Spectral Methods For Volatility Derivatives
AbstractIn 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.
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Bibliographic InfoPaper provided by University Library of Munich, Germany in its series MPRA Paper with number 5244.
Date of creation: 01 Mar 2006
Date of revision:
Volatility derivatives; operator methods;
Other versions of this item:
- G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
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- Peter Friz & Jim Gatheral, 2005. "Valuation of volatility derivatives as an inverse problem," Quantitative Finance, Taylor and Francis Journals, vol. 5(6), pages 531-542.
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- Albanese, Claudio, 2006. "Operator Methods, Abelian Processes And Dynamic Conditioning," MPRA Paper 5246, University Library of Munich, Germany, revised 06 Nov 2007.
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