This article develops a method for analytically deriving RNDs (risk-neutral densities) of future asset prices from volatility smiles. It extends an existing analytical method, which is for volatility smiles with respect to the strike price, to cover smiles with respect to option delta. A worked-out example on currency options shows that the analytically derived RNDs are free of approximation errors that would arise with numerical methods. The proposed method should be useful to practitioners. TOPICS: Options, futures and forward contracts Key Findings • This article develops an analytical method for deriving RNDs (risk-neutral probability densities) of future asset prices from volatility smiles with respect to option delta. • A worked-out example shows that exact RNDs computed by the method are free of approximation errors associated with numerical derivatives. • The proposed method should be useful to practitioners.

中文翻译：

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从波动率微笑中解析得出风险中性密度

本文提出了一种方法，可以从波动性微笑中分析得出未来资产价格的RND（风险中性密度）。它扩展了现有分析方法的适用范围，该方法适用于针对行使价的波动性微笑，涵盖针对期权增量的微笑。一个关于货币期权的算例表明，分析得出的RND不存在数值方法会产生的近似误差。所提出的方法对从业者应该是有用的。主题：期权，期货和远期合约主要发现•本文开发了一种分析方法，该方法可从相对于期权增量的波动性微笑中得出未来资产价格的RND（风险中性概率密度）。•一个实例说明该方法计算出的精确RND不存在与数值导数相关的近似误差。•提议的方法对从业者应该有用。