European Option Under Jump-Diffusion and Stochastic Interest Rate

A one-dimensional partial differential-difference equation (pdde) under forward measure is developed to value European option under jump-diffusion, stochastic interest rate and local volatility. The corresponding forward Kolmogorov partial differential-difference equation for transition probability density is a also developed to value the options for various strikes at a given maturity time.The mathematical formulation of those equations is verified numerically by comparing their finite difference computation results with those of the Monte Carlo simulations. For the Kolmogorov equation, an alternate numerical method called the redistribution method is also developed. The redistribution method is based on the moments of the transition probability density and avoids some of the difficulties of a finite difference method.