I don't understand the words "risk-neutral density". Please explain what it is, and how it can be estimated in practice.
My guess would be that we have an underlying probability space $(\Omega, P)$. We are interested in the equivalent martingale measure $Q$, i.e. the space $(\Omega, Q)$.
For every stock price $S_t$, we wish to determine the $Q$-distribution. The risk-neutral density, if it exists, is the density $\phi$ of $S_t$ with respect to the Lebesque measure, i.e. we can write for some fixed $t$, $$E^Q S_t = \int \phi \cdot s \ ds.$$
Is this guess correct? If so, how do we estimate $\phi$ from stock prices?