Volatility measures the variability and ascertains the unpredictable and uncertain behavior of the asset price. As a concept and phenomenon, it has remained central area of research in modern financial markets and academics. The importance of volatility in stock market can’t be undermined in financial economics, as it plays a significant role in investment and risk management decisions. This paper attempts to examine the dynamics of time varying volatility of Indian stock market with reference to BSE and S&P CNX Nifty. Using daily observation, data have been taken for period of 2000-2014. To examine the characteristics of Indian Stock Market Volatility, GARCH models are being employed. EGARCH and TARCH are employed to find possibility of Asymmetry or Leverage effects in the market.