Extreme Value Estimation Of Value-at-Risk

Recently, with the daily volatility of financial market and some financial catastrophic events happened one by one, which put a challenge for risk management, we have been long for some more appropriate models to deal with such events. Because the conventional method to measure is based on normal distribution assumption which has been proved to underestimate risk, in order to measure the risk more accurate, more and more researchers put forward using extreme-value-theory to measure market risk which has been used in engineering field widely. Because extreme-value-distribution need not to put any hypothesis on the whole distribution of return but only by data themselves, fitting the tail of distribution, which is fitted to measure risk. Currently, the Generalized Pareto distribution based on extreme-value-theory was widely used to study market risk. In the paper, we prepare to use Generalized Pareto distribution based on extreme-value-theory to study Chinese stock market. At the same time, because various risk factors is dependent, how to measure multi-risk factors has been concerned increasingly. Based on the study of extreme-value-theory by forefathers, we will discuss how to use extreme-value-theory to estimate financial risk precisely faced by Chinese stock market. The center of this paper is using the method similar to those used in McNeil and Frey to select the threshold, and adopted the method of probability weighted moment(PWM) but not the maximum likelihood method(MLM) which is widely used in the last years to estimate the parameters of Generalized Pareto distribution after make an comparative analysis between several methods. The paper discuss how to make a Monte Carlo simulation to forecast the value-at-risk of multi-risk factors by using copula function based on the theory of single risk factor. In the end, the paper make a numerical simulation study for ShangHai stock market and Shenzhen stock market and conducted the back comparative analysis with the assumption of multi-norm distribution. A satisfactory result is taken out in the end of the paper.The paper organized as follow. Chapter one is the summary of the paper that gives the origin, aim and meanings as well as the technique and content. In the chapter two describe the definition of VaR and methods used to calculate VaR. Chapter three introduce extreme value theory emphasizing on POT model and illuminate the advantages and disadvantages. Which method should be adopted to select the threshold value of generalized Pareto distribution is discussed and we have numerical simulation study based on selecting a good method among several methods to estimate parameters of generalized Pareto distribution in chapter four. The chapter five extend to multi-variable risk factors, adopting Monte Carlo simulation estimate the VaR of multi-variable risk factors based on Copula function and extreme value theory, and has a numerical simulation study. The last chapter is our results and conclusions, and in the end I provide some suggestions for farther research.