Spectral Methods for Volatility Derivatives

40 Pages Posted: 2 Oct 2007 Last revised: 13 May 2009

See all articles by Claudio Albanese

Claudio Albanese

Global Valuation

Aleksandar Mijatovic

Imperial College London

Harry Lo

Imperial College London

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-diagonalisation. 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.

Suggested Citation

Albanese, Claudio and Mijatovic, Aleksandar and Lo, Harry, Spectral Methods for Volatility Derivatives. Available at SSRN: https://ssrn.com/abstract=1018608 or http://dx.doi.org/10.2139/ssrn.1018608

Claudio Albanese (Contact Author)

Global Valuation ( email )

9 Devonshire Sq.
London, London EC2M 4YF
United Kingdom

Aleksandar Mijatovic

Imperial College London ( email )

Department of Mathematics
180 Queen's Gate
London, SW7 2AZ
United Kingdom

HOME PAGE: http://www3.imperial.ac.uk/people/a.mijatovic

Harry Lo

Imperial College London ( email )

South Kensington Campus
Exhibition Road
London, Greater London SW7 2AZ
United Kingdom

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