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

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