EXTREME VALUE THEORY IN FINANCE: A SURVEY

Extreme value theory is concerned with the study of the asymptotic distribution of extreme events, that is to say events which are rare in frequency and huge in magnitude with respect to the majority of observations. Statistical methods derived from it have been employed increasingly in finance, especially for risk measurement. This paper surveys some of those main applications, namely for testing different distributional assumptions for the data, for Value‐at‐Risk and Expected Shortfall calculations, for asset allocation under safety‐first type constraints, and for the study of contagion and dependence across markets under conditions of stress.     

Value at risk models for volatile emerging markets equity portfolios

This paper investigates the issue of market risk quantification for emerging and developed market equity portfolios. A very wide spectrum of popular and widely used in practice Value at Risk (VaR) models are evaluated and compared with Extreme Value Theory (EVT) and adaptive filtered models, during normal, crises, and post-crises periods. The results are interesting and indicate that despite the documented differences between emerging and developed markets, the most successful VaR models are common for both asset classes. Furthermore, in the case of the (fatter tailed) emerging market equity portfolios, most VaR models turn out to yield conservative risk forecasts, in contrast to developed market equity portfolios, where most models underestimate the realized VaR. VaR estimation during periods of financial turmoil seems to be a difficult task, particularly in the case of emerging markets and especially for the higher loss quantiles. VaR models seem to be affected less by crises periods in the case of developed markets. The performance of the parametric (non-parametric) VaR models improves (deteriorates) during post-crises periods due to the inclusion of extreme events in the estimation sample.     

On downside risk predictability through liquidity and trading activity: A dynamic quantile approach

Most downside risk models implicitly assume that returns are a sufficient statistic with which to forecast the daily conditional distribution of a portfolio. In this paper, we analyze whether the variables that proxy for market-wide liquidity and trading conditions convey valid information for forecasting the quantiles of the conditional distribution of several representative market portfolios, including volume- and value-weighted market portfolios, and several Book-to-Market- and Size-sorted portfolios. Using dynamic quantile regression techniques, we report evidence of conditional tail predictability in terms of these variables. A comprehensive backtesting analysis shows that this link can be exploited in dynamic quantile modelling, in order to considerably improve the performances of day-ahead Value at Risk forecasts.     

Forecast encompassing tests for the expected shortfall

We introduce new forecast encompassing tests for the risk measure Expected Shortfall (ES). The ES has received much attention since its introduction into the Basel III Accords, which stipulate its use as the primary market risk measure for international banking regulation. We utilize joint loss functions for the pair ES and Value at Risk to set up three ES encompassing test variants. The tests are built on an asymptotic theory that is robust to misspecifications. We investigate the finite sample properties of the tests in an extensive simulation study. Finally, we use the encompassing tests to illustrate the potential of forecast combination methods for different financial assets.     

Subsampling the distribution of diverging statistics with applications to finance

In this paper we propose a subsampling estimator for the distribution of statistics diverging at either known or unknown rates when the underlying time series is strictly stationary and strong mixing. Based on our results we provide a detailed discussion of how to estimate extreme order statistics with dependent data and present two applications to assessing financial market risk. Our method performs well in estimating Value at Risk and provides a superior alternative to Hill's estimator in operationalizing Safety First portfolio selection.     

GARCH模型族在沪深300中的比较研究

风险测量一直是金融研究领域的热门话题,而如何构建合适的模型来衡量风险自然而然成为众多学者研究的关注点.VaR方法是当今应用最广泛的衡量金融风险的方法之一,其核心又在构建良好的波动率估计模型.GARCH模型族能很好地描述股指波动率呈现的重尾、波动性聚集、杠杆效用等,是当前效果比较好的条件异方差性的模型.本文着重研究基于GARCH模型族(GARCH、EGARCH、PGARCH)在不同分布假定下(高斯分布、t分布、广义误差分布)的表现,从而计算出沪深300的在险价值( VaR),比较分析模型拟合效果,选出适合的模型,对规范国内沪深300的风险管理提供了理论依据.     

基于VaR的我国证券投资基金绩效评价方法

证券投资基金绩效的评价,不仅要考察基金的收益率,而且还要看它所承担的风险。投资基金绩效评价传统经典方法主要有特雷诺指数法、夏普指数法、詹森指数法及T-M模型、H-M模型。基于VaR的证券投资基金绩效评价方法——RAROC,这种经风险调整后的绩效评价方法能更客观、准确地反映证券投资基金的绩效。     

Extreme returns and intensity of trading

Asymmetric information models of market microstructure claim that variables such as trading intensity are proxies for latent information on the value of financial assets. We consider the interval‐valued time series (ITS) of low/high returns and explore the relationship between these extreme returns and the intensity of trading. We assume that the returns (or prices) are generated by a latent process with some unknown conditional density. At each period of time, from this density, we have some random draws (trades) and the lowest and highest returns are the realized extreme observations of the latent process over the sample of draws. In this context, we propose a semiparametric model of extreme returns that exploits the results provided by extreme value theory. If properly centered and standardized extremes have well‐defined limiting distributions, the conditional mean of extreme returns is a nonlinear function of the conditional moments of the latent process and of the conditional intensity of the process that governs the number of draws. We implement a two‐step estimation procedure. First, we estimate parametrically the regressors that will enter into the nonlinear function, and in a second step we estimate nonparametrically the conditional mean of extreme returns as a function of the generated regressors. Unlike current models for ITS, the proposed semiparametric model is robust to misspecification of the conditional density of the latent process. We fit several nonlinear and linear models to the 5‐minute and 1‐minute low/high returns to seven major banks and technology stocks, and find that the nonlinear specification is superior to the current linear models and that the conditional volatility of the latent process and the conditional intensity of the trading process are major drivers of the dynamics of extreme returns.     

