Long-Memory,Asymmetry and Fat-Tailed GARCH Models in Value-at-Risk Estimation: Empirical Evidence from the Global Real Estate Markets

In this paper, we address the question of whether long memory, asymmetry, and fat-tails in global real estate markets volatility matter when forecasting the two most popular measures of risk in financial markets, namely Value-at-risk (VaR) and Expected Shortfall (ESF), for both short and long trading positions. The computations of both VaR and ESF are conducted with three long memory GARCH-class models including the Fractionally Integrated GARCH (FIGARCH), Hyperbolic GARCH (HYGARCH), and Fractionally Integrated Asymmetric Power ARCH (FIAPARCH). These models are estimated under three alternative innovation’s distributions: normal, Student, and skewed Student. To test the efficacy of the forecast, we employ various backtesting methodologies. Our empirical findings show that considering for long memory, fat-tails, and asymmetry performs better in predicting a one-day-ahead VaR and ESF for both short and long trading positions. In particular, the forecasting ability analysis points out that the FIAPARCH model under skewed Student distribution turns out to improve substantially the VaR and ESF forecasts. These results may have several potential implications for the market participants, financial institutions, and the government.

     

Tail risk in emerging markets of Southeastern Europe

This article examines the left-tail behaviour of returns on stocks in Southeastern Europe (SEE). We apply conditional extreme value theory (EVT) approach on daily returns of six stock market indices from SEE between 2004 and 2013. Predictive performance of value-at-risk (VaR) and expected shortfall (ES) based on EVT is compared against several alternatives, such as historical simulation and analytical approach based on GARCH with a single conditional distribution. Model backtesting with daily returns shows that EVT-based models provide more reliable VaR and ES forecasts than the alternative models in all six markets. Unlike the alternatives, the EVT-based models cannot be rejected as VaR confidence level is increased. This emphasizes the importance of extreme events in SEE markets and indicates that the ability of a model to capture volatility clustering accurately is not sufficient for a correct assessment of risk in these markets.     

Early warning signals using AVaRs of infinitely divisible GARCH models — evidence from stock index markets

Classical time series models have failed to properly assess the risks that are associated with large adverse stock price behaviour. This article contributes to autoregressive moving average model–GARCH (ARMA–GARCH) models with standard infinitely divisible innovations and assesses the performance of these models by comparing them with other time series models that have normal innovation. We discuss the limitations of value at risk (VaR) and aim to develop early warning signal models using average value at risk (AVaRs) based on the ARMA–GARCH model with standard infinitely divisible innovations. Empirical results for the daily Dow Jones Industrial Average Index, the England Financial Times Stock Exchange 100 Index and the Japan Nikkei 225 Index reveal that estimating AVaRs for the ARMA–GARCH model with standard infinitely divisible innovations offers an improvement over prevailing models for evaluating stock market risk exposure during periods of distress in financial markets and provides a suitable early warning signal in both extreme events and highly volatile markets.     

Forecasting the volatility of S&P depositary receipts using GARCH-type models under intraday range-based and return-based proxy measures

We employ four various GARCH-type models, incorporating the skewed generalized t (SGT) errors into those returns innovations exhibiting fat-tails, leptokurtosis and skewness to forecast both volatility and value-at-risk (VaR) for Standard & Poor's Depositary Receipts (SPDRs) from 2002 to 2008. Empirical results indicate that the asymmetric EGARCH model is the most preferable according to purely statistical loss functions. However, the mean mixed error criterion suggests that the EGARCH model facilitates option buyers for improving their trading position performance, while option sellers tend to favor the IGARCH/EGARCH model at shorter/longer trading horizon. For VaR calculations, although these GARCH-type models are likely to over-predict SPDRs' volatility, they are, nevertheless, capable of providing adequate VaR forecasts. Thus, a GARCH genre of model with SGT errors remains a useful technique for measuring and managing potential losses on SPDRs under a turbulent market scenario.     

