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.     

Intraday risk management in International stock markets: A conditional EVT approach

The study compares the predictive ability of various models in estimating intraday Value-at-Risk (VaR) and Expected Shortfall (ES) using high frequency share price index data from sixteen different countries across the world for a period of seven and half months from September 20, 2013 to May 07, 2014. The main emphasis of the study has been given to Extreme Value Theory (EVT) and to evaluate how well Conditional EVT model performs in modeling tails of distributions and in estimating and forecasting intraday VaR and ES measures. We have followed McNeil and Frey's (2000) two stage approach called Conditional EVT to estimate dynamic intraday VaR and ES. We have compared the accuracy of Conditional EVT approach to intraday VaR and ES estimation with other competing models. The best performing model is found to be the Conditional EVT in estimating both the quantiles for the entire sample. The study is useful for market participants (such as intraday traders and market makers) involved in frequent intraday trading in such equity markets.     

基于实现极差和实现波动率的中国金融市场风险测度研究

目前比较流行的金融市场风险价值研究一般采用日收益数据,并基于GARCH类模型进行估计和预测。本文利用沪深股指日内高频数据,分别通过ARFIMA模型和CARR模型对实现波动率和较新的实现极差建模,计算风险价值。通过对VaR的似然比和动态分位数等回测检验,实证分析了各种模型的VaR预测能力。结果显示,使用日内高频数据的实现波动率和实现极差模型的预测能力强于采用日数据的各种GARCH类模型。     

Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach

We propose a method for estimating Value at Risk (VaR) and related risk measures describing the tail of the conditional distribution of a heteroscedastic financial return series. Our approach combines pseudo-maximum-likelihood fitting of GARCH models to estimate the current volatility and extreme value theory (EVT) for estimating the tail of the innovation distribution of the GARCH model. We use our method to estimate conditional quantiles (VaR) and conditional expected shortfalls (the expected size of a return exceeding VaR), this being an alternative measure of tail risk with better theoretical properties than the quantile. Using backtesting of historical daily return series we show that our procedure gives better 1-day estimates than methods which ignore the heavy tails of the innovations or the stochastic nature of the volatility. With the help of our fitted models we adopt a Monte Carlo approach to estimating the conditional quantiles of returns over multiple-day horizons and find that this outperforms the simple square-root-of-time scaling method.     

A fully parametric approach to return modelling and risk management of hedge funds

This paper examines the empirical properties of hedge fund returns and proposes a fully parametric model capable of adequately describing both univariate and multivariate return properties. The suggested model is based on the multivariate extension of the Normal Inverse Gaussian (NIG) distribution and will be shown to be capable of capturing the characteristic distributional features of hedge fund returns. Drawing on recent research in the area of Generalized Hyperbolic distributions and their calibration, we will elaborate on the application of the NIG-model for risk management purposes, and highlight the differences between the NIG and the Gaussian model.      

High-frequency financial data modeling using Hawkes processes

Intraday Value-at-Risk (VaR) is one of the risk measures used by market participants involved in high-frequency trading. High-frequency log-returns feature important kurtosis (fat tails) and volatility clustering (extreme log-returns appear in clusters) that VaR models should take into account. We propose a marked point process model for the excesses of the time series over a high threshold that combines Hawkes processes for the exceedances with a generalized Pareto distribution model for the marks (exceedance sizes). The conditional approach features intraday clustering of extremes and is used to calculate instantaneous conditional VaR. The models are backtested on real data and compared to a competitor approach that proposes a nonparametric extension of the classical peaks-over-threshold method. Maximum likelihood estimation is computationally intensive; we use a differential evolution genetic algorithm to find adequate starting values for the optimization process.     

Asymmetries of the intraday return-volatility relation

This study investigates the asymmetry of the intraday return-volatility relation at different return horizons ranging from 1, 5, 10, 15, up to 60 min and compares the empirical results with results for the daily return horizon. Using data on the S&P 500 (SPX) and the VIX from September 25, 2003 to December 30, 2011 and a Quantile-Regression approach, we observe strong negative return-volatility relation over all return horizons. However, this negative relation is asymmetric in three different aspects. First, the effects of positive and negative returns on volatility are different and more pronounced for negative returns. Second, for both positive and negative returns, the effect is conditional on the distribution of volatility changes. The absolute effect is up to five times larger in the extreme tails of the distribution. Third, at the intraday level, there is evidence of both autocorrelation in volatility changes and cross-autocorrelation with returns. This lead-lag relation with returns is also very asymmetric and more pronounced in the tails of the distribution. These effects are, however, not observed at the daily return horizon.     

