Stepping through Fourier space

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