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The stock market is a very complicated versatile system. It is an integrated part of nowadays financial markets. Also, stock market's movements are influenced by many aspects like firm's policies, political events, general economic conditions, investors' expectations, movement of other stock market, institutional investors' choices, psychology of investors, etc. Financial markets follow time-series sequence of data which are measured at uniform time intervals. It is a sequence of vectors 
, where y represents past value which varies with the time t (Omidi, et al., 2011). It has high volatility and it changes with time. Prediction techniques which are based on these time series data have been used in many real world applications such as electric utility load forecasting, financial market prediction, weather and environmental state prediction, and reliability forecasting. Modeling stock market volatility and forecasting has been an active topic for theoretical as well as practical investigation and empirical studies. Volatility refers to the uncertainty or risk, taking the degree and size of variation in the value of a security in consideration, in other words we can say that it is a statistical evaluation of the dispersion of returns for a market index. In other words volatility refers to the degree to which financial prices fluctuate. The measure of volatility grabs the concept of uncertainty for future returns. Stock index prediction is regarded as a challenge for the financial time series prediction process. With few researches in the past it was assumed that fundamental information available in the past has some important relationships to the future stocks. It includes exchange rates and interest rate economic variable, industry and company specific information. But later it was stated that to forecast future returns is next to impossible since they already reflect current information about the stocks. But there are proofs that provide evidence that its state is influenced by some visible stock market information including returns (Hagenau, et al., 2012). The emerging markets’ volatility have also been observed before and after the market in US crashed 1987 using GARCH-M model acknowledging volatility clustering in stock returns (Choudhry, 1996). Major of the works have been done on the developed markets (Schwert, 1989; Poon and Taylor, 1992, Hansen et al., 2003). There have been works which analyses volatility of stock returns on emerging and mature markets but forecasting using GARCH and ANN and comparing both the economies on same factor will be an all new dealing. Authors (Koutmos et al., 1993; Negakis and Kambouris, 1994) performed similar investigation on emerging and matured market. In this paper we are dealing with the behavior of both the economies on stock market volatility and forecasting factor that how both the economies are affected by it and which one is more predictable. The stock indices used in this work allow a fair comparison between both the economies thus taking two mature economies stock indices(S & P 500, Nikkei) and two emerging economies stock indices(BSE 30, Hang Seng) and then comparing among them will give a better and judgmental comparison graph. BSE 30 and HANG SENG is used for financial study in this paper because the economies of India and China are considered to be the largest emerging markets in today’s scenario and BSE 30 or Bombay stock exchange being stock market index of 30 financially sound companies and HANG SENG being the main basis of the complete market efficacy in Hong Kong. Under it The 48 constituent companies correspond to about 60% of capitalization of the Hong Kong Stock Exchange. And last but not the least we are also using Japan stock exchange i.e. Nikkei stock index as a mature market example. BSE 30, HANG SENG and S&P 500 both are free-float capitalization-weighted stock indices.