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Spectral methods for volatility derivatives

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  • Claudio Albanese
  • Harry Lo
  • Aleksandar Mijatovi\'c

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 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.

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Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 0905.2091.

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Date of creation: May 2009
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Handle: RePEc:arx:papers:0905.2091

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  1. Peter Friz & Jim Gatheral, 2005. "Valuation of volatility derivatives as an inverse problem," Quantitative Finance, Taylor & Francis Journals, vol. 5(6), pages 531-542.
  2. Claudio Albanese & Aleksandar Mijatović, 2009. "A Stochastic Volatility Model For Risk-Reversals In Foreign Exchange," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(06), pages 877-899.
  3. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51, October.
  4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
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Cited by:
  1. Albanese, Claudio, 2006. "Operator Methods, Abelian Processes And Dynamic Conditioning," MPRA Paper 5246, University Library of Munich, Germany, revised 06 Nov 2007.
  2. Nicolas Merener, 2009. "Swap Rate Variance Swaps," Business School Working Papers 2009-02, Universidad Torcuato Di Tella.
  3. Cheng, Jun & Ibraimi, Meriton & Leippold, Markus & Zhang, Jin E., 2012. "A remark on Lin and Chang's paper ‘Consistent modeling of S&P 500 and VIX derivatives’," Journal of Economic Dynamics and Control, Elsevier, vol. 36(5), pages 708-715.
  4. Gabriel G. Drimus, 2012. "Options on realized variance by transform methods: a non-affine stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 12(11), pages 1679-1694, November.
  5. Albanese, Claudio, 2007. "Callable Swaps, Snowballs And Videogames," MPRA Paper 5229, University Library of Munich, Germany, revised 01 Oct 2007.

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