指数Levy过程及其波动微笑的渐近性； 调查和新结果

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摘要翻译：
指数L\'evy过程可以用来模拟各种金融变量，如汇率、股票价格等的演化。在这类过程的支配下，衍生工具的定价已经得到了相当多的研究，并对相应的隐含波动率面进行了详细的分析。在非渐近状态下，期权价格由Lewis-Lipton公式描述，可以表示为Fourier积分；价格可以用隐含波动率来表示。最近，在计算隐含波动率的渐近极限方面的尝试已经得到了几种短时渐近、长时渐近和翼渐近的表达式。为了更详细地研究波动率面，本文利用外汇惯例，将隐含波动率描述为Black-Scholes delta的函数。令人惊讶的是，这一约定与代数几何中经常使用的奇异点的消解密切相关。在这个框架下，我们回顾了文献，重新表述了关于隐含波动率渐近行为的一些已知事实，并给出了几个新的结果。我们强调了分数阶微分在研究回火稳定指数Levy过程中的作用，并基于分数阶导数的有限差分逼近导出了新的数值方法。我们还简要地证明了如何推广我们的结果以研究局部和随机波动模型的重要情况，当使用Lewis-Lipton公式时，这些模型与基于L\'evy过程的模型的密切关系尤其明显。我们的主要结论是，研究隐含波动率的渐近性质虽然在理论上令人兴奋，但并不总是有用的，因为许多渐近表达式的有效域很小。
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英文标题：
《Asymptotics for Exponential Levy Processes and their Volatility Smile:
  Survey and New Results》
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作者：
Leif Andersen and Alexander Lipton
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最新提交年份：
2012
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Pricing of Securities        证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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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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英文摘要：
  Exponential L\'evy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, etc. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes, and the corresponding implied volatility surfaces have been analyzed in some detail. In the non-asymptotic regimes, option prices are described by the Lewis-Lipton formula which allows one to represent them as Fourier integrals; the prices can be trivially expressed in terms of their implied volatility. Recently, attempts at calculating the asymptotic limits of the implied volatility have yielded several expressions for the short-time, long-time, and wing asymptotics. In order to study the volatility surface in required detail, in this paper we use the FX conventions and describe the implied volatility as a function of the Black-Scholes delta. Surprisingly, this convention is closely related to the resolution of singularities frequently used in algebraic geometry. In this framework, we survey the literature, reformulate some known facts regarding the asymptotic behavior of the implied volatility, and present several new results. We emphasize the role of fractional differentiation in studying the tempered stable exponential Levy processes and derive novel numerical methods based on judicial finite-difference approximations for fractional derivatives. We also briefly demonstrate how to extend our results in order to study important cases of local and stochastic volatility models, whose close relation to the L\'evy process based models is particularly clear when the Lewis-Lipton formula is used. Our main conclusion is that studying asymptotic properties of the implied volatility, while theoretically exciting, is not always practically useful because the domain of validity of many asymptotic expressions is small.
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PDF链接：
https://arxiv.org/pdf/1206.6787

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关键词：Levy Quantitative Applications Exponential derivatives 逼近 been 行为 Exponential 指数