A duscrete-time model of high-frequency stock returns

A model of intraday financial time series is developed. The model is a dynamic factor model consisting of two equations. First, a rate of return of a ‘stock’ in a single day is assumed to be generated by serveral common factors plus some additive erros (‘intraday equation’). Secondly, the joint distribution of those common factors is assumed to depend on the hidden state of the day, which fluctuates according to a Markov chain (‘day-by-day equation’). Together the equations compose a hidden Markov model.

We investigate properties of the model. Among them is a central limit theorem for cumulative returns, which agrees with the well-known empirical phenomenon in the stock markets that the distributions of longer-horizon returns are closer to the normal. We propose a two-step procedure consisting of the method of principal components and the EM algorithm to estimate the model parameters as well as the unboservable states. In addition, we propose a procedure for predicting intraday returns. Finally, the model is fitted to empirical data, the Standard&Poors 500 Index 5 min return data, to see if the model is capable of describing intraday movements of the index.     

A Bayesian encompassing test using combined value-at-risk estimates

The Value at Risk (VaR) is a risk measure that is widely used by financial institutions in allocating risk. VaR forecast estimation involves the conditional evaluation of quantiles based on the currently available information. Recent advances in VaR evaluation incorporate conditional variance into the quantile estimation, yielding the Conditional Autoregressive VaR (CAViaR) models. However, the large number of alternative CAViaR models raises the issue of identifying the optimal quantile predictor. To resolve this uncertainty, we propose a Bayesian encompassing test that evaluates various CAViaR models predictions against a combined CAViaR model based on the encompassing principle. This test provides a basis for forecasting combined conditional VaR estimates when there are evidences against the encompassing principle. We illustrate this test using simulated and financial daily return data series. The results demonstrate that there are evidences for using combined conditional VaR estimates when forecasting quantile risk.     

基于条件极值分布的金融高频数据VaR动态估计模型

高频数据由于自身数量大、周期短、信息丰富的特点而受到关注。基于高频数据。对金融时间序列的厚尾特征进行条件极值分布下的VaR估计。在对条件均值和条件波动率估计时,以往采用一阶自回归模型和GARCH模型,但基于高频数据的估计较为繁复。为了充分利用日内信息,基于高频样本观测值,建立已实现均值RM模型,在考虑市场异质性的基础上,对条件均值进行估计。通过对TCL股票价格进行实证分析,估计出VaR风险值,验证模型是合理的。     

期货投资组合交易保证金设置研究——基于Copula模型和极值理论的分析

本文以棉花、铜、天然橡胶三个期货合约为研究对象,基于t-Copula模型,利用Monte Carlo模拟法计算在一定权重下由三个品种构成的期货投资组合的VaR和ES值作为投资组合的保证金数值。Kupiec回溯测试结果表明,t-Copula模型结合极值理论计算出的期货投资组合保证金相比其他方法能够在较好覆盖极端风险的同时降低投资成本。     

Financial crisis,Value-at-Risk forecasts and the puzzle of dependency modeling

Forecasting Value-at-Risk (VaR) for financial portfolios is a crucial task in applied financial risk management. In this paper, we compare VaR forecasts based on different models for return interdependencies: volatility spillover (Engle & Kroner, 1995), dynamic conditional correlations (Engle, 2002, 2009) and (elliptical) copulas (Embrechts et al., 2002). Moreover, competing models for marginal return distributions are applied. In particular, we apply extreme value theory (EVT) models to GARCH-filtered residuals to capture excess returns.Drawing on a sample of daily data covering both calm and turbulent market phases, we analyze portfolios consisting of German Stocks, national indices and FX-rates. VaR forecasts are evaluated using statistical backtesting and Basel II criteria. The extensive empirical application favors the elliptical copula approach combined with extreme value theory (EVT) models for individual returns. 99% VaR forecasts from the EVT-GARCH-copula model clearly outperform estimates from alternative models accounting for dynamic conditional correlations and volatility spillover for all asset classes in times of financial crisis.     

Nonlinear portfolio selection using approximate parametric Value-at-Risk

As the skewed return distribution is a prominent feature in nonlinear portfolio selection problems which involve derivative assets with nonlinear payoff structures, Value-at-Risk (VaR) is particularly suitable to serve as a risk measure in nonlinear portfolio selection. Unfortunately, the nonlinear portfolio selection formulation using VaR risk measure is in general a computationally intractable optimization problem. We investigate in this paper nonlinear portfolio selection models using approximate parametric Value-at-Risk. More specifically, we use first-order and second-order approximations of VaR for constructing portfolio selection models, and show that the portfolio selection models based on Delta-only, Delta–Gamma-normal and worst-case Delta–Gamma VaR approximations can be reformulated as second-order cone programs, which are polynomially solvable using interior-point methods. Our simulation and empirical results suggest that the model using Delta–Gamma-normal VaR approximation performs the best in terms of a balance between approximation accuracy and computational efficiency.     

