Forward Density Approach to European Option Under Stochastic Interest Rate

The forward Kolmogorov ( Fokker-Planck ) partial differential equation for the transition density under forward measure is developed to value european option under stochastic interest rate and local volatility. The advantage of this approach is that the density needs to be computed just once out to the longest maturity to obtain the option value of any shorter maturity and of any strike. This reduces the computational time in the optimization process for calibration of local volatility surface to implied volatility. The forward Kolmogorov equation is one dimensional and computed using standard finite difference methods. The singular initial condition (delta-function) is smoothed by a Gaussian. An alternate method called the Vorticity Redistribution Method is also developed to handle the delta function exactly. The results of both the finite difference and the vorticity redistribution methods are verified using the results of the forward measure Black-Scholes equation for the option developed previously. The conservation laws of the density are used to examine the errors in the computations