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 created 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 use a regime switching
model with jumps and local volatility defined in \cite{FXrev} and calibrate it
to the European options on the S&P 500 for a broad range of strikes and
maturities. The main idea of this paper is to "lift" (i.e. extend) the
generator of the underlying process to keep track of the relevant path
information, namely the realized variance. The lifted generator is too large a
matrix to be diagonalized numerically. We overcome this difficulty by applying
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 European options, general accrued variance payoffs and
forward-starting and VIX options.Comment: to appear in Quantitative Financ