从微笑渐近到市场风险测度

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摘要翻译：
隐含波动率倾斜的左尾来自于货币外看跌期权的报价，可以认为反映了市场对股价大幅下跌风险的评估。我们分析了如何将这些市场信息整合到凸货币风险度量的理论框架中。特别地，我们利用以倒向随机微分方程解形式给出的动态凸风险测度的无差别定价，建立了这两种风险度量方法之间的联系。我们用非线性偏微分方程的解导出了隐含波动率的刻画，并给出了一个小的到期时间展开式和数值解。该方法允许在我们引入的一类参数化方便的扭曲熵动态风险测度中选择凸风险测度，从而使无差异定价下的渐近波动率偏差与市场偏差相匹配。我们在对市场隐含波动率数据的校准练习中证明了这一点。
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英文标题：
《From Smile Asymptotics to Market Risk Measures》
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作者：
Ronnie Sircar and Stephan Sturm
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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：Risk Management        风险管理
分类描述：Measurement and management of financial risks in trading, banking, insurance, corporate and other applications
衡量和管理贸易、银行、保险、企业和其他应用中的金融风险
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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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一级分类：Quantitative Finance        数量金融学
二级分类：General Finance        一般财务
分类描述：Development of general quantitative methodologies with applications in finance
通用定量方法的发展及其在金融中的应用
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英文摘要：
  The left tail of the implied volatility skew, coming from quotes on out-of-the-money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations (BSDEs), to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear PDE and provide a small time-to-maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parametrized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.
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PDF链接：
https://arxiv.org/pdf/1107.4632

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关键词：市场风险 Applications Quantitative Indifference Differential implied risk market dynamic 反映