We propose an affine extension of the linear Gaussian term structure model (LGM) such that the instantaneous covariation of the factors is given by an affine process on semidefinite positive matrixes. We begin by setting up the model and presenting some important properties concerning the Laplace transform of the factors and the ergodicity of the model. Then, we present two main numerical tools for implementing the model in practice. First, we obtain an expansion of caplet and swaption prices around the LGM. Such a fast and accurate approximation is useful in assessing model behavior on the implied volatility smile. Second, we provide a second-order scheme for the weak error, which enables us to calculate exotic options by a Monte Carlo algorithm. These two pricing methods are compared with the standard method based on Fourier inversion.