英文标题：
《Hilbert transform, spectral filters and option pricing》
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作者：
Carolyn E. Phelan, Daniele Marazzina, Gianluca Fusai, Guido Germano
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最新提交年份：
2020
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英文摘要：
We show how spectral filters can improve the convergence of numerical schemes which use discrete Hilbert transforms based on a sinc function expansion, and thus ultimately on the fast Fourier transform. This is relevant, for example, for the computation of fluctuation identities, which give the distribution of the maximum or the minimum of a random path, or the joint distribution at maturity with the extrema staying below or above barriers. We use as examples the methods by Feng and Linetsky (2008) and Fusai, Germano and Marazzina (2016) to price discretely monitored barrier options where the underlying asset price is modelled by an exponential L\\\'evy process. Both methods show exponential convergence with respect to the number of grid points in most cases, but are limited to polynomial convergence under certain conditions. We relate these rates of convergence to the Gibbs phenomenon for Fourier transforms and achieve improved results with spectral filtering.
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中文摘要：
我们展示了谱滤波器如何提高基于sinc函数展开的离散Hilbert变换的数值格式的收敛性，从而最终提高了快速傅立叶变换的收敛性。例如，这与波动恒等式的计算有关，波动恒等式给出了随机路径的最大值或最小值的分布，或在极值保持在障碍下方或上方的成熟期的联合分布。我们以Feng和Linetsky（2008）以及Fusai、Germano和Marazzina（2016）的方法为例，对离散监控的障碍期权进行定价，其中基础资产价格由指数列维过程建模。这两种方法在大多数情况下都显示出相对于网格点数量的指数收敛性，但在某些条件下仅限于多项式收敛。我们将这些收敛速度与傅里叶变换的吉布斯现象联系起来，并通过频谱滤波获得了更好的结果。
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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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