Valuation of stochastic interest rate securities with time-dependent variance

We consider the problem of how to prize general securities whose payoff at maturity only depends on the interest rate rT at the time of exercise, where rt is supposed to be a stochastic Feller process. We show how to generalize the results of Cox et al. [Econometrica 53 (2) (1985) 385] regarding bond valuation to a situation where the stochastic evolution of rt under the martingale probability involves time-dependent coefficients and the payoff is arbitrary. The solution to this problem is given in terms of the propagator for the heat operator with a potential. This propagator is constructed in terms of a classical harmonic oscillator with time-dependent frequency.