Pensee Journal Vol 76, No. 9;Sep 2014 52 office@penseejournal.com Modeling Volatility with GARCH Family Models: An Application to Daily Stock Log-returns in Pharmaceutical Companies Dina Hassan Abdel Hady Dept. Of Statistics, Mathematics and Insurance Faculty of Commerce, Tanta University E-mail: dinaabdelhady44@Gmail.com Abstract Financial log-returns suffer from volatility clustering; that causes positive autocorrelation coefficients of squared returns, with a relatively slowly decreasing pattern starting from a first small value. Volatility measures the ―dispersion‖ not the direction of data points. In this study, the ARCH model and some GARCH family models, namely, GARCH, TGARCH and EGARCH models are discussed applied to data. Steps to conduct tests and methodologies for the applications for each model are given. While ARCH model allows the conditional variance to change over time as a function of past errors leaving the unconditional variance constant, GARCH model specifies the conditional variance to be a linear combination of (q) lags of the squared residuals, TGARCH and EGARCH models allow for leverage effect. The Daily log- returns of three pharmaceutical companies registered in Egypt Stock Market covering the period from 2th January 2000 to 23th Jul 2008 were used for the application of the three models. Results show that the OLS regression model for Arab Drugs Company is sufficient since no serial correlation and no ARCH effect is depicted in the data. For Alex Pharma the best fit model is TGARCH (p,q) model while for EPICO Company, the best fit model is EGARCH (p ,q). The effect of the lags (p ,q) on the fitness of the model were studied for one company, it has been found that fitness is better as q increases and p=1, and as p increases and q=1 and for p>1 fitness is improved for q≤ p. Keywords: Stationary Volatility, ARCH Effects, ARCH, GARCH, TGARCH, EGARCH, Volatility Clustering, Conditional Volatility, Stock Market volatility, Leverage effect, Heteroskedastic models, Asymmetric Models. Introductıon An analytical solution for volatility has become a critical issue in many applications in financial decision making modeling that may include risk management, portfolio management, assets allocation, and foreign exchange. Volatility does not measure the direction but the dispersion of price changes. Stock market volatility has been studied by various researchers ( Engle ,1982; Bollerslev , 1986, 1992; Poon and Granger, 2003 ; Engle, Lilien, and Robins ,1987; Nelson (1991; Glosten, Jagannathan and Runkle ,1993 ; Engle and Ng ,1993 ; Black (1976), Christie (1982), FSS (1987), Schwert (1990) and Pagan and Schwert (1990); Batra ,2004 ; Kumar ; 2006 ). The unconditional probability distribution of financial log-returns are found to be leptokurtic due to " Volatility Clustering" that causes positive autocorrelation coefficients of squared log-returns, with a relatively slowly decreasing pattern starting from a first small value (Mandelbrot, 1963). Conventional time series and econometric models operate under an assumption of constant variance and independence of error terms (Bollerslev et al. ,1994); however, Engle (2001) concluded that