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

## Author Info

Listed author(s):
• Brace, Alan
• Fabbri, Giorgio
• Goldys, Benjamin

## 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.

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File URL: https://mpra.ub.uni-muenchen.de/6321/1/MPRA_paper_6321.pdf
File Function: original version

## Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 6321.

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## References

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1. H. Berestycki & J. Busca & I. Florent, 2002. "Asymptotics and calibration of local volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 61-69.
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