Extensions of Dupire Formula: Stochastic Interest Rates and Stochastic Local Volatility

We derive generalizations of the Dupire formula to the case of general stochastic drift and/or the case of general stochastic local volatility. First, we handle the case in which the drift is given as a difference of two stochastic short rates. Such a setting is natural in a foreign exchange context where the short rates correspond to the short rates of the two currencies, in an equity single-currency context with stochastic dividend yield, or in a commodity context with stochastic convenience yield. We present the formula in both a call surface formulation and a total implied variance formulation where the latter avoids calendar spread arbitrage by construction. We provide derivations for the case where both short rates are given as single factor processes, and we present limits for a single stochastic rate or all deterministic short rates. The limits agree with published results. Then we derive a formulation that allows a more general stochastic drift and diffusion, including one or more stochastic local volatility terms. In the general setting, our derivation allows for the computation and cali ration of the leverage function for stochastic local volatility models. Despite being implicit, the generalized Dupire formula can be used numerically in a fixed-point iterative scheme.