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.

中文翻译：

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隐含波动率的切比雪夫方法

隐含波动率是任何金融工具箱的关键要素，因为它用于报价和期权对冲以及模型校准。与 Black-Scholes 公式相反，它的反函数隐含波动率不是明确可用的，需要数值近似。我们提出了基于切比雪夫多项式的隐含波动率表面的双变量插值。这产生了隐含波动率的封闭形式近似值，易于实现和维护。我们证明了次指数误差衰减。这使我们能够通过低次多项式获得接近机器精度的精度。我们将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比较。与现有的插值方法相比，所提出的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着提高，尤其是对于大型数据集。