  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 matrices. First, we set up the model
and present some important properties concerning the Laplace transform of the
factors and the ergodicity of the model. Then, we present two main numerical
tools to implement the model in practice. First, we obtain an expansion of
caplets and swaptions prices around the LGM. Such a fast and accurate
approximation is useful for assessing the model behavior on the implied
volatility smile. Second, we provide a second order scheme for the weak error,
which enables to calculate exotic options by a Monte-Carlo algorithm.