Optimal portfolio selection via conditional convex risk measures on L p

We consider conditional convex risk measures on L ^p and show their robust representation in a standard way. Such measures are used as evaluation functionals for optimal portfolio selection in a Black&Scholes setting. We study this problem focusing on the conditional Average Value at Risk and the conditional entropic risk measure and compare the respective optimizers.     

Upside/Downside spillovers between oil and Chinese stock sectors: From the global financial crisis to global pandemic

This paper examines the effects of the COVID-19 outbreak, recent oil price fall, and both global and European financial crises on dependence structure and asymmetric risk spillovers between crude oil and Chinese stock sectors. Using time-varying symmetric and asymmetric copula functions and the conditional Value at Risk measure, we provide evidence of positive tail dependence in most sectors using copula and conditional Value-at-Risk techniques. We can see the average dependence between oil and industries during the oil crisis. Moreover, we find strong evidence of bidirectional risk spillovers for all oil-sector pairs. The intensity of risk spillovers from oil to all stock sectors varies across sectors. The risk spillovers from sectors to oil are substantially larger than those from oil to sectors during COVID-19. Furthermore, the return spillover is time varying and sensitive to external shocks. The spillover strengths are higher during COVID-19 than financial and oil crises. Finally, oil do not exhibit neither hedge nor safe-haven characteristics irrespective of crisis periods.     

Using extreme value theory to measure value-at-risk for daily electricity spot prices

The recent deregulation in electricity markets worldwide has heightened the importance of risk management in energy markets. Assessing Value-at-Risk (VaR) in electricity markets is arguably more difficult than in traditional financial markets because the distinctive features of the former result in a highly unusual distribution of returns—electricity returns are highly volatile, display seasonalities in both their mean and volatility, exhibit leverage effects and clustering in volatility, and feature extreme levels of skewness and kurtosis. With electricity applications in mind, this paper proposes a model that accommodates autoregression and weekly seasonals in both the conditional mean and conditional volatility of returns, as well as leverage effects via an EGARCH specification. In addition, extreme value theory (EVT) is adopted to explicitly model the tails of the return distribution. Compared to a number of other parametric models and simple historical simulation based approaches, the proposed EVT-based model performs well in forecasting out-of-sample VaR. In addition, statistical tests show that the proposed model provides appropriate interval coverage in both unconditional and, more importantly, conditional contexts. Overall, the results are encouraging in suggesting that the proposed EVT-based model is a useful technique in forecasting VaR in electricity markets.     

High quantiles estimation with Quasi-PORT and DPOT: An application to value-at-risk for financial variables

Recurrent ⿿black swans⿿ financial events are a major concern for both investors and regulators because of the extreme price changes they cause, despite their very low probability of occurrence. In this paper, we use unconditional and conditional methods, such as the recently proposed high quantile (HQ) extreme value theory (EVT) models of DPOT (Duration-based Peak Over Threshold) and quasi-PORT (peaks over random threshold), to estimate the Value-at-Risk with very small probability values for an adequately long and major financial time series to obtain a reasonable number of violations for backtesting. We also compare these models and other alternative strategies through an out-of-sample accuracy investigation to determine their relative performance within the HQ context. Policy implications relevant to estimation of risk for extreme events are also provided.     

基于极值分布理论的VaR与ES度量

本文应用极值分布理论对金融收益序列的尾部进行估计,计算收益序列的在险价值VaR和预期不足ES来度量市场风险。通过伪最大似然估计方法估计的GARCH模型对收益数据进行拟合,应用极值理论中的GPD对新息分布的尾部建模,得到了基于尾部估计产生收益序列的VaR和ES值。采用上证指数日对数收益数据为样本,得到了度量条件极值和无条件极值下VaR和ES的结果。实证研究表明:在置信水平很高(如99%)的条件下,采用极值方法度量风险值效果更好。而置信水平在95%下,其他方法和极值方法结合效果会很好。用ES度量风险能够使我们了解不利情况发生时风险的可能情况。     

Assessing value at risk with CARE, the Conditional Autoregressive Expectile models

In this paper we propose a downside risk measure, the expectile-based Value at Risk (EVaR), which is more sensitive to the magnitude of extreme losses than the conventional quantile-based VaR (QVaR). The index θ of an EVaR is the relative cost of the expected margin shortfall and hence reflects the level of prudentiality. It is also shown that a given expectile corresponds to the quantiles with distinct tail probabilities under different distributions. Thus, an EVaR may be interpreted as a flexible QVaR, in the sense that its tail probability is determined by the underlying distribution. We further consider conditional EVaR and propose various Conditional AutoRegressive Expectile models that can accommodate some stylized facts in financial time series. For model estimation, we employ the method of asymmetric least squares proposed by Newey and Powell [Newey, W.K., Powell, J.L., 1987. Asymmetric least squares estimation and testing. Econometrica 55, 819–847] and extend their asymptotic results to allow for stationary and weakly dependent data. We also derive an encompassing test for non-nested expectile models. As an illustration, we apply the proposed modeling approach to evaluate the EVaR of stock market indices.     

