Modeling stock market returns using heavy tailed distributions

Abstract

Stock market returns have been assumed to be normal is several financial models including Black - Scholes and almost every other model that tries to capture the stochastic nature of the stock movement. Though such an assumption is taken for simplicity sake, various studies have proven that stock market returns may not be normal . several deviations that the daily stock returns exhibit from a normal distribution are high kurtosis, heavy tails and skewness. This has spawned several new models of distributions such as Generalized secant Hyperbolic(GSH) and Skewed Generalized Secant Hyperbolic (SGSH) that tries to closely mimic the daily stock returns. Our study was on modeling the stock returns based on three distributions: GSH, SGSH and Voigt profile. Voigt is a relatively new entrant in the modeling field, original used in spectroscopy studies. The voigt profile doesn't have any closed form and hence is not as simple to model as its secant hyperbolic counterparts. However, it can be modeled to have very high kurtosis, Though it is incapable of modeling skewness. Out studies on two stocks in the National Stock Exchange(NSE) for the last 12 years showed that the returns showed negligible kurtosis and hence not much advantage of the SGSH distribution over GSH . Moreever , both GSH and SGSH failed to account for the very high peakness of returns near the mean(zero) value. No matter what kurtosis parameter we took, the models didn't peak enough to mimic the observed returns. on the other hand, voigt profile provided some interesting results. Though majority of our studies on the profile was empirical in the absence of any closed form equation, the distribution closely resembled the actual returns exhibiting very high peakness near zero. Moreover, They showed the ability to generate sub maximas on either side of the mean which in some cases might prove useful to model actual returns because the observed data showed similar tendency too.