The Black-Scholes formula assumes the price of the underlying asset evolves as a geometric Brownian motion with constant volatility, and so the probability distribution of the asset price at a fixed time in the future is lognormal. However, a single volatility parameter cannot describe the prices of all traded options in general, due to volatility smiles.
To use the Black-Scholes formula, an implied volatility surface is required. Given such a surface, call options with any strike and time to expiry can be priced, and these prices are sufficient to infer the risk-neutral implied probability distribution for the asset price at any time.
This paper obtains expressions for the cumulative distribution function and probability density function implied by an arbitrary choice of smile, by transforming to a dimensionless smile parameterised by log-moneyness. Since a cumulative distribution function has obvious constraints, the smile must necessarily obey certain constraints which are presented here.