Forecasting the volatility of S&P depositary receipts using GARCH-type models under intraday range-based and return-based proxy measures

We employ four various GARCH-type models, incorporating the skewed generalized t (SGT) errors into those returns innovations exhibiting fat-tails, leptokurtosis and skewness to forecast both volatility and value-at-risk (VaR) for Standard & Poor's Depositary Receipts (SPDRs) from 2002 to 2008. Empirical results indicate that the asymmetric EGARCH model is the most preferable according to purely statistical loss functions. However, the mean mixed error criterion suggests that the EGARCH model facilitates option buyers for improving their trading position performance, while option sellers tend to favor the IGARCH/EGARCH model at shorter/longer trading horizon. For VaR calculations, although these GARCH-type models are likely to over-predict SPDRs' volatility, they are, nevertheless, capable of providing adequate VaR forecasts. Thus, a GARCH genre of model with SGT errors remains a useful technique for measuring and managing potential losses on SPDRs under a turbulent market scenario.     

Energy portfolio risk management using time-varying extreme value copula methods

This work is concerned with the statistical modeling of the dependence structure between three energy commodity markets (WTI crude oil, natural gas and heating oil) using the concept of copulas and proposes a method for estimating the Value at risk (VaR) of energy portfolio based on the combination of time series models with models of the extreme value theory before fitting a copula. Each return series is modeled by AR-(FI) GARCH univariate model. Then, we fit the GPD distribution to the tails of the residuals to model marginal residuals distributions. The extreme value copula to the iid residuals is fitted and we simulate from it to construct N portfolios and estimate VaR. As a first step, the method is applied to a two-dimensional energy portfolio. In second step, we extend method in trivariate context to measure VaR of three-dimensional energy portfolio. Dependences between residuals are modeled using a trivariate nested Gumbel copulas. Methods proposed are compared with various univariate and multivariate conventional VaR methods. The reported results demonstrate that GARCH-t, conditional EVT and FIGARCH extreme value copula methods produce acceptable estimates of risk both for standard and more extreme VaR quantiles. Generally, copula methods are less accurate compared with their predictive performances in the case of portfolio composed of exchange market indices.     

我国银行间债券回购市场利率风险的实证研究

为提高我国银行间债券回购市场利率风险测定的准确性和实用性,本文针对我国银行间债券回购市场隔夜回购利率进行了基本特征分析,探讨了如何利用混合正态分布对利率数据进行拟合并据此计算VaR;作为对比组,本文同时采用GARCH模型族对利率数据进行处理。实证结果表明:与GARCH模型族相比,混合正态分布拟合方法计算VaR在准确性和实用性方面均有所提高。     

Risk assessment of oil price from static and dynamic modelling approaches

The price gap between West Texas Intermediate (WTI) and Brent crude oil markets has been completely changed in the past several years. The price of WTI was always a little larger than that of Brent for a long time. However, the price of WTI has been surpassed by that of Brent since 2011. The new market circumstances and volatility of oil price require a comprehensive re-estimation of risk. Therefore, this study aims to explore an integrated approach to assess the price risk in the two crude oil markets through the value at risk (VaR) model. The VaR is estimated by the extreme value theory (EVT) and GARCH model on the basis of generalized error distribution (GED). The results show that EVT is a powerful approach to capture the risk in the oil markets. On the contrary, the traditional variance–covariance (VC) and Monte Carlo (MC) approaches tend to overestimate risk when the confidence level is 95%, but underestimate risk at the confidence level of 99%. The VaR of WTI returns is larger than that of Brent returns at identical confidence levels. Moreover, the GED-GARCH model can estimate the downside dynamic VaR accurately for WTI and Brent oil returns.     

Forecasting Irregularly Spaced UHF Financial Data: Realized Volatility vs UHF-GARCH Models

A new literature has been recently devoted to the modeling of ultra-high-frequency (UHF) data. Our first aim is to develop an empirical application of UHF-GARCH models to forecast future volatilities on irregularly spaced data. We also compare the out-sample performance of these generalized autoregressive conditional heteroskedastic (GARCH) models with the realized volatility method. We propose a procedure to account for the time deformation problem and show how to use these models for computing daily Value at Risk (VaR).     

