An Hilbert space approach for a class of arbitrage free implied volatilities models

Alan Brace, Giorgio Fabbri and Benjamin Goldys

MPRA Paper from University Library of Munich, Germany

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 existence-and-uniqueness 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.

Keywords: Implied volatility; Option pricing; Stochastic SPDE; Hilbert space (search for similar items in EconPapers)
JEL-codes: C02 G13 (search for similar items in EconPapers)
Date: 2007-12-17
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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Working Paper: An Hilbert space approach for a class of arbitrage free implied volatilities models (2007)
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