Quantile estimation and application of the Value at Risk

• Main Authors: Weng,Yi-Chun, 翁鐿純
• Other Authors: Chang, Yoeng-Kuan
• Format: Others
• Language: zh-TW
• Published: 2013
• Online Access: http://ndltd.ncl.edu.tw/handle/63517319856150377454

碩士 === 東吳大學 === 財務工程與精算數學系 === 101 === Risk management becomes even more critical due to the increasing varieties of financial instruments. Among all the measurement tools, the most significant is Value at Risk (VaR), commonly including variance-covariance, historical simulation and monte carlo simulation methods. Under those above-mentioned methodologies, the normal distribution is assumed or the past historical data is used as the distribution. However, if the actual distribution differs from the assumption, model risk will be generated and consequently influence the estimation of VaR. Generalized Tukey λ distribution, applied in this research, can estimate percentile by calculating the first four moments without assuming the probability distribution and this methodology is the most obvious contribution of this research. This study tests 9 kinds of distributions that are normally used in calculating the different quantiles and sample amounts successively. Providing each random variable of the distribution is X, the testing results are all similar to the theoretical quantiles, and the deviation gets smaller when the sample amount gets larger. When -∞<X<∞, most of the tested distributions are assumed at the left tail of quantile, and the average deviation falls in the reasonable range of 9%. Since the Generalized Tukey λ distribution doesn’t assume the real data follows any distribution, the multi-day VaR can’t be calculated directly by multiplying√t with the one-day VaR. As usual thus the study derives the theorem of the transformation between the one-day VaR and the multi-day VaR. As for the nonlinear pricing models, the price change in the option pricing models used to be calculated to the first three moments. In this study, the price change in the option pricing model is calculated to the first four moments, and the VaRs of the nonlinear pricing models are also estimated. According to the results of the empirical research, the Generalized Tukey λ distribution is more adapted and easier to get the fat tails than variance-covariance, historical simulation and monte carlo simulation methods, when testing the single asset, asset portfolio, one-day and multi-day VaR on the assets of the linear models. For empirical test, we take the listed stocks such as Hon Hai, Shan Yuan, Service & Quality Co., Ltd., Macro Well OMG Co., Ltd. and the TWSE index as the standards. As for the nonlinear models, the TXO (Taiwan Stock Exchange Capitalization Weighted Stock Index Options) and TEO (Taiwan Stock Exchange Electronic Sector Index Options) are selected as the subjects. Compared with the monte carlo simulation method that are commonly used in the option calculation, the Generalized Tukey λ distribution is more adapted to the nonlinear models and is more efficient in getting the fat tails.