Implied Distribution as a Function of the Volatility Smile

In this paper we obtain the risk-neutral density of an underlying asset price as a function of its option implied volatility. We derive a known expression for the density and decompose it into a sum of lognormal and adjustment terms. We also derive no-arbitrage conditions on the volatility smile. We then explain how to use the results. Our methodology is applied to the pricing of a portfolio of digital options. It is then applied to the fitting of a distribution for log-return modelling.
Keywords: Option pricing; No arbitrage; Risk-neutral distribution; Implied volatility smile.
JEL: C14; C52; G13