Applications For Extreme Value Theory In VaR And Empirical Analysis

Value at risk (VaR) describes the problems of measuring market risk faced by financial institutions. After proposed by G30Group in1993, VaR has become the main method to measure financial market risk. The key idea of different methods is to describe the distribution of the rate on assets, and then calculate the quantile at a given confident level.One of the most basic and simplest method is history simulation (HS). The thought and operation are quite simple, but the shortcomings are obvious. Its result is easily influenced by the length of the data and history volatility. And it also ignores the distribution of abnormal values. As a result, VaR by HS method cannot reflect the real market risk. We then introduce Monte Carlo method, assuming that the asset price satisfies geometric Brown motion, and then we calculate the VaR at the strike time. There exists model risk, and it takes a long time for a multi-asset portfolio to calculate the VaR. In empirical experiments, economists show that there exists volatility cluster phenomenon, which is called auto-regression conditional heteroskedasticity effect, i.e. ARCH effect. To describe it conditional volatility model which is named GARCH model was proposed, and then volatility is a stochastic process. Since risk is caused by extreme events, what only should we do is to study the tail of rate distribution. Based on this idea, extreme value theory (EVT) was raised. EVT method is separated by block maximum model (BMM) and peaks over threshold (POT) model which have a good behavior. But VaR cannot tell the regulators how much a company will loss if risk happens, expected shortfall (ES) gives this question a good describe. Besides checking whether the models are right or not, we should also show if the models can reflect the real VaR. Hence, we use Kupiec’s test to calculate the failure rate based on the VaR confident level.At the end of this paper, we give two examples, Index of Shanghai Stock Exchange and Compositional Index of Shenzhen Stock Market, to simulate the VaR calculated by the above methods. Financial data has a thick tail and doesn’t satisfy normal distribution. With comparison, HS method is easily affected by the length of history datum. And Kupicc’s test shows the VaRs are not effective. ARCH effect is verified after the datum is tested, proving that GARCH models are appropriate to describe the conditional heteroskedasticity. The accuracy is reduced because of too many parameters in ARCH model. GARCH model can effectively avoid this shortcoming. Kupicc’s test value is dependable. EVT method is effective. including block maximum method (BMM) and peaks over threshold (POT) method. But attention must be paid for BMM can be influenced by the number of "group". POT method is much better than EVT, showing that POT is a usefull method to measure the tail. In practice, because shortcomings exist in each method, we should take couples of methods and have a backtest to get a dependable VaR value.