SPECTRAL METHODS FOR VOLATILITY DERIVATIVES

Abstract. 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 difficulty 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 payoffs as well
as forward-starts and VIX options.

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