A regime-switching real-time copula GARCH model for optimal futures hedging

A regime-switching real-time copula GARCH (RSRTCG) model is suggested for optimal futures hedging. The specification of RSRTCG is to model the margins of asset returns with state-dependent real-time GARCH and the dependence structure of asset returns with regime switching copula functions. RSRTCG is faster in adjusting to the new level of volatility under different market regimes which is a regime-switching multivariate generalization of the state-independent univariate real-time GARCH. RSRTCG is applied to cross hedge the price risk of S&P 500 sector indices with crude oil futures. The empirical results show that RSRTCG possesses superior hedging performance compared to its nested non-real-time or state-independent copula GARCH models based on the criterion of percentage variance reduction, utility gain, model confidence set, model combination strategy, risk-adjusted return and reward-to-semivariance ratio.     

On accurate and provably efficient GARCH option pricing algorithms

The GARCH model has been very successful in capturing the serial correlation of asset return volatilities. As a result, applying the model to options pricing attracts a lot of attention. However, previous tree-based GARCH option pricing algorithms suffer from exponential running time, a cut-off maturity, inaccuracy, or some combination thereof. Specifically, this paper proves that the popular trinomial-tree option pricing algorithms of Ritchken and Trevor (Ritchken, P. and Trevor, R., Pricing options under generalized GARCH and stochastic volatility processes. J. Finance, , 54(1), 377–402.) and Cakici and Topyan (Cakici, N. and Topyan, K., The GARCH option pricing model: a lattice approach. J. Comput. Finance, , 3(4), 71–85.) explode exponentially when the number of partitions per day, n, exceeds a threshold determined by the GARCH parameters. Furthermore, when explosion happens, the tree cannot grow beyond a certain maturity date, making it unable to price derivatives with a longer maturity. As a result, the algorithms must be limited to using small n, which may have accuracy problems. The paper presents an alternative trinomial-tree GARCH option pricing algorithm. This algorithm provably does not have the short-maturity problem. Furthermore, the tree-size growth is guaranteed to be quadratic if n is less than a threshold easily determined by the model parameters. This level of efficiency makes the proposed algorithm practical. The surprising finding for the first time places a tree-based GARCH option pricing algorithm in the same complexity class as binomial trees under the Black–Scholes model. Extensive numerical evaluation is conducted to confirm the analytical results and the numerical accuracy of the proposed algorithm. Of independent interest is a simple and efficient technique to calculate the transition probabilities of a multinomial tree using generating functions.     

Forecasting the return distribution using high-frequency volatility measures

The aim of this paper is to forecast (out-of-sample) the distribution of financial returns based on realized volatility measures constructed from high-frequency returns. We adopt a semi-parametric model for the distribution by assuming that the return quantiles depend on the realized measures and evaluate the distribution, quantile and interval forecasts of the quantile model in comparison to a benchmark GARCH model. The results suggest that the model outperforms an asymmetric GARCH specification when applied to the S&P 500 futures returns, in particular on the right tail of the distribution. However, the model provides similar accuracy to a GARCH (1, 1) model when the 30-year Treasury bond futures return is considered.     

Forecasting VaR using analytic higher moments for GARCH processes

It is widely accepted that some of the most accurate Value-at-Risk (VaR) estimates are based on an appropriately specified GARCH process. But when the forecast horizon is greater than the frequency of the GARCH model, such predictions have typically required time-consuming simulations of the aggregated returns distributions. This paper shows that fast, quasi-analytic GARCH VaR calculations can be based on new formulae for the first four moments of aggregated GARCH returns. Our extensive empirical study compares the Cornish–Fisher expansion with the Johnson SU distribution for fitting distributions to analytic moments of normal and Student t, symmetric and asymmetric (GJR) GARCH processes to returns data on different financial assets, for the purpose of deriving accurate GARCH VaR forecasts over multiple horizons and significance levels.     

Intraday Return Volatility Process: Evidence from NASDAQ Stocks

This paper presents a comprehensive analysis of the distributional and time-series properties of intraday returns. The purpose is to determine whether a GARCH model that allows for time varying variance in a process can adequately represent intraday return volatility. Our primary data set consists of 5-minute returns, trading volumes, and bid-ask spreads during the period January 1, 1999 through March 31, 1999, for a subset of thirty stocks from the NASDAQ 100 Index. Our results indicate that the GARCH(1,1) model best describes the volatility of intraday returns. Current volatility can be explained by past volatility that tends to persist over time. These results are consistent with those of Akgiray (1989) who estimates volatility using the various ARCH and GARCH specifications and finds the GARCH(1,1) model performs the best. We add volume as an additional explanatory variable in the GARCH model to examine if volume can capture the GARCH effects. Consistent with results of Najand and Yung (1991) and Foster (1995) and contrary to those of Lamoureux and Lastrapes (1990), our results show that the persistence in volatility remains in intraday return series even after volume is included in the model as an explanatory variable. We then substitute bid-ask spread for volume in the conditional volatility equation to examine if the latter can capture the GARCH effects. The results show that the GARCH effects remain strongly significant for many of the securities after the introduction of bid-ask spread. Consistent with results of Antoniou, Homes and Priestley (1998), intraday returns also exhibit significant asymmetric responses of volatility to flow of information into the market.     

