英文标题：
《Dirichlet Forms and Finite Element Methods for the SABR Model》
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作者：
Blanka Horvath and Oleg Reichmann
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最新提交年份：
2018
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英文摘要：
We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via non-symmetric Dirichlet forms. Our pricing method is valid under mild assumptions on parameter configurations of the process both in moderate interest rate environments and in near-zero interest rate regimes such as the currently prevalent ones. The parabolic Kolmogorov pricing equations for the SABR model are degenerate at the origin, yielding non-standard partial differential equations, for which conventional pricing methods ---designed for non-degenerate parabolic equations--- potentially break down. We derive here the appropriate analytic setup to handle the degeneracy of the model at the origin. That is, we construct an evolution triple of suitably chosen Sobolev spaces with singular weights, consisting of the domain of the SABR-Dirichlet form, its dual space, and the pivotal Hilbert space. In particular, we show well-posedness of the variational formulation of the SABR-pricing equations for vanilla and barrier options on this triple. Furthermore, we present a finite element discretization scheme based on a (weighted) multiresolution wavelet approximation in space and a $\\theta$-scheme in time and provide an error analysis for this discretization.
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中文摘要：
基于非对称Dirichlet形式对Kolmogorov定价方程进行有限元离散，我们提出了一种SABR随机波动率模型下香草期权定价的确定性数值方法。我们的定价方法在中等利率环境和近零利率制度（如当前普遍的利率制度）下，在对过程参数配置的温和假设下有效。SABR模型的抛物型Kolmogorov定价方程在原点处退化，产生了非标准偏微分方程，而针对非退化抛物型方程设计的传统定价方法可能会崩溃。我们在这里导出了适当的分析设置，以处理模型在原点处的简并度。也就是说，我们构造了一个适当选择的具有奇异权重的Sobolev空间的演化三元组，由SABR-Dirichlet形式的域、其对偶空间和关键Hilbert空间组成。特别地，我们证明了在这个三元组上香草期权和障碍期权的SABR定价方程的变分公式的适定性。此外，我们提出了一种基于空间（加权）多分辨率小波近似和时间$\\θ$-格式的有限元离散化方案，并对这种离散化进行了误差分析。
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分类信息：

一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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