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摘要翻译：
我们给出了一组隐含波动率模型的Hilbert空间公式，其中作者研究了一族欧式看涨期权在不同的到期时间和执行价$T$a$K$下无套利的条件。无套利条件给出了隐含波动率面${\hat\sigma}_T(T，K)$演化的随机偏微分方程组。我们将重点关注家庭获得的固定罢工$K$和变化$T$。为了给出证明系统解的存在唯一性结果的条件，这里用正向隐含波动率的平方根表示，并在Hilbert空间中改写。证明了一类模型的正向隐含波动率（无套利）演化的存在性和唯一性，进而证明了一类模型的隐含波动率（无套利）演化的存在唯一性。文中还给出了具体实例。
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英文标题：
《An Hilbert space approach for a class of arbitrage free implied
volatilities models》
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作者：
A. Brace, G. Fabbri, B. Goldys
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最新提交年份：
2007
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分类信息：
一级分类：Quantitative Finance 数量金融学
二级分类：Computational Finance 计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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英文摘要：
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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PDF链接：
https://arxiv.org/pdf/0712.1343
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