In this paper, we propose a Partial Differential Equation formulation for the value of an option when the underlying asset's price is described by a discrete-time GARCH process. Our numerical approach involves a spectral Fourier-Chebyshev interpolation in a finite difference approximation. Numerical illustrations are provided, and the results are compared with other available valuation methods. Our numerical procedure shows exponential convergence and allows for the efficient computation of option prices, reaching high precision in a few seconds of computing time.