Extreme observations in the metal market and their implication for risk measure
It is well known that financial data often exhibit features like high kurtosis and skewness that are incompatible with the normality assumption which makes the inference more challenging and problematic. A natural approach to overcome these inconsistencies is to assume that returns follow a stable law. Models based on stable laws have the potential to provide more realistic estimates of the frequency of large price movements, and therefore they seem preferable to classical models based on the assumption of normally distributed returns. The second approach for tail modeling is extreme value theory based models. The most important difference between these approaches is that EVT-based methods discard a large amount of observed data, while stable-based ones use all of the data points in the time series. Applications of stable distributions in the field of finance have a long history. Since they represent a model for the entire distribution and not just the tails, reliable model parameter estimation methods exist. The instability of parameter estimation, inherent in EVT, is not present for stable distributions, because the entire sample is taken into account, rather than just the tail of the distribution. The purpose of this chapter is to compare and contrast the more popular fattailed methodologies based on value theory (EVT) and a-stable distributions, and obtain deeper insight into risk management for financial stock returns of precious metal market traded on the London Metal Exchange. We concern on a particular measure of market risk Value at Risk, the amount of money necessary to provide the institution with coverage against losses that occur with probability over some holding period. However, EVT provides natural approach to VaR estimation, given that VaR is primarily concerned with the tails of our return distributions. Research findings show that EVT models outperform more widely used VaR forecast methods, such as Risk Metrics and historical simulation for the data set of daily returns. The use of a-stable distributions in the measurement of VaR has also already conducted. Several empirical studies have shown that a-stable distribution give better results than traditional methods that rely on the normal distribution. The paper is structured as follows. In section 1 we describe four precious metals. In sections 2 and 3 we present some general results of extreme value theory and a-stable distribution. Next, we recall how to compute VaR in a standard univariate Gaussian setting and using only past observation (historical simulation). Section 4 derives information about stable and EVT-based VaR measures. The approaches are empirically tested in section 5 where the risk of financial stock returns is analyzed via VaR. The last section concludes. (fragment of text)
- Blattberg R., Gonedes N. (1974), A Comparison of the Stable and Student Distributions as Statistical Models of Stock Prices, "Journal of Business", 7.
- Coles S. (2001), An Introduction to Statistical Modeling of Extreme Values, Springer Series in Statistics, Springer-Verlag, London.
- Fama E. (1965), The Behaviour Of Stock Market Prices, "Journal ofBusiness", 38.
- Jondeau E., Poon S.H., RockingerM. (2007), Financial Modelling Under Non-Gaussian Distributions, Springer-Verlag, London.
- Jorion P. (2006), Value At Risk: The New Benchmark for Managing Financial Risk, McGraw-Hill, New York.
- Kupiec P.H. (1995), Techniques for Verifying the Accuracy of Risk Measurement Models, "The Journal of Derivatives", Vol. 3, No. 2.
- Mandelbrot B. (1963), The Variation of Certain Speculative Prices, "Journal of Business", Vol. 36, No. 4.
- McNeil A.J. (1999), Extreme Value Theoiy for Risk Managers, Internal Modelling and CAD II, RISK Books.
- Mittnik S., Rachev S.T., Schwartz E. (2002), Value-at-Risk and Asset Allocation with Stable Return Distributions, "Allgemeines Statistisches Archiv", No. 86, Physica-Verlag.
- Rachev S.T., Stoyanov S.V., Fabozzi F.J. (2011), A Probability Metrics Approach to Financial Risk Measures, Wiley-Blackwell, Oxford.
- Samorodnitsky G., Taqqu M.S. (1994), Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance, Chapman & Hall, New York.
- Smith R.L. (1987), Estimating Tails of Probability Distributions, "Annals of Statistics", 15.
- Villasenor-Alva J.A., Gonzales-Estrada E. (2009), A Bootstrap Goodness of Fit Test for the Generalized Pareto Distribution, "Computational Statistics and Data Analysis", 53, 11.