英文标题：
《Left-wing asymptotics of the implied volatility in the presence of atoms》
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作者：
Archil Gulisashvili
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最新提交年份：
2013
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英文摘要：
We consider the asymptotic behavior of the implied volatility in stochastic asset price models with atoms. In such models, the asset price distribution has a singular component at zero. Examples of models with atoms include the constant elasticity of variance model, jump-to-default models, and stochastic models described by processes stopped at the first hitting time of zero. For models with atoms, the behavior of the implied volatility at large strikes is similar to that in models without atoms. On the other hand, the behavior of the implied volatility at small strikes is influenced significantly by the atom at zero. S. De Marco, C. Hillairet, and A. Jacquier found an asymptotic formula for the implied volatility at small strikes with two terms and also provided an incomplete description of the third term. In the present paper, we obtain a new asymptotic formula for the left wing of the implied volatility, which is qualitatively different from the De Marco-Hillairet-Jacquier formula. The new formula contains three explicit terms and an error estimate. We show how to derive the De Marco-Hillairet-Jacquier formula from our formula, and compare the performance of the two formulas in the case of the CEV model. The resulting graphs show that the new formula provides a notably better approximation to the smile in the CEV model than the De Marco-Hillairet-Jacquier formula.
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中文摘要：
我们考虑了原子随机资产价格模型中隐含波动率的渐近行为。在这样的模型中，资产价格分布在零处有一个奇异分量。原子模型的例子包括恒定弹性方差模型、跳转到默认模型，以及在第一次达到零时停止的过程所描述的随机模型。对于有原子的模型，隐含波动率在大冲击下的行为与没有原子的模型相似。另一方面，小冲击下隐含波动率的行为受到零原子的显著影响。S.De Marco、C.Hillairet和A.Jacquier发现了两项小规模冲击下隐含波动率的渐近公式，并提供了第三项的不完整描述。在本文中，我们得到了隐含波动率左翼的一个新的渐近公式，它与De Marco Hillairet-Jacquier公式在性质上不同。新公式包含三个显式项和一个误差估计。我们展示了如何从我们的公式推导出De Marco Hillairet-Jacquier公式，并在CEV模型的情况下比较了这两个公式的性能。结果图显示，新公式比De Marco Hillairet-Jacquier公式更接近CEV模型中的微笑。
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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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