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

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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 
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URI:  https://mpra.ub.unimuenchen.de/id/eprint/6321 