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In this paper, we study the performance of the Artificial Neural Networks (ANNs) and GARCH modelsto predict the volatility of the Indian stock market indices namely, NIFTY 50, NIFTY Bank and NIFTYFMCG. We have used the GARCH (1,1) and Recurrent Neural Network, a type of neural network whichis widely used for predicting time series data. The purpose of the study is to investigate if the ArtificialNeural Networks perform better than the traditional GARCH (1,1) model. An out of sample testingmethodology is applied to the most recent 20 percent of the observations for all the three indices. Wehave used Root Means Squared Error (RMSE) and Mean Absolute Error (MAE) as metrics to evaluatethe volatility predicting performances of the models. The results show no clear evidence of ANN modelperforming better than GARCH model for any of the three indices. ANNs may prove to be betterindicators in periods with low volatility while its performance deteriorated in periods with highvolatility.