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.     

Learning,large deviations and rare events

We examine the role of generalized stochastic gradient constant gain (SGCG) learning in generating large deviations of an endogenous variable from its rational expectations value. We show analytically that these large deviations can occur with a frequency associated with a fat-tailed distribution even though the model is driven by thin-tailed exogenous stochastic processes. We characterize these large deviations, driven by sequences of consistently low or consistently high shocks and then apply our model to the canonical asset pricing framework. We demonstrate that the tails of the stationary distribution of the price–dividend ratio will follow a power law.     

Asymptotic comparison at optimal levels of reduced‐bias extreme value index estimators

In this article we are interested in the asymptotic comparison, at optimal levels, of a set of semi‐parametric reduced‐bias extreme value (EV) index estimators, valid for a wide class of heavy‐tailed models, underlying the available data. Again, as in the classical case, there is not any estimator that can always dominate the alternatives, but interesting clear‐cut patterns are found. Consequently, and in practice, a suitable choice of a set of EV index estimators will jointly enable us to better estimate the EV index γ, the primary parameter of extreme events.     

Growth Optimal Investment Strategy: The Impact of Reallocation Frequency and Heavy Tails

The strategy to maximize the long‐term growth rate of final wealth (maximum expected log strategy, maximum geometric mean strategy, Kelly criterion) is based on probability theoretic underpinnings and has asymptotic optimality properties. This article reviews the allocation of wealth in a two‐asset economy with one risky asset and a risk‐free asset. It is also shown that the optimal fraction to be invested in the risky asset (i) depends on the length of the basic return period and (ii) is lower for heavy‐tailed log returns than for light‐tailed log returns.     

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.     

Distribution of parallel exchange rates in African countries

This paper investigates the distribution of parallel exchange rates in African countries using exploratory data analysis techniques and model fitting. Stable laws are fitted to empirical distributions using the maximum likelihood estimation method. Empirical evidence supports the stable hypothesis these distributions are positively skewed and have tails that are much heavier than Gaussian counterparts. The stable hypothesis is further supported by the “converging variance test,” which suggests that these distributions have infinite variance.     

不同萃取试剂检测羽绒残脂率的差异性分析

选用乙醚、二氯甲烷和石油醚作为羽绒残脂率的萃取试剂,对3种试剂检测结果进行了比对分析。结果显示：二氯甲烷检测结果高于乙醚的检测结果,而石油醚的检测结果低于乙醚的检测结果。从统计学角度分析得出这两种试剂的检测结果与乙醚的检测结果都存在显著性差异,二氯甲烷和石油醚无法直接作为乙醚萃取的替代性试剂。     

Cross-ownership as a structural explanation for over- and underestimation of default probability

Based on the work of Suzuki, we consider a generalization of Merton’s asset valuation approach in which two firms are linked by cross-ownership of equity and liabilities. Suzuki’s results then provide no arbitrage prices of firm values, which are derivatives of exogenous asset values. In contrast to the Merton model, the assumption of lognormally distributed assets does not result in lognormally distributed firm values, which also affects the corresponding probabilities of default. In a simulation study we see that, depending on the type of cross-ownership, the lognormal model can lead to both over- and underestimation of the actual probability of default of a firm under cross-ownership. In the limit, i.e. if the levels of cross-ownership tend to their maximum possible value, these findings can be shown theoretically as well. Furthermore, we consider the default probability of a firm in general, i.e. without a distributional assumption, and show that the lognormal model is often able to yield only a limited range of probabilities of default, while the actual probabilities may take any value between 0 and 1.     

低能量焙烤食品的研究与开发

低能量焙烤食品的市场潜力很大,目前国际市场上虽有此类产品上市,但发展还很不理想。焙烤食品的低能量化主要在于蔗糖和脂肪的替代。随着各种低能量配料的使用,会使其产品的含水性、口感及风味等发生明显的变化,同时导致产品生产成本提高,这些都是开发低能量焙烤食品必须解决的技术难题。合理使用各种高品质的蔗糖替代品和脂肪替代品,对原有生产技术加以合理化改造,是开发高品质的低能量焙烤食品的关键。     

Value-at-Risk Prediction: A Comparison of Alternative Strategies

Given the growing need for managing financial risk, risk predictionplays an increasing role in banking and finance. In this studywe compare the out-of-sample performance of existing methodsand some new models for predicting value-at-risk (VaR) in aunivariate context. Using more than 30 years of the daily returndata on the NASDAQ Composite Index, we find that most approachesperform inadequately, although several models are acceptableunder current regulatory assessment rules for model adequacy.A hybrid method, combining a heavy-tailed generalized autoregressiveconditionally heteroskedastic (GARCH) filter with an extremevalue theory-based approach, performs best overall, closelyfollowed by a variant on a filtered historical simulation, anda new model based on heteroskedastic mixture distributions.Conditional autoregressive VaR (CAViaR) models perform inadequately,though an extension to a particular CAViaR model is shown tooutperform the others.