We construct explicitly computable approximations to the Green's function of certain second-order parabolic equations, using Dyson series, Taylor expansions and exact commutator formulas. These approximations are accurate to arbitrary order in the short-time limit, and can be extended to large time by combining with numerical bootstrapping. The method applies also to certain kinds of degenerate equations, such as Fokker-Planck equations arising in pricing of contingent claims. A main motivation for this work is parameter estimation via Bayesian inference.