A systematic investigation of the effect of different neural network architecture alternatives for predicting the future course of stock prices in the Cyprus Stock Exchange (CSE) market is conducted. This market exhibited an abrupt increase in the general index price during the last year, thus forming a very interesting case for research purposes. The influence of various economic and political factors, from both the local and the international scene, has also been examined. The main conclusion drawn is that the CSE market is governed by unique conditions which when properly modeled can yield successful predictions.