  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.
