Stepping through Fourier space

Diverse finite-difference schemes for solving pricing problems with Levy underlyings appear in financial literature. Invariably, the integral and diffusive terms are treated asymmetrically, large jumps are truncated, and the methods are difficult to extend to higher dimensions and cannot easily incorporate regime switching or stochastic volatility. Sebastian Jaimungal and Vladimir Surkov present a new, efficient approach that switches between Fourier and real space as time propagates backwards, which they call Fourier space time-stepping

Jump-diffusion models, and more generally Levy models, have been extensively applied in practice to partially correct the defects in the Black-Scholes-Merton model and explain the implied volatility surface's shape and dynamics. Popular types include the variance gamma (VG) and normal inverse Gaussian models. Under such models, the pricing partial differential equation transforms into a partial-integro differential equation (PIDE) with a non-local integral term. By writing the price as a