Fourier transform algorithms for pricing and hedging discretely sampled exotic variance products and volatility derivatives under additive processes

We develop efficient fast Fourier transform algorithms for pricing and hedging discretely sampled variance products and volatility derivatives under additive processes (time-inhomogeneous Lévy processes). Our numerical algorithms are non-trivial versions of the Fourier space time stepping method to nonlinear path dependent payoff structures, like those in variance products and volatility derivatives. The exotic path dependency associated with the discretely sampled realized variance is captured in the numerical procedure by updating two path dependent state variables across monitoring dates. The time stepping procedure between successive monitoring dates can be performed using fast Fourier transform calculations without the usual tedious time stepping calculations in typical finite difference algorithms. We also derive effective numerical procedures that compute the hedge parameters of variance products and volatility derivatives. Numerical tests on pricing various variance products and volatility derivatives were performed that illustrate efficiency, accuracy, reliability and robustness of the proposed Fourier transform algorithms.