Abstract
In this study, we model the returns of a stock index using various parametric distribution models. There are four indices used in this study: HSCEI, KOSPI 200, S&P 500, and EURO STOXX 50. We applied 12 distributions to the data of these stock indices—Cauchy, Laplace, normal, Student’s t, skew normal, skew Cauchy, skew Laplace, skew Student’s t, hyperbolic, normal inverse Gaussian, variance gamma, and general hyperbolic—for the parametric distribution model. In order to choose the best-fit distribution for describing the stock index, we used the information criteria, goodness-of-fit test, and graphical tail test for each stock index. We estimated the value-at-risk (VaR), one of the most popular management concepts in the area of risk management, for the return of stock indices. Furthermore, we applied the parametric distributions to the risk analysis of equity-linked securities (ELS) as they are a very popular financial product on the Korean financial market. Relevant risk measures, such as VaR and conditional tail expectation, are calculated using various distributions. For calculating the risk measures, we used Monte Carlo simulations under the best-fit distribution. According to the empirical results, investing in ELS is more risky than investing in securities, and the risk measure of the ELS heavily depends on the type of security.
Highlights
Introduction e normal orGaussian distribution is a widespread distribution for modeling in finance
Much empirical research has shown that real data for stock price returns are generally characterized by skewness, kurtosis, and fat tails. erefore, multiple distributions have been used as an alternative to the normal one
Skew distributions allow us to take advantage of the skewness value. e skew normal and skew Student’s t are typical skew distributions. ird, a normal variance-mean mixture with generalized inverse Gaussian distribution can generate, for example, the generalized hyperbolic distribution introduced by Barndorff-Nielsen [14]. ese types of distributions can be both symmetric and skewed, and their tails are heavier than those of the Gaussian distribution. e generalized hyperbolic distribution was used in many studies to fit a series of stock index returns
Summary
We used 12 parametric distributions in this study: Cauchy, Laplace, normal, Student’s t, skew normal, skew Cauchy, skew Laplace, skew Student’s t, hyperbolic, normal inverse Gaussian, variance gamma, and general hyperbolic. e Cauchy distribution is named after the mathematician A.L. U admits the Student’s t distribution with mean μ, scale σ, and degrees of freedom parameter ] (see [37]). E Cauchy, Laplace, normal, and Student’s t distribution cannot show the asymmetry of the pdf. According to Nurminen et al [42], the univariate skew Student’s t distribution is parameterized by the location μ, scale σ, skew α, and degrees of freedom ]. Ese features are normality, kurtosis, and skewness, and they can be controlled by the skew Student’s t, hyperbolic, NIG, variance gamma, and generalized hyperbolic distributions
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