In this paper, we discuss the pricing of stock index options. Under a generalized assumption that the underlying individual stock price which is an element of the stock index follows a semi-martingale process with correrlation, we derive the partial differential equation which the corresponding European stock index option prices satisfy. More precisely, we calculate the respective equilibrium prices for the value line index (VLI) option and the average stock index (ASI) option analytically and numerically. Finally, we perform the sensitivity analysis of model parameters in the option prices and the position analysis. Throughout numerical examples, it is shown quantitatively that the correlation for underlying assets is one of the most important factors to price stock index options.