How to choose the return model for market risk? Getting towards a right magnitude of stressed VaR

Value at Risk (VaR) and stressed value at Risk (SVaR) or expected shortfall are important risk measures widely used in the financial services industry for risk management and market risk capital computation. Fundamental to any (S)VaR model is the choice of the return type model for each risk factor. Because the resulting SVaR numbers are highly sensitive to the chosen return type model it is important to make a prudent choice on the return type modelling. We propose to estimate the return type model from historic data without making an a priori model assumption on the return model. We explain the fundamentals of return type modelling and how it impacts the magnitude of SVaR. We further show how to obtain a global return type model from a set of similar return type models by using geometric calculus. Numerical simulations and illustrations are provided. In this paper, we consider interest rate data, but the proposed methodology is general and can be applied to any other asset class such as inflation, credit spread, equity or fx.     

Deutsche Mark–Dollar Volatility: Intraday Activity Patterns, Macroeconomic Announcements, and Longer Run Dependencies

This paper provides a detailed characterization of the volatility in the deutsche mark–dollar foreign exchange market using an annual sample of five-minute returns. The approach captures the intraday activity patterns, the macroeconomic announcements, and the volatility persistence (ARCH) known from daily returns. The different features are separately quantified and shown to account for a substantial fraction of return variability, both at the intraday and daily level. The implications of the results for the interpretation of the fundamental "driving forces" behind the volatility process is also discussed.     

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.     

A Characterization of the Daily and Intraday Behavior of Returns on Options

The daily and intraday behavior of returns on Chicago Board Options Exchange options is examined. Option returns contain systematic patterns even after adjusting for patterns in the means and variances of the underlying assets. This is consistent with the hypothesis that informed trading in options can make the order flow in the options market informative about the value of the underlying asset, making options nonredundant. The intraday patterns in adjusted option return variances are further consistent with a model of strategic trading by informed and discretionary liquidity traders.     

Heteroskedasticity in Stock Return Data: Volume versus GARCH Effects

This paper provides empirical support for the notion that Autoregressive Conditional Heteroskedasticity (ARCH) in daily stock return data reflects time dependence in the process generating information flow to the market. Daily trading volume, used as a proxy for information arrival time, is shown to have significant explanatory power regarding the variance of daily returns, which is an implication of the assumption that daily returns are subordinated to intraday equilibrium returns. Furthermore, ARCH effects tend to disappear when volume is included in the variance equation.     

Model averaging in risk management with an application to futures markets

This paper considers the problem of model uncertainty in the case of multi-asset volatility models and discusses the use of model averaging techniques as a way of dealing with the risk of inadvertently using false models in portfolio management. Evaluation of volatility models is then considered and a simple Value-at-Risk (VaR) diagnostic test is proposed for individual as well as ‘average’ models. The asymptotic as well as the exact finite-sample distribution of the test statistic, dealing with the possibility of parameter uncertainty, are established. The model averaging idea and the VaR diagnostic tests are illustrated by an application to portfolios of daily returns on six currencies, four equity indices, four ten year government bonds and four commodities over the period 1991–2007. The empirical evidence supports the use of ‘thick’ model averaging strategies over single models or Bayesian type model averaging procedures.     

典型事实、极值理论与金融市场动态风险测度研究

本文在金融市场典型事实约束下,运用ARFIMA模型对金融市场条件收益率建模,运用GARCH、GJR、FIGARCH、APARCH、FIAPARCH等5种模型对金融波动率进行建模,进而运用极值理论(EVT)对标准收益的极端尾部风险建模来测度各股市的动态风险,并用返回测试(Back-testing)方法检验模型的适应性。实证结果表明,总的来说,FIAPARCH-EVT模型对各个市场具有较强的适应性,风险测度能力较为优越。进一步,本文在ARFIMA-FIAPARCH模型下,假定标准收益分别服从正态分布(N)、学生t分布(st)、有偏学生t分布(skst)、广义误差分布(GED)共4种分布,对各股市的动态风险测度的准确性进行检验,并和EVT方法的测度结果进行对比分析。结果表明,EVT方法风险测度能力优于其他方法,有偏学生t分布假设下的风险测度模型虽然略逊于EVT方法,但也不失为一种较好的方法;ARFIMA-FI-APARCH-EVT不仅在中国大陆沪深股市表现最为可靠,而且在其他市场也表现出同样的可靠性。