Pricing European-style options under jump diffusion processes with stochastic volatility: applications of Fourier transform

We develop a general methodology for pricing European-style options under various stochastic processes via the Fourier transform. We generalize previous work in this field and present two approaches for solving the pricing problem: the characteristic formula which is an extension of Lewis (2001) work, and the Black-Scholes-style formula which is an extension and generalization of previous works by Heston (1993) and Bates (1996). We show how to apply our formulas for two types of asset price dynamics: 1) stochastic volatility models with price jumps at a stochastic jump intensity rate, 2) stochastic volatility models with price and volatility jumps. Convergence properties of Fourier integrals arising from both approaches are studied.