Tempered stable and tempered infinitely divisible GARCH models

In this paper, we introduce a new GARCH model with an infinitely divisible distributed innovation. This model, which we refer to as the rapidly decreasing tempered stable (RDTS) GARCH model, takes into account empirical facts that have been observed for stock and index returns, such as volatility clustering, non-zero skewness, and excess kurtosis for the residual distribution. We review the classical tempered stable (CTS) GARCH model, which has similar statistical properties. By considering a proper density transformation between infinitely divisible random variables, we can find the risk-neutral price process, thereby allowing application to option-pricing. We propose algorithms to generate scenarios based on GARCH models with CTS and RDTS innovations. To investigate the performance of these GARCH models, we report parameter estimates for the Dow Jones Industrial Average index and stocks included in this index. To demonstrate the advantages of the proposed model, we calculate option prices based on the index.     

Forecasting Value-at-Risk using high frequency data: The realized range model

Current studies on financial market risk measures usually use daily returns based on GARCH type models. This paper models realized range using intraday high frequency data based on CARR framework and apply it to VaR forecasting. Kupiec LR test and dynamic quantile test are used to compare the performance of VaR forecasting of realized range model with another intraday realized volatility model and daily GARCH type models. Empirical results of Chinese Stock Indices show that realized range model performs the same with realized volatility model, which performs much better than daily models.     

Dynamic Asymmetric GARCH

This article develops the dynamic asymmetric GARCH (or DAGARCH)model that generalizes asymmetric GARCH models such as thatof Glosten, Jagannathan, and Runkle (GJR), introduces multiplethresholds, and makes the asymmetric effect time dependent.We provide the stationarity conditions for the DAGARCH modeland show how GJR can be obtained as a special case. Furthermore,we derive the news impact curve implied by the DAGARCH modeland demonstrate its flexibility. An application to daily stockmarket indices is presented to demonstrate the practical usefulnessof the new model.     

Robust outlier detection for Asia–Pacific stock index returns

Outliers can lead to model misspecifications, poor forecasts and invalid inferences. Their identification and correction is therefore an important objective of financial modeling.This paper introduces a simple method to detect outliers in a financial series. It uses an AR(1)–GARCH(1,1) model to calculate interval forecasts for one-step ahead returns that are then compared to realized returns to determine whether or not we are in the presence of an aberrant observation. The GARCH model, however, is only used as a filter and the identification algorithm remains robust to model misspecifications.The efficiency of this outlier-correction technique is first tested with a simulation study, before being applied to five Asian stock market returns to identify the outlying observations. After an analysis of these extreme fluctuations, the out-of-sample forecasting performance of our outlier-corrected model is then compared to the classical forecasts of a GARCH model in which no account is taken of outliers.     

On the distribution and conditional heteroscedasticity in Taiwan stock prices

This paper studies the distribution and conditional heteroscedasticity in stock returns on the Taiwan stock market. Apart from the normal distribution, in order to explain the leptokurtosis and skewness observed in the stock return distribution, we also examine the Student-t, the Poisson–normal, and the mixed-normal distributions, which are essentially a mixture of normal distributions, as conditional distributions in the stock return process. We also use the ARMA (1,1) model to adjust the serial correlation, and adopt the GJR–generalized autoregressive conditional heteroscedasticity (GARCH (1,1)) model to account for the conditional heterscedasticity in the return process. The empirical results show that the mixed–normal–GARCH model is the most probable specification for Taiwan stock returns. The results also show that skewness seems to be diversifiable through portfolio. Thus the normal–GARCH or the Student-t–GARCH model which involves symmetric conditional distribution may be a reasonable model to describe the stock portfolio return process¹.     

Stock market dynamics in a regime-switching asymmetric power GARCH model

This paper analyzes the dynamics of Asian stock index returns through a Regime-Switching Asymmetric Power GARCH model (RS-APGARCH). The model confirms some stylized facts already discussed in former studies but also highlights interesting new characteristics of stock market returns and volatilities. Mainly, it improves the traditional regime-switching GARCH models by including an asymmetric response to news and, above all, by allowing the power transformations of the heteroskedasticity equations to be estimated directly from the data. Several mixture models are compared where a first-order Markov process governs the switching between regimes.     

Positivity constraints on the conditional variances in the family of conditional correlation GARCH models

In this article, we derive a set of necessary and sufficient conditions for positivity of the vector conditional variance equation in multivariate GARCH models with explicit modelling of conditional correlation. These models include the constant conditional correlation GARCH model of Bollerslev [1990. Review of Economics and Statistics 72, 498–505] and its extensions. Under the new conditions, it is possible to introduce negative volatility spillovers in the model. An empirical example illustrates usefulness of having such conditions in practice.     

