Brace, Alan and Fabbri, Giorgio and Goldys, Benjamin (2007): An Hilbert space approach for a class of arbitrage free implied volatilities models.

PDF
MPRA_paper_6321.pdf Download (202kB)  Preview 
Abstract
We present an Hilbert space formulation for a set of implied volatility models introduced in \cite{BraceGoldys01} in which the authors studied conditions for a family of European call options, varying the maturing time and the strike price $T$ an $K$, to be arbitrage free. The arbitrage free conditions give a system of stochastic PDEs for the evolution of the implied volatility surface ${\hat\sigma}_t(T,K)$. We will focus on the family obtained fixing a strike $K$ and varying $T$. In order to give conditions to prove an existenceanduniqueness result for the solution of the system it is here expressed in terms of the square root of the forward implied volatility and rewritten in an Hilbert space setting. The existence and the uniqueness for the (arbitrage free) evolution of the forward implied volatility, and then of the the implied volatility, among a class of models, are proved. Specific examples are also given.
Item Type:  MPRA Paper 

Original Title:  An Hilbert space approach for a class of arbitrage free implied volatilities models 
Language:  English 
Keywords:  Implied volatility; Option pricing; Stochastic SPDE; Hilbert space 
Subjects:  C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods G  Financial Economics > G1  General Financial Markets > G13  Contingent Pricing; Futures Pricing 
Item ID:  6321 
Depositing User:  Giorgio Fabbri 
Date Deposited:  17. Dec 2007 14:58 
Last Modified:  18. Feb 2013 06:54 
References:  H. Berestycki, J. Busca, and I. Florent. Asymptotics and calibration of local volatility models. Quantitative Finance, 2(1):6169, 2002. H. Berestycki, J. Busca, and I. Florent. Computing the implied volatility in stochastic volatility models. Communications on Pure and Applied Mathematics, 57(10):1352?1373, 2004. A. Brace, B. Goldys, F. Klebaner, and R. Womersley. Market model of stochastic implied volatility with application to bgm model. Preprint, available at the page http://www.maths.unsw.edu.au/statistics/files/preprint200101.pdf. A. Brace, B. Goldys, J. van der Hoek, and R. Wormersley. Markovian models in implied volatility framework. Statistics preprints, University of NSW, 2002. R. Cont. Modeling term structure dynamics: an infinite dimensional approach. Int. J. Theor. Appl. Finance, 8(3):357?380, 2005. G. Da Prato and J. Zabczyk. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and Its Applications. 44. Cambridge etc.: Cambridge University Press. xviii, 454 p., 1992. D. Filipovic. Consistency Problems for HeathJarrowMorton Interest Rate Models. Springer, 2001. B. Goldys and M. Musiela. Infinite dimensional diffusions, kolmogorov equations and interest rate models. In Option pricing, interest rates and risk management, Handb. Math. Finance, pages 314?335. Cambridge Univ. Press, Cambridge, 2001. M. Musiela. Stochastic pdes and term structure hlodels. Journees Internationales de Finance, IGRAFFI, La Baule, 1993. M. Musiela and M. Rutkowski. Martingale Methods In Financial Modelling. Springer, 2005. P.E. Protter. Stochastic integration and differential equations, volume 21 of Stochastic Modelling and Applied Probability. SpringerVerlag, Berlin, 2005. Second edition. Version 2.1, Corrected third printing. N. Ringer and M. Tehranchi. Optimal portfolio choice in the bond market. Finance and Stochastics, 10(4):553?573, 2006. B. L. Rozovskiff. The ItoVentcel' formula. Vestnik Moskov. Univ. Ser. I Mat. Meh., 28(1):26?32, 1973. M. Schweizer and J. Wissel. Term structures of implied volatilities: absence of arbitrage and existence results. Mathematical ?nance. to appear, M. Schweizer and J. Wissel. Arbitragefree market models for option prices: The multistrike case. Preprint, ETH Zurich, 2007. P.J. Shombucher. A market model for stochastic implied volatility. Phil. Trans. of the Royal Soc. Series A, 357:2071?2092, 1999. J. Wissel. Some results on strong solutions of SDEs with applications to interest rate models. Stochastic Process. Appl., 117(6):720?741, 2007 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/6321 