Analytically Deriving Risk-Neutral Densities from Volatility Smiles in Delta

Abstract

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.