Empirical analysis of GARCH models in value at risk estimation

This paper studies seven GARCH models, including RiskMetrics and two long memory GARCH models, in Value at Risk (VaR) estimation. Both long and short positions of investment were considered. The seven models were applied to 12 market indices and four foreign exchange rates to assess each model in estimating VaR at various confidence levels. The results indicate that both stationary and fractionally integrated GARCH models outperform RiskMetrics in estimating 1% VaR. Although most return series show fat-tailed distribution and satisfy the long memory property, it is more important to consider a model with fat-tailed error in estimating VaR. Asymmetric behavior is also discovered in the stock market data that t-error models give better 1% VaR estimates than normal-error models in long position, but not in short position. No such asymmetry is observed in the exchange rate data.     

On measuring volatility and the GARCH forecasting performance

We apply a new algorithm based on Fourier analysis to compute the volatility of a diffusion process. By using simulations of the continuous-time GARCH model, we show that our method performs well in computing integrated volatility. We show that linear interpolation of high frequency observations induces a downward bias in estimating integrated volatility. By measuring ex post volatility with our method, we find that the forecasting performance of the GARCH model is improved with respect to what is established when classical methods are employed. These results are confirmed by the analysis of exchange rate high frequency time series.     

Estimating GARCH models using support vector machines*

Abstract

Support vector machines (SVMs) are a new nonparametric tool for regression estimation. We will use this tool to estimate the parameters of a GARCH model for predicting the conditional volatility of stock market returns. GARCH models are usually estimated using maximum likelihood (ML) procedures, assuming that the data are normally distributed. In this paper, we will show that GARCH models can be estimated using SVMs and that such estimates have a higher predicting ability than those obtained via common ML methods.     

Orthogonal GARCH and covariance matrix forecasting: The Nordic stock markets during the Asian financial crisis 1997–1998

In risk management, modelling large numbers of assets and their variances and covariances in a unified framework is often important. In such multivariate frameworks, it is difficult to incorporate GARCH models and thus a new member of the ARCH-family, Orthogonal GARCH, has been suggested as a remedy to inherent estimation problems in multivariate ARCH modelling. Orthogonal GARCH creates positive definite covariance matrices of any size but builds on assumptions that partly break down during stress scenarios. This article therefore assesses the stress performance of the model by looking at four Nordic stock indices and covariance matrix forecasts during the highly volatile years of 1997 and 1998. Overall, Orthogonal GARCH is found to perform significantly better than traditional historical variance and moving average methods. Out-of-sample evaluation measures include symmetric loss functions (RMSE), asymmetric loss functions, operational methods suggested by the Basle Committee on Banking Supervision, as well as a forecast evaluation methodology based on pricing of simulated ‘rainbow options’.     

An examination of causality and predictability between Australian domestic and offshore interest rates

This paper studies the causality and predictability between Australian domestic and offshore short term interest rates in both the first and second moments during the period 1987 to 1996. Causality flow is observed to be stronger from the domestic to the offshore market in the earlier sub periods but characterised by significant two-way causality flow in the latter sub-periods. Volatility tests show that the volatility in one market spills over to the other market simultaneously, which is consistent with Australian markets being well integrated with global markets. The predictability across the two markets in the first moments is examined through an error correction model, whose forecasting performance is assessed relative to a benchmark random walk model. To test the predictability of volatility, four different models are compared: A GARCH model, A GARCH model incorporating contemporaneous spillover effects, a GARCH model with lagged spillover effects, and a benchmark random walk model. Results indicate that the error correction model and the GARCH model with contemporaneous volatility spillover are the superior models for forecasting changes in interest rates and for forecasting volatility, respectively.     

Forecasting value-at-risk and expected shortfall using fractionally integrated models of conditional volatility: International evidence

The present study compares the performance of the long memory FIGARCH model, with that of the short memory GARCH specification, in the forecasting of multi-period value-at-risk (VaR) and expected shortfall (ES) across 20 stock indices worldwide. The dataset is composed of daily data covering the period from 1989 to 2009. The research addresses the question of whether or not accounting for long memory in the conditional variance specification improves the accuracy of the VaR and ES forecasts produced, particularly for longer time horizons. Accounting for fractional integration in the conditional variance model does not appear to improve the accuracy of the VaR forecasts for the 1-day-ahead, 10-day-ahead and 20-day-ahead forecasting horizons relative to the short memory GARCH specification. Additionally, the results suggest that underestimation of the true VaR figure becomes less prevalent as the forecasting horizon increases. Furthermore, the GARCH model has a lower quadratic loss between actual returns and ES forecasts, for the majority of the indices considered for the 10-day and 20-day forecasting horizons. Therefore, a long memory volatility model compared to a short memory GARCH model does not appear to improve the VaR and ES forecasting accuracy, even for longer forecasting horizons. Finally, the rolling-sampled estimated FIGARCH parameters change less smoothly over time compared to the GARCH models. Hence, the parameters' time-variant characteristic cannot be entirely due to the news information arrival process of the market; a portion must be due to the FIGARCH modelling process itself.