摘要翻译：
本文对一类具有或不具有布朗分量的指数L{e}vy模型，导出了ATM期权价格的一个新的二阶近似。这些结果进一步揭示了连续分量和跳跃参数的波动性与ATM期权临近到期时的价格行为之间的联系。在存在布朗分量的情况下，二阶项时间-$T$的形式为$D_{2}\，t^{(3-y)/2}$，其中$D_{2}$只取决于$Y$,跳跃活动的程度，取决于$\sigma$,连续分量的波动性，以及控制“小”跳跃强度的附加参数（不管它们的符号如何）。这扩展了前导一阶项是$\sigma t^{1/2}/\sqrt{2\pi}$这一众所周知的结果。相反，在纯跳跃模型下，对$y$的依赖以及对负和正小跳跃的独立强度的依赖已经反映在前导项中，其形式为$d_{1}t^{1/y}$。二阶项的形式为$\tilde{d}_{2}t$,因此它的衰变阶数与$y$无关。本文还讨论了相应的Black-Scholes隐含挥发的渐近性态。我们的方法是足够普遍的，可以覆盖满足后一性质的L\'{e}vy过程的广泛类别，其L\'{e}vy密度可以用原点附近的稳定密度近似。我们的数值结果表明，一阶项通常表现出相当差的性能，而二阶项可以显著地提高逼近的精度，尤其是在没有布朗分量的情况下。
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英文标题：
《High-order short-time expansions for ATM option prices of exponential
  L\'evy models》
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作者：
Jos\'e E. Figueroa-L\'opez, Ruoting Gong, Christian Houdr\'e
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最新提交年份：
2014
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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：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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英文摘要：
  In the present work, a novel second-order approximation for ATM option prices is derived for a large class of exponential L\'{e}vy models with or without Brownian component. The results hereafter shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In the presence of a Brownian component, the second-order term, in time-$t$, is of the form $d_{2}\,t^{(3-Y)/2}$, with $d_{2}$ only depending on $Y$, the degree of jump activity, on $\sigma$, the volatility of the continuous component, and on an additional parameter controlling the intensity of the "small" jumps (regardless of their signs). This extends the well known result that the leading first-order term is $\sigma t^{1/2}/\sqrt{2\pi}$. In contrast, under a pure-jump model, the dependence on $Y$ and on the separate intensities of negative and positive small jumps are already reflected in the leading term, which is of the form $d_{1}t^{1/Y}$. The second-order term is shown to be of the form $\tilde{d}_{2} t$ and, therefore, its order of decay turns out to be independent of $Y$. The asymptotic behavior of the corresponding Black-Scholes implied volatilities is also addressed. Our approach is sufficiently general to cover a wide class of L\'{e}vy processes which satisfy the latter property and whose L\'{e}vy densitiy can be closely approximated by a stable density near the origin. Our numerical results show that the first-order term typically exhibits rather poor performance and that the second-order term can significantly improve the approximation's accuracy, particularly in the absence of a Brownian component.
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PDF链接：
https://arxiv.org/pdf/1208.5520