This paper concerns the design of a Fourier based pseudospectral numerical method for the model of European Option Pricing with transaction costs under Exponential Utility derived by Davis, Panas and Zariphopoulou. Computing the option price involves solving two stochastic optimal control problems. With a Exponential Utility function, the dimension of the problem can be reduced, but one has to deal with high absolute values in the objective function. In this paper, we propose two changes of variables that reduce the impact of the exponential growth. We propose a Fourier pseudospectral method to solve the resulting non linear equation. Numerical analysis of the stability, consistency, convergence and localization error of the method are included. Numerical experiments support the theoretical results. The effect of incorporating transaction costs is also studied.