence, which could be a topic corresponding to the well-known notions such as conditional mutual information and polymatroid functions, however, with many practically useful approximate conditional independence models unmanageable within the information theoretic framework. The major paper’s contribution lays in extending the means for understanding the degrees of approximate conditional dependence, with appropriately generalized semi-graphoid properties formulated and with the mathematical soundness of the Bayesian network-like representation of the approximate conditional independence statements thoroughly proved. As an additional contribution, we provide a case study of the approximate conditional independence model, which would not be manageable without the above-mentioned extensions.     

An empirical evaluation of fat-tailed distributions in modeling financial time series

There is substantial evidence that many financial time series exhibit leptokurtosis and volatility clustering. We compare the two most commonly used statistical distributions in empirical analysis to capture these features: the t distribution and the generalized error distribution (GED). A Bayesian approach using a reversible-jump Markov chain Monte Carlo method and a forecasting evaluation method are adopted for the comparison. In the Bayesian evaluation of eight daily market returns, we find that the fitted t error distribution outperforms the GED. In terms of volatility forecasting, models with t innovations also demonstrate superior out-of-sample performance.     

Parameterisation and efficient MCMC estimation of non-Gaussian state space models

The impact of parameterisation on the simulation efficiency of Bayesian Markov chain Monte Carlo (MCMC) algorithms for two non-Gaussian state space models is examined. Specifically, focus is given to particular forms of the stochastic conditional duration (SCD) model and the stochastic volatility (SV) model, with four alternative parameterisations of each model considered. A controlled experiment using simulated data reveals that relationships exist between the simulation efficiency of the MCMC sampler, the magnitudes of the population parameters and the particular parameterisation of the state space model. Results of an empirical analysis of two separate transaction data sets for the SCD model, as well as equity and exchange rate data sets for the SV model, are also reported. Both the simulation and empirical results reveal that substantial gains in simulation efficiency can be obtained from simple reparameterisations of both types of non-Gaussian state space models.     

A class of nonlinear stochastic volatility models and its implications for pricing currency options

A class of stochastic volatility (SV) models is proposed by applying the Box-Cox transformation to the volatility equation. This class of nonlinear SV (N-SV) models encompasses all standard SV models, including the well-known lognormal (LN) SV model. It allows to empirically compare and test all standard specifications in a very convenient way and provides a measure of the degree of departure from the classical models. A likelihood-based technique is developed for analyzing the model. Daily dollar/pound exchange rate data provide some evidence against LN model and strong evidence against all the other classical specifications. An efficient algorithm is proposed to study the economic importance of the proposed model on pricing currency options.     

Executives’ perceived environmental uncertainty shortly after 9/11

Environmental uncertainty refers to situations when decision makers experience difficulty in predicting their organizations’ environments. Prediction difficulty is mapped by closeness of decision makers’ probability distributions of environmental variables to the uniform distribution. A few months after the 9/11 terrorist attacks, we solicited probabilities for three environmental variables from 93 business executives by a mail survey. Each executive assigned probabilities to the future state of the economy specified as categories of growth projected for a year after the 9/11 jolt, conditional probabilities of its effect on her/his organization, and conditional probabilities of her/his organizational response capability to each economic condition. Shannon entropy maps uncertainty, but the data do not provide trivariate state-effect-response distribution. We use maximum entropy method to impute the trivariate distributions from the data on state-effect and state-response bivariate probabilities. Uncertainty about each executive’s probability distribution is taken into account in two ways: using a Dirichlet model with each executive’s distribution as its mode, and using a Bayesian hierarchical model for the entropy. Both models reduce the observed heterogeneity among the executives’ environmental uncertainty. A Bayesian regression examines the effects of two organizational characteristics on uncertainty. Presentation of results includes uncertainty tableaux for visualizations of the joint and marginal entropies and mutual information between variables.     

Bayesian estimation of the Gaussian mixture GARCH model

Bayesian inference and prediction for a generalized autoregressive conditional heteroskedastic (GARCH) model where the innovations are assumed to follow a mixture of two Gaussian distributions is performed. The mixture GARCH model can capture the patterns usually exhibited by many financial time series such as volatility clustering, large kurtosis and extreme observations. A Griddy-Gibbs sampler implementation is proposed for parameter estimation and volatility prediction. Bayesian prediction of the Value at Risk is also addressed providing point estimates and predictive intervals. The method is illustrated using the Swiss Market Index.     

Bayesian inference with stochastic volatility models using continuous superpositions of non-Gaussian Ornstein-Uhlenbeck processes

Continuous superpositions of Ornstein-Uhlenbeck processes are proposed as a model for asset return volatility. An interesting class of continuous superpositions is defined by a Gamma mixing distribution which can define long memory processes. In contrast, previously studied discrete superpositions cannot generate this behaviour. Efficient Markov chain Monte Carlo methods for Bayesian inference are developed which allow the estimation of such models with leverage effects. The continuous superposition model is applied to both stock index and exchange rate data. The continuous superposition model is compared with a two-component superposition on the daily Standard and Poor’s 500 index from 1980 to 2000.