Albanese, Claudio and Mijatovic, Aleksandar (2006): SPECTRAL METHODS FOR VOLATILITY DERIVATIVES.

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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, forwardstarts, 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 semianalytic algorithm for blockdiagonalization. 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 forwardstarts and VIX options.
Item Type:  MPRA Paper 

Institution:  Independent Consultant 
Original Title:  SPECTRAL METHODS FOR VOLATILITY DERIVATIVES 
Language:  English 
Keywords:  Volatility derivatives; operator methods 
Subjects:  G  Financial Economics > G1  General Financial Markets > G13  Contingent Pricing ; Futures Pricing 
Item ID:  5244 
Depositing User:  Claudio Albanese 
Date Deposited:  10. Oct 2007 
Last Modified:  21. Feb 2013 08:53 
References:  Albanese, C. & A. Mijatovi´c (2005), ‘A stochastic volatility model for riskreversals in foreign exchange’. Submitted for publication. Black, F. & M. Scholes (1973), ‘The pricing of options and corporate liabilities’, Journal of Political Economy 81, 637–654. Bottcher, A. & B. Silbermann (2006), Analysis of Toeplitz operators, Springer monographs in mathematics, 2nd edn, Springer. Breeden, D. & R. Litzenberger (1978), ‘Prices of state contingent claims implicit in option prices’, Journal of Business 51, 621–651. Brockhaus, O. & D. Long (1999), ‘Volatility swaps made simple’, Risk 2 1(1), 92–95. Buehler, H. (2006), ‘Consistent variance curve models’. Working paper, Deutsche Bank. Carr, P. & D. Madan (1998), Towards a theory of volatility trading, in R.Jarrow, ed., ‘Volatility: New Estimation Techniques for Pricing Derivatives’, Risk publication, Risk, pp. 417–427. Carr, P., H. Geman, D. B. Madan & M. Yor (2005), ‘Pricing options on realized variance’, Finance and Stochastics IX(4), 453–475. Carr, P. & K. Lewis (February 2004), ‘Corridor variance swaps’, Risk . Carr, P. & R. Lee (2004), ‘Robust hedging of volatility derivatives’. Presentation of Roger Lee at Columbia Financial Engineering seminar. CBOE, Publication (2003), ‘Vix white paper’. see http://www.cboe.com/micro/vix/vixwhite.pdf. Chriss, N. & W. Moroko (October 1999), ‘Market risk for volatility and variance swaps’, Risk . Demeterfi, K., E. Derman, M. Kamal & J. Zou (1999a), ‘A guide to volatility and variance swaps’, Journal of derivatives 6(4), 9–32. Demeterfi, K., E. Derman, M. Kamal & J. Zou (1999b), ‘More than you ever wanted to know about volatility swaps’. Quantitative Strategies Research Notes, Goldman Sachs. Detemple, J. & C. Osakwe (2000), ‘The valuation of volatility options’. Working paper, Boston University. Friz, P. & J. Gatheral (2005), ‘Valuation of volatility derivatives as an inverse problem’, Quantitative Finance 5(6), 531–542. Grimmett, J. & D. Stirzaker (2001), Probability and random processes, 3nd edn, Oxford University Press. Heston, S. L. & S. Nandi (November 2000), ‘Derivatives on volatility: some sample solutions based on observables’. Technical report, Federal Reserve Bank of Atlanta. Howison, S., A. Rafailidis & H. Rasmussen (2004), ‘On the pricing and hedging of volatility derivatives’, Applied Mathematical Finance 11, 317–346. Karatzas, I. & S. E. Shreve (1998), Brownian motion and stochastic calculus, Graduate texts in mathematics, 2nd edn, Springer. Madan, D., P. Carr & E.C. Chang (1998), ‘The variance gamma process and option pricing’, European Finance Review 2(1), 79–105. Neuberger, A. (1994), ‘The log contract’, The Journal of Portfolio Management pp. 74–80. Phillips, R.S. (1952), ‘On the generation of semigroups of linear operators’, Pacific Journal of Mathematics 2(3), 343–369. Schoutens, W. (2005), ‘Moment swaps’, Quantitative Finance 5(6), 525–530. Windcli, H., P.A. Forsyth & K.R. Vetzal (2006), ‘Pricing methods and hedging strategies for volatility derivatives’, Journal of Banking and Finance 30, 409–431. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/5244 