This article explores the recovery of underlying asset risk-neutral probability distributions for European options on the S&P 500 index using nonparametric methods. The authors aim to minimize an objective function that ensures the probabilities are consistent with observed option and underlying asset prices. They propose a new optimization technique based on maximizing the smoothness of the resulting distribution, which is faster and more efficient than existing methods. The analysis reveals that since the 1987 stock market 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 lognormal assumption. The study also examines the shapes of implied probability distributions over time, finding a significant change in skewness and kurtosis after the crash, with distributions becoming more left-skewed and leptokurtic. The results suggest that investors concerned with extreme stock market events should use methods that allow for flexibility in the shape of the lower left-hand tail of the distribution.This article explores the recovery of underlying asset risk-neutral probability distributions for European options on the S&P 500 index using nonparametric methods. The authors aim to minimize an objective function that ensures the probabilities are consistent with observed option and underlying asset prices. They propose a new optimization technique based on maximizing the smoothness of the resulting distribution, which is faster and more efficient than existing methods. The analysis reveals that since the 1987 stock market 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 lognormal assumption. The study also examines the shapes of implied probability distributions over time, finding a significant change in skewness and kurtosis after the crash, with distributions becoming more left-skewed and leptokurtic. The results suggest that investors concerned with extreme stock market events should use methods that allow for flexibility in the shape of the lower left-hand tail of the distribution.