我国银行间同业拆借市场利率风险度量

本文利用1996年1月5日至2008年9月17日的我国银行间隔日同业拆借利率序列,通过GARCH模型对收益数据中的自相关和异方差现象进行了实证研究,采用MLE方法估计模型参数,再利用所得参数分别计算了不同收益率分布假设下的不同置信水平的VaR值;在此基础上,进行回测检验,比较了各种模型估计效果,并进一步分析了我国同业拆借利率市场的系统性风险历史波动趋势;最后提出了相关结论与政策建议。     

Pricing and hedging options with GARCH-stable proxy volatilities

This article considers modelling nonnormality in return with stable Paretian (SP) innovations in generalized autoregressive conditional heteroskedasticity (GARCH), exponential generalized autoregressive conditional heteroskedasticity (EGARCH) and Glosten-Jagannathan-Runkle generalized autoregressive conditional heteroskedasticity (GJR-GARCH) volatility dynamics. The forecasted volatilities from these dynamics have been used as a proxy to the volatility parameter of the Black–Scholes (BS) model. The performance of these proxy-BS models has been compared with the performance of the BS model of constant volatility. Using a cross section of S&P500 options data, we find that EGARCH volatility forecast with SP innovations is an excellent proxy to BS constant volatility in terms of pricing. We find improved performance of hedging for an illustrative option portfolio. We also find better performance of spectral risk measure (SRM) than value-at-risk (VaR) and expected shortfall (ES) in estimating option portfolio risk in case of the proxy-BS models under SP innovations.

Abbreviation: generalized autoregressive conditional heteroskedasticity (GARCH), exponential generalized autoregressive conditional heteroskedasticity (EGARCH) and Glosten-Jagannathan-Runkle generalized autoregressive conditional heteroskedasticity (GJR-GARCH)     

Breaks or long range dependence in the energy futures volatility: Out-of-sample forecasting and VaR analysis

The aim of this paper is to propose an empirical strategy that allows the discrimination between true and spurious long memory behaviors. That strategy is based on the comparison between the estimated long memory parameter before and after filtering out the breaks. To date the breaks, we use the probability smoothing of the Markov Switching GARCH model of Haas et al. (2004). Application of this strategy to the crude oil, heating oil, RBOB regular gasoline and the propane futures energy with the one, two, three and four months maturities show strong evidence for the presence of long range dependence in all futures energy prices volatility¹ time series. This result of long range dependence in the volatility is confirmed by the superiority of the FIGARCH and FIEGARCH models compared with the Markov switching GARCH models in terms of out-of-sample forecasting and value at risk (VaR) performances. Moreover, we show that the proposed empirical strategy is robust to different data frequency. Practical implications of the results for market participants are proposed and discussed.     

基于事件研究法的“大小非”解禁关于股票市场风险研究

通过比较＂大小非＂解禁事件前后不同时期的风险价值VaR,来评价大小非解禁对证券市场风险的影响。首先针对股票收益率序列具有波动聚集以及尖峰、厚尾的分布形态,应用GARCH类模型计算解禁前后各一段时期内沪深两市不同解禁量股票的VaR;其次应用多种定性、定量统计方法对所计算的VaR值进行前后分析比较,分析结果表明,采用的方法能够很好地捕捉到＂大小非＂解禁事件增大股票市场风险趋势这一现象。     

Empirical analysis of jump dynamics, heavy-tails and skewness on value-at-risk estimation

This study provides a comprehensive analysis of the possible influences of jump dynamics, heavy-tails, and skewness with regard to VaR estimates through the assessment of both accuracy and efficiency. To this end, the ARJI model, and its degenerative GARCH model with normal, GED, and skewed normal (SN) distributions were adopted to capture the properties of time-varying volatility, time-varying jump intensity, heavy-tails and skewness, for a range of stock indices across international stock markets during the period of the U.S. subprime mortgage crisis. Empirical results show that, with regard to the evaluation of accuracy, the role of jump dynamics is more substantial than heavy-tails or skewness as it pertains to VaR accuracy at the 90% and 95% levels, while heavy-tails become more important at the 99% level for a long position. However, the influence of the abovementioned properties on VaR estimation does not appear substantial for a short position. In addition, the properties of jump dynamics and skewness appear to be beneficial for the improvement of efficiency.     

