Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models
Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models
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大偏差和跳跃的随机波动率：仿射模型的渐近隐含波动率
DOI：
10.1080/17442508.2012.720687
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发表时间：
2011
期刊：
影响因子：
0.9
通讯作者：
A. Mijatović
中科院分区：
文献类型：
--
作者：
A. Jacquier;Martin Keller;A. Mijatović
令表示到期日t的隐含波动率，其中和是标的资产的当前价值。我们表明，有一个统一的（在x）的限制，成熟度t趋于无穷大，由公式给出，x在一些紧邻域的零类仿射随机波动率模型。函数是标度对数点过程的极限累积生成函数h的凸对偶。我们用底层模型的功能特性来表示h。极限公式的证明依赖于当时间趋于无穷大时标度对数点过程的大偏差行为。我们应用我们的结果获得了几类随机波动模型的极限微笑的应用程序中使用的跳跃（例如赫斯顿与状态无关的跳跃，贝茨与状态相关的跳跃和Barndorff-Nielsen-Shephard模型）。
Let denote the implied volatility at maturity t for a strike , where and is the current value of the underlying. We show that has a uniform (in x) limit as maturity t tends to infinity, given by the formula , for x in some compact neighbourhood of zero in the class of affine stochastic volatility models. Function is the convex dual of the limiting cumulant-generating function h of the scaled log-spot process. We express h in terms of the functional characteristics of the underlying model. The proof of the limiting formula rests on the large deviation behaviour of the scaled log-spot process as time tends to infinity. We apply our results to obtain the limiting smile for several classes of stochastic volatility models with jumps used in applications (e.g. Heston with state-independent jumps, Bates with state-dependent jumps and Barndorff-Nielsen–Shephard model).