α-Stable Distribution and its Application to Value at Risk and Financial Forecasting, in Comparison with Student's t-Distribution

In practice, business people used to deal with financial data as if they follow the normal distribution. However, researches have shown that most financial assets returns possess fat-tailed property, which is contradictory to that of the normal distribution. Both the t-distribution and α-stable distribution are attractive alternatives. Past study have stated that the t-distribution dominates the normal distribution, but there is no definite dominance of either the t-distribution or the α-stable distribution over the other. They both carry unique features when fitted to financial data. This paper compares the fitness of the t-distribution and the α-stable distribution to the stock indices returns in Asia, since most past researches of this kind focus on the equity indices in Europe and America. The analysis in this paper is classified into two parts, first the time independent part and followed by the time dependent part. In the first part, the Value at Risk (VaR) estimated by the unconditional t-distribution and the α-stable distribution are discussed. In the second part, the time series GARCH models with t-innovation and α-stable innovation respectively are also investigated. The main finding is that in the sense of VaR, the unconditional α-stable distribution provides better estimates of VaR at moderate levels, and extreme VaR less than 1% with α-stable distribution tends to be conservative, with comparison to t-distribution. This is a valuable feature of the application of α-stable distribution to risk management, because it allows risk managers to preserve more reservation in advance for the potential upcoming losses. Moreover, this paper also shows that the time series GARCH models with α-stable innovation always have smaller RMSE than those with t-innovation when the out-of-sample forecasting is conducted, indicating that the models with α-stable innovation may have better forecasting accuracy than those with t-innovation, though the degrees of significance are different due to the property of the data. Finally, the 95% forecasting intervals are constructed in this paper and they can be connected to the dynamic VaR, making it possible for us to estimate the VaR in accordance with time.