European Option Under Stochastic Interest Rate and Local Volality

A partial differential equation formulation is developed to value an european option under stochastic interest rate and local volatility. The partial differential equation is one dimensional under forward measure.The formulation follows the usual techniques based on replication or the martingale methods, but require modifications. A monte carlo simulation is also performed to verify a posteriori the final partial differential formulation. Finite difference method is used to solve the partial differential equation, and a family of implicit methods are used for time-stepping in the monte carlo simulations