  We obtain a first order extension of the large deviation estimates in the
G\"{a}rtner-Ellis theorem. In addition, for a given family of measures, we find
a special family of functions having a similar Laplace principle expansion up
to order one to that of the original family of measures. The construction of
the special family of functions mentioned above is based on heat kernel
expansions. Some of the ideas employed in the paper come from the theory of
affine stochastic processes. For instance, we provide an explicit expansion
with respect to the homogenization parameter of the rescaled cumulant
generating function in the case of a generic continuous affine process. We also
compute the coefficients in the homogenization expansion for the Heston model
that is one of the most popular stock price models with stochastic volatility.
