This article derives risk-neutral probability distributions for European options on the S&P 500 index using nonparametric methods. The approach minimizes an objective function while ensuring probabilities align with observed option and underlying asset prices. Alternative specifications yield similar results. A new optimization technique is proposed to maximize distribution smoothness. Post-crash, the risk-neutral probability of a three (four) standard deviation decline in the index is 10 (100) times more likely than under lognormality. Recent interest in derivatives and risk management has led to efforts to measure sensitivity of portfolios to extreme events. Historical methods often fail to capture extreme events, which can have significant financial impacts. The lognormal assumption is inadequate for capturing extreme events like the 1987 crash, which had a -27 standard deviation event with probability 10^-160. The 1987 crash also affects historical volatility estimates, showing that volatility varies with sample size and includes the crash. The article presents data on S&P 500 options from 1986 to 1993, including implied and historical volatilities. It discusses challenges in using historical data, such as nonstationarity and biases in sample statistics. The paper introduces nonparametric methods to derive risk-neutral probabilities from option prices, avoiding assumptions about functional relations. The method uses quadratic programming to minimize differences between prior and posterior probabilities, ensuring consistency with market prices. The paper also addresses issues with implied volatility smiles, showing that they are persistent and irregular. It proposes methods to smooth these smiles and improve accuracy. The results show that implied probability distributions are more leptokurtic and left-skewed post-crash compared to pre-crash. The cumulative probabilities indicate that large declines are more likely post-crash. The article concludes that maximizing smoothness is a suitable objective for nonparametric methods. The results show that implied probabilities are more likely to reflect extreme events post-crash. The method is robust and nonparametric, requiring no assumptions about stochastic processes or investor behavior. However, the absence of trading costs is a key assumption that may affect the results. The paper suggests that the observed smile effects could be due to changes in investor beliefs or risk aversion post-crash.This article derives risk-neutral probability distributions for European options on the S&P 500 index using nonparametric methods. The approach minimizes an objective function while ensuring probabilities align with observed option and underlying asset prices. Alternative specifications yield similar results. A new optimization technique is proposed to maximize distribution smoothness. Post-crash, the risk-neutral probability of a three (four) standard deviation decline in the index is 10 (100) times more likely than under lognormality. Recent interest in derivatives and risk management has led to efforts to measure sensitivity of portfolios to extreme events. Historical methods often fail to capture extreme events, which can have significant financial impacts. The lognormal assumption is inadequate for capturing extreme events like the 1987 crash, which had a -27 standard deviation event with probability 10^-160. The 1987 crash also affects historical volatility estimates, showing that volatility varies with sample size and includes the crash. The article presents data on S&P 500 options from 1986 to 1993, including implied and historical volatilities. It discusses challenges in using historical data, such as nonstationarity and biases in sample statistics. The paper introduces nonparametric methods to derive risk-neutral probabilities from option prices, avoiding assumptions about functional relations. The method uses quadratic programming to minimize differences between prior and posterior probabilities, ensuring consistency with market prices. The paper also addresses issues with implied volatility smiles, showing that they are persistent and irregular. It proposes methods to smooth these smiles and improve accuracy. The results show that implied probability distributions are more leptokurtic and left-skewed post-crash compared to pre-crash. The cumulative probabilities indicate that large declines are more likely post-crash. The article concludes that maximizing smoothness is a suitable objective for nonparametric methods. The results show that implied probabilities are more likely to reflect extreme events post-crash. The method is robust and nonparametric, requiring no assumptions about stochastic processes or investor behavior. However, the absence of trading costs is a key assumption that may affect the results. The paper suggests that the observed smile effects could be due to changes in investor beliefs or risk aversion post-crash.