Statistics of extreme events in risk management: The impact of the subprime and global financial crisis on the German stock market

Given the growing need for managing financial risk and the recent global crisis, risk prediction is a crucial issue in banking and finance. In this paper, we show how recent advances in the statistical analysis of extreme events can provide solid methodological fundamentals for modeling extreme events. Our approach uses self-exciting marked point processes for estimating the tail of loss distributions. The main result is that the time between extreme events plays an important role in the statistical analysis of these events and could therefore be useful to forecast the size and intensity of future extreme events in financial markets. We illustrate this point by measuring the impact of the subprime and global financial crisis on the German stock market in extenso, and briefly as a benchmark in the US stock market. With the help of our fitted models, we backtest the Value at Risk at various quantiles to assess the likeliness of different extreme movements on the DAX, S&P 500 and Nasdaq stock market indices during the crisis. The results show that the proposed models provide accurate risk measures according to the Basel Committee and make better use of the available information.     

Forecasting value at risk and expected shortfall with mixed data sampling

I propose applying the Mixed Data Sampling (MIDAS) framework to forecast Value at Risk (VaR) and Expected shortfall (ES). The new methods exploit the serial dependence on short-horizon returns to directly forecast the tail dynamics of the desired horizon. I perform a comprehensive comparison of out-of-sample VaR and ES forecasts with established models for a wide range of financial assets and backtests. The MIDAS-based models significantly outperform traditional GARCH-based forecasts and alternative conditional quantile specifications, especially in terms of multi-day forecast horizons. My analysis advocates models that feature asymmetric conditional quantiles and the use of the Asymmetric Laplace density to jointly estimate VaR and ES.     

Realized extreme quantile: A joint model for conditional quantiles and measures of volatility with EVT refinements

We propose a new framework exploiting realized measures of volatility to estimate and forecast extreme quantiles. Our realized extreme quantile (REQ) combines quantile regression with extreme value theory and uses a measurement equation that relates the realized measure to the latent conditional quantile. Model estimation is performed by quasi maximum likelihood, and a simulation experiment validates this estimator in finite samples. An extensive empirical analysis shows that high‐frequency measures are particularly informative of the dynamic quantiles. Finally, an out‐of‐sample forecast analysis of quantile‐based risk measures confirms the merit of the REQ.     

Sensitivity-based Conditional Value at Risk (SCVaR): An efficient measurement of credit exposure for options

Counterparty Credit Risk (CCR) has received extensive attention in the Over-The-Counter (OTC) derivative markets. This paper proposes a credit risk exposure measurement for European options: Sensitivity-based Conditional Value at Risk (SCVaR), which can cover the future credit risk by a stable sensitivity weight, and improve the accuracy of risk tracking in most cases. Compared with VaR and CVaR, SCVaR has superiority in extensibility, computational efficiency and stability. We further derive the tendency and upper bound of sensitivity weights, consequently obtaining a practical value of price weight for long-term stability. The simulation and empirical analysis in the Chinese options market also show good applicability of SCVaR. The risk exposures are efficiently covered during periods of fluctuation, which alleviates the procyclicality to some extent. These results provide a useful guidance for the development of financial risk management.     

Wavelet analysis of change-points in a non-parametric regression with heteroscedastic variance

In this paper we develop wavelet methods for detecting and estimating jumps and cusps in the mean function of a non-parametric regression model. An important characteristic of the model considered here is that it allows for conditional heteroscedastic variance, a feature frequently encountered with economic and financial data. Wavelet analysis of change-points in this model has been considered in a limited way in a recent study by Chen et al. (2008) with a focus on jumps only. One problem with the aforementioned paper is that the test statistic developed there has an extreme value null limit distribution. The results of other studies have shown that the rate of convergence to the extreme value distribution is usually very slow, and critical values derived from this distribution tend to be much larger than the true ones. Here, we develop a new test and show that the test statistic has a convenient null limit N(0,1) distribution. This feature gives the proposed approach an appealing advantage over the existing approach. Another attractive feature of our results is that the asymptotic theory developed here holds for both jumps and cusps. Implementation of the proposed method for multiple jumps and cusps is also examined. The results from a simulation study show that the new test has excellent power and the estimators developed also yield very accurate estimates of the positions of the discontinuities.     

价值工程在企业文化创新中的应用研究

通过对价值工程和文化创新的对比分析,说明了在企业文化创新中应用价值工程的必要性和重要性。然后结合企业组织文化创新过程,利用价值工程研究原理,探讨了价值工程的具体应用,为文化创新决策提供可靠的理论依据。