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摘要翻译：
本文研究了具有L\'evy跳跃的L\'evy随机波动率模型，其形式为$Z=u+x$,其中$u=(Z_{t}）_{t\geq0}$是经典随机波动率过程，$x=(X_{t}）_{t\geq0}$是具有绝对连续L\'evy测度$\nu$的独立L\'evy过程。对于尾盘$\bbp(Z_{t}\geqz)$,$z>0$和看涨期权价格$\bbe(E_{z+Z_{t}}-1)_{+}$,$z\neq0$,在时间上得到了任意多项式阶的小时间展开式。对于光滑函数$\phi$和$z>0$，我们的方法允许统一处理形式为$\phi(x){\bf1}_{x\geq{}z}$的一般支付函数。作为尾展开式的结果，在温和的条件下，跃迁密度$f_{t}$的$t$中的多项式展开式也是{\green greated}。
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英文标题：
《Small-time expansions of the distributions, densities, and option prices
of stochastic volatility models with L\'evy jumps》
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作者：
J. E. Figueroa-L\'opez, R. Gong, and C. Houdr\'e
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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：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：
We consider a stochastic volatility model with L\'evy jumps for a log-return process $Z=(Z_{t})_{t\geq 0}$ of the form $Z=U+X$, where $U=(U_{t})_{t\geq 0}$ is a classical stochastic volatility process and $X=(X_{t})_{t\geq 0}$ is an independent L\'evy process with absolutely continuous L\'evy measure $\nu$. Small-time expansions, of arbitrary polynomial order, in time-$t$, are obtained for the tails $\bbp(Z_{t}\geq z)$, $z>0$, and for the call-option prices $\bbe(e^{z+Z_{t}}-1)_{+}$, $z\neq 0$, assuming smoothness conditions on the {\PaleGrey density of $\nu$} away from the origin and a small-time large deviation principle on $U$. Our approach allows for a unified treatment of general payoff functions of the form $\phi(x){\bf 1}_{x\geq{}z}$ for smooth functions $\phi$ and $z>0$. As a consequence of our tail expansions, the polynomial expansions in $t$ of the transition densities $f_{t}$ are also {\Green obtained} under mild conditions.
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PDF链接：
https://arxiv.org/pdf/1009.4211
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