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摘要翻译：
我们证明了在一类具有附加斜函数的随机波动率模型（局部随机波动率模型）中，对数收益累积分布的尾部表现为exp（-cy）,其中c是依赖于时间和模型参数的正常数。我们得到了这个估计，证明了一个更强的结果：使用Bally等人对Ito过程保持在确定性曲线附近的概率的一些估计。'09年，我们降低了对偶(X，V)在给定的成熟度下保持在二维曲线周围的概率，X是对数回归过程，V是它的瞬时方差。然后，我们找到了最优的曲线，导致在终端CDF上的界。我们所依赖的方法不需要特征函数的反演，但适用于底层SDE的一般系数（特别是，不需要仿射结构）。尽管所涉及的常数不如对具有特定结构的随机波动率模型所导出的常数尖锐，但我们的下界导致矩爆炸，从而暗示Black-Scholes隐含波动率在所考虑的模型类中总是表现出翅膀。在本文的第二部分，利用Malliavin演算技术，我们证明了对数返回密度的类似估计也成立。
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英文标题：
《Bounds on Stock Price probability distributions in Local-Stochastic
Volatility models》
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作者：
Vlad Bally and Stefano De Marco
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最新提交年份：
2010
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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 show that in a large class of stochastic volatility models with additional skew-functions (local-stochastic volatility models) the tails of the cumulative distribution of the log-returns behave as exp(-c|y|), where c is a positive constant depending on time and on model parameters. We obtain this estimate proving a stronger result: using some estimates for the probability that Ito processes remain around a deterministic curve from Bally et al. '09, we lower bound the probability that the couple (X,V) remains around a two-dimensional curve up to a given maturity, X being the log-return process and V its instantaneous variance. Then we find the optimal curve leading to the bounds on the terminal cdf. The method we rely on does not require inversion of characteristic functions but works for general coefficients of the underlying SDE (in particular, no affine structure is needed). Even though the involved constants are less sharp than the ones derived for stochastic volatility models with a particular structure, our lower bounds entail moment explosion, thus implying that Black-Scholes implied volatility always displays wings in the considered class of models. In a second part of this paper, using Malliavin calculus techniques, we show that an analogous estimate holds for the density of the log-returns as well.
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PDF链接：
https://arxiv.org/pdf/1006.3337
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