Previous estimation results of risk-neutral densities explain in rather general terms that the tails of the resulting distribution “look fat,” and a way has to be found to model the tails of the estimated distribution. The author uses deep out-of-the-money S&P 500 index options to examine model mispricing of the tails of daily estimated risk-neutral densities. Out-of-sample tests show that model mispricing increases as one moves farther into the tails of the distribution. Across most moneyness groups, model mispricing increases as the option reaches maturity. The author compares two curve-fitting methods that have been proposed in the literature to estimate risk-neutral densities. The first method interpolates with a fourth-order spline and attaches tails from the general extreme value distribution (Figlewski 2010). The second method extends the available implied volatility space by balancing smoothness and fit of the estimated risk-neutral density (Jackwerth 2004). Fitting a fourth-order spline produces a closer fit to the observed implied volatilities. Examining the ability to replicate the implied volatility with the complete estimated option-implied risk-neutral density by looking at mean root-mean-square error, the method by Jackwerth (2004) resulted in lower in- and out-of-sample model mispricing, except for the deepest out-of-the-money put options. TOPICS: Tail risks, options Key Findings • This article compares two methods from the curve-fitting literature to estimate option-implied risk-neutral densities and looks at the accuracy to recover implied volatilities. • Model mispricing, measured by the root-mean-square error, increases for deeper out-of-the-money options. • Model mispricing increases as the option reaches its maturity across most out-of-sample moneyness groups.

中文翻译：

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风险中性密度估计：观察尾巴

风险中性密度的先前估计结果相当笼统地解释了所得分布的尾部“看起来很胖”，因此必须找到一种对估计分布的尾部建模的方法。作者使用了超值的标准普尔500指数期权来检验每日估计的风险中性密度尾部的模型定价错误。样本外测试表明，模型定价错误会随着人们向分布尾部的进一步移动而增加。在大多数货币群体中，随着期权到期，模型定价错误会增加。作者比较了文献中提出的两种曲线拟合方法，以估计风险中性密度。第一种方法使用四阶样条进行插值，并附加来自一般极值分布的尾部（Figlewski 2010）。第二种方法通过平衡估计的风险中性密度的平滑度和拟合来扩展可用的隐含波动率空间（Jackwerth 2004）。拟合四阶样条曲线可以更紧密地拟合所观察到的隐含波动率。通过查看平均均方根误差，检验具有完整估计期权隐含风险中性密度的隐含波动率的能力，Jackwerth（2004）的方法导致样本内和样本外模型错误定价降低，但最便宜的认沽期权除外。主题：尾部风险，期权主要发现•本文比较了曲线拟合文献中的两种方法，以估计期权隐含的风险中性密度，并研究了回收隐含波动率的准确性。•模型的定价错误，以均方根误差衡量，增加用于更深层的价外选择。•随着期权在大多数非抽样货币组中到期，模型定价错误会增加。