|Abstract:||In this thesis, we show how to deploy machine learning techniques such as Gaussian process regression to approximate the European basket option prices. For the underlying asset of European basket option, we assume it follows multivariate Black \&\ Scholes model, and we can derive the PDE for the option price. In order to deal with the curse of dimensionality, we assume that the basket consists of several comonotonic groups, in each comonotonic group, the stock prices are driven by a single random source. Then we can derive an approximation for the price of European basket option. Next, we introduce the finite difference scheme to price the European basket option for given parameters such as risk-free interest rates, maturities and so on. However, for approximating the European basket option for different risk-free interest rates, maturities and strikes, using finite difference scheme to get corresponding approximations costs much time. Hence, in order to save time for approximating the European basket option for different risk-free interest rates, maturities and strikes, we deploy Gaussian process regression to fit the training set produced by finite difference scheme and after comparing the results, we can conclude that the errors are often well within reasonable limits and hence very acceptable from a practical point of view and the Gaussian process regression truly save much time.