This article proposes a derivative pricing framework when the asset returns and the short term rate process are modelled with affine GARCH models driven by correlated Gaussian innovations. The risk neutral dynamics are derived based on a co-variance dependent pricing kernel and semi-closed form solutions are derived for European style options and bond prices. We further derive the weak diffusion limits of the underlying processes under both physical and risk-neutral measure and we investigate the consistency between the proposed pricing kernel with the well-known Girsanov principle in continuoustime. A variety of numerical exercises are provided to analyze the validity of our pricing formulae, the sensitivity of the option prices relative to the pricing kernel parameters, and the convergence of option prices to those based on the limiting diffusions. Finally, we illustrate an empirical analysis based on a joint estimation using historical asset returns and short-term rates, and cross sections of options and bond prices.