No abstract

A search for a distribution which adequately describes the dynamics of log returns has been a subject of study for many years. Empirical evidence has resulted in stylized facts of returns. Arguably, in this study, the three components of returns, mean equation part, the changing variance part and the resulting residuals are determined and their corresponding parameters estimated within the proposed framework. Spectral density analysis is used to trace the seasonality component inherent in the standardized residuals. Empirical data sets from eight different indexes and common stock are applied to the model, and results tabulated in support of the resulting framework.

This article seeks to model daily asset returns using log-ARCH-Lévy type model which is expected to reproduce most of the stylized features of financial time series data (such as volatility clustering, leptokurtic nature of log returns, joint covariance structure and aggregational Gaussianity) that are empirically found in different types of market. In addition, unconditional variance of daily log returns in risk neutral world of different conditional heteroscedastic models is derived. A key observation is that liquid markets and illiquid market may not have the same underlying dynamics. For instance empirical analysis based on S&P500 index log returns as a liquid market do not have autoregressive part in their first moments while in Nairobi Securities Exchange NSE20 index there is strong presence of autoregressive dynamics of order three, i.e. AR(3). Higher moments of both markets are serially correlated.

Generalized hyperbolic distribution and some of its subclasses like normal, hyperbolic and variance gamma distributions are used to fit daily log returns of eight listed companies in Nairobi Securities Exchange and Montréal Exchange. EM-based maximum likelihood estimation procedure is used to estimate parameters of the model. Kernel densities and empirical distribution of data are compared. The goodness of fit statistics of proposed distributions are used to measure how well model fits the data. Empirical results show that Generalized hyperbolic Distribution seems to improve partially, the geometric Brownian assumption on modeling returns of the underlying process, both in a developed and emerging market. Both markets seem to have different stochastic time.

Abstract. This paper seeks to model daily, weekly and monthly stock indices returns using GARCH (1,1) model which is expected to reproduce most of the stylized facts of financial time series data which, in most cases, are found in different types of market. In addition, the distributional behavior of returns as the data changes from daily through to monthly returns is investigated by performing the JB and K-S tests. The results indicate evidence of volatility clustering, leverage effects, Gaussianity and leptokurtic distribution in the stock returns. A key observation is that the monthly returns of the three indices follow a Gaussian distribution (i.e. as the data changes from daily through to monthly returns it follows a normal distribution).