This paper proposes a nonparametric estimator for the state price density implied by a single cross-section of European options with different strikes and the same maturity. The proposed estimator has two distinctive features. First, it extracts information from both call and put options, as opposed to only call options. Second, it does not require estimating any second-order derivative; instead, it solves a constrained and penalized linear regression. The asymptotic analysis faces two challenges because the state price density is defined by a Fredholm integral equation of the first kind with an unbounded support, and the kernel function is unbounded and non-differentiable. We address these challenges by exploiting the structure of the option pricing problem. After establishing the estimator’s consistency and convergence rate, we apply it to estimate the state price densities implied by the S&P500 index options and those by the VIX options. The sample period includes the recent financial crisis and the Great Recession, during which the turbulent market conditions imposed substantial challenges on the estimation. We show that the estimator can work with both daily and high-frequency observations. We also study whether the various features of this density can predict future asset returns and obtain positive findings. Finally, we apply the method to examine the causal effects of monetary policy announcements on the financial market, using high-frequency data.

本文针对具有不同行使价和相同到期日的欧式期权的单一横截面所隐含的状态价格密度提出了一种非参数估计器。建议的估计器有两个显着特征。首先，它从看涨期权和看跌期权中提取信息，而不仅仅是看涨期权。其次，它不需要估计任何二阶导数；相反，它解决了受约束和惩罚的线性回归。渐近分析面临两个挑战，因为状态价格密度由具有无界支持的第一类 Fredholm 积分方程定义，并且核函数是无界且不可微的。我们通过利用期权定价问题的结构来解决这些挑战。建立估计器的一致性和收敛速度后，我们应用它来估计标准普尔 500 指数期权和 VIX 期权隐含的状态价格密度。样本期间包括最近的金融危机和大衰退，在此期间动荡的市场环境给估计带来了巨大挑战。我们表明估计器可以处理日常和高频观察。我们还研究了这种密度的各种特征是否可以预测未来的资产回报并获得积极的发现。最后，我们使用高频数据应用该方法来检验货币政策公告对金融市场的因果影响。在此期间，动荡的市场环境对估计构成了重大挑战。我们表明估计器可以处理日常和高频观察。我们还研究了这种密度的各种特征是否可以预测未来的资产回报并获得积极的发现。最后，我们使用高频数据应用该方法来检验货币政策公告对金融市场的因果影响。在此期间，动荡的市场环境对估计构成了重大挑战。我们表明估计器可以处理日常和高频观察。我们还研究了这种密度的各种特征是否可以预测未来的资产回报并获得积极的发现。最后，我们使用高频数据应用该方法来检验货币政策公告对金融市场的因果影响。