In the first quarter of 2006 Chicago Board Options Exchange (CBOE) introduced, as one of the listed products, options on its implied volatility index (VIX). This opened the challenge of developing a pricing framework that can simultaneously handle European options, forward-starts, options on the realized variance and options on the VIX. In this paper we propose a new approach to this problem using spectral methods. We define a stochastic volatility model with jumps and local volatility, which is almost stationary, and calibrate it to the European options on the S&P 500 for a broad range of strikes and maturities. We then extend the model, by lifting the corresponding Markov generator, to keep track of relevant path information, namely the realized variance. The lifted generator is too large a matrix to be diagonalized numerically. We overcome this diculty by developing a new semi-analytic algorithm for block-diagonalization. This method enables us to evaluate numerically the joint distribution between the underlying stock price and the realized variance which in turn gives us a way of pricing consistently the European options, general accrued variance payos as well as forward-starts and VIX options.