基于ARMA-GARCH模型的股指波动压力测试情景设计研究

针对股指收益率时间序列某期间的异方差、尖峰厚尾以及序列自相关等特性,将ARMA模型与GARCH模型相结合,回归建模测算相关股指年度收益率VaR值,可以有效预测类似市场条件下股指的波动以及相伴概率。因此,在证券公司压力测试实践中,基于相伴概率合理设计股指下跌的压力测试情景,可以进一步提高压力测试情景设计的科学性,增强压力测试结果的现实指导意义。同时,可以将本文研究思路推广应用于利率、汇率、市场交易量等历史数据较充分的金融时间序列的实证分析,借以指导债市波动、汇市波动以及市场交易量波动等压力测试情景的设计工作。     

Principal Component Models for Generating Large GARCH Covariance Matrices

The implementation of multivariate GARCH models in more than a few dimensions is extremely difficult: because the model has many parameters, the likelihood function becomes very flat, and consequently the optimization of the likelihood becomes practicably impossible. There is simply no way that full multivariate GARCH models can be used to estimate directly the very large covariance matrices that are required to net all the risks in a large trading book. This paper begins by describing the principal component GARCH or 'orthogonal GARCH' (O-GARCH) model for generating large GARCH covariance matrices that was first introduced in Alexander and Chibumba (1996) and subsequently developed in Alexander (2000, 2001b). The O-GARCH model is an accurate and efficient method for generating large covariance matrices that only requires the estimation of univariate GARCH models. Hence, it has many practical advantages, for example in value–at–risk models. It works best in highly correlated systems, such as term structures. The purpose of this paper is to show that, if sufficient care is taken with the initial calibration of the model, equities and foreign exchange rates can also be included in one large covariance matrix. Simple conditions for the final covariance matrix to be positive semi-definite are derived.
(J.E.L.: C32, C53, G19, G21, G28).     

基于VaR的沪深300股指期货风险管理实证研究

我国以沪深300为标的指数的股指期货即将推出。股指期货在具有控制风险功能的同时,也与其他金融衍生产品一样,具有风险性,且其风险远远大于股票现货市场。因此,必须采用积极的风险管理技术,加强对股指期货的风险防范。在GARCH模型的基础上,采用VaR方法对我国的沪深300股指期货仿真交易进行定量研究,计算出它们的VaR值,并将其与期望值进行比较。经过对比分析可以得出:基于GARCH模型的VaR方法适合我国的股指期货风险管理。     

VaR模型在套期保值比率计算中的实证分析

提出了考虑套期保值期内不同期限价格风险的最小平均VaR套期保值比率计算模型。基于我国外汇市场及股票市场数据,用最小平均VaR套期保值模型进行了实证分析,并同常用的最小方差及最小VaR套期保值模型进行了对比,得出了最小平均VaR模型在套期保值过程中的效果要优于其他两种模型,并能更有效地降低投资者提前终止套期保值可能面临额外风险的结论。     

Warrant pricing under GARCH diffusion model

The GARCH diffusion model has attracted a great deal of attention in recent years, as it is able to describe financial time series better, when comparing to many other models. This paper considers the problem of warrant pricing when the underlying asset follows the GARCH diffusion model. An analytical approximate solution for European option prices is derived by means of Fourier transform. The approximate solution can be quickly computed by the fast Fourier transform (FFT) algorithm. Monte Carlo simulations show that this approximate solution is correct and the FFT is accurate and efficient, and hence it enables us to investigate the volatility smile implied by the GARCH diffusion model. Then a method is developed to provide the maximum likelihood (ML) estimation of the GARCH diffusion model based on the efficient importance sampling (EIS) procedure. Furthermore, the empirical performance of the GARCH diffusion model applied to the valuation of Hang Seng Index (HSI) warrants traded on the Hong Kong Stock Exchange (HKEx) is investigated. Empirical results show that the GARCH diffusion model outperforms the Black–Scholes (B–S) model in terms of the pricing accuracy, indicating that the pricing model incorporated with stochastic volatility can improve the pricing of warrants.     

Modelling conditional heteroskedasticity: Application to the “IBEX-35” stock-return index

This paper compares alternative time-varying volatility models for daily stock-returns using data from Spanish equity index IBEX-35. Specifically, we estimate a parametric family of models of generalized autoregressive heteroskedasticity (which nests the most popular symmetric and asymmetric GARCH models), a semiparametric GARCH model, the generalized quadratic ARCH model, the stochastic volatility model, the Poisson Jump Diffusion model and, finally, a nonparametric model. Those models which use conditional standard deviation (specifically, TGARCH and AGARCH models) produce better fits than all other GARCH models. We also compare the within sample predictive power of all models using a standard efficiency test. Our results show that the asymmetric behaviour of responses is a statistically significant characteristic of these data. Moreover, we observe that specifications with a distribution which allows for fatter tails than a normal distribution do not necessarily outperform specifications with a normal distribution.