Recovering Probability Distributions from Option Prices
用非参数方法从标普500指数欧式期权价格中提取风险中性概率分布，提出一种基于平滑性最大化的快速优化技术，并发现股灾后极端下跌概率远高于对数正态假设。
This article derives underlying asset risk-neutral probability distributions of European options on the S&P 500 index. Nonparametric methods are used to choose probabilities that minimize an objective function subject to requiring that the probabilities are consistent with observed option and underlying asset prices. Alternative optimization specifications produce approximately the same implied distributions. A new and fast optimization technique for estimating probability distributions based on maximizing the smoothness of the resulting distribution is proposed. Since the crash, the risk-neutral probability of a three (four) standard deviation decline in the index (about −36 percent (−46 percent) over a year) is about 10 (100) times more likely than under the assumption of lognormality.