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

TL;DR

This paper develops an Hilbert space framework for arbitrage-free implied volatility models, proving existence and uniqueness of solutions for the evolution of implied volatility surfaces using stochastic PDEs.

Contribution

It introduces an Hilbert space formulation for implied volatility models, establishing existence and uniqueness results for arbitrage-free implied volatility evolution.

Findings

Proved existence and uniqueness of implied volatility surface evolution

Established conditions for arbitrage-free implied volatility models

Provided specific examples of the models

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.