英文标题：
《The Chebyshev method for the implied volatility》
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作者：
Kathrin Glau, Paul Herold, Dilip B. Madan, Christian P\\\"otz
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最新提交年份：
2017
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英文摘要：
The implied volatility is a crucial element of any financial toolbox, since it is used for quoting and the hedging of options as well as for model calibration. In contrast to the Black-Scholes formula its inverse, the implied volatility, is not explicitly available and numerical approximation is required. We propose a bivariate interpolation of the implied volatility surface based on Chebyshev polynomials. This yields a closed-form approximation of the implied volatility, which is easy to implement and to maintain. We prove a subexponential error decay. This allows us to obtain an accuracy close to machine precision with polynomials of a low degree. We compare the performance of the method in terms of runtime and accuracy to the most common reference methods. In contrast to existing interpolation methods, the proposed method is able to compute the implied volatility for all relevant option data. In this context, numerical experiments confirm a considerable increase in efficiency, especially for large data sets.
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中文摘要：
隐含波动率是任何金融工具箱的关键元素，因为它用于期权报价和对冲以及模型校准。与布莱克-斯科尔斯公式相反，它的反比，即隐含波动率，并不明确可用，需要数值近似。我们提出了一种基于切比雪夫多项式的隐含波动率曲面的二元插值方法。这产生了隐含波动率的闭合形式近似值，易于实施和维护。我们证明了一个次指数误差衰减。这使我们能够使用低阶多项式获得接近机器精度的精度。我们将该方法在运行时间和准确性方面的性能与最常见的参考方法进行了比较。与现有的插值方法相比，该方法能够计算所有相关期权数据的隐含波动率。在这种情况下，数值实验证实了效率的显著提高，尤其是对于大型数据集。
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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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