For a European call option with payoff function \(h(S_{T})=(S_{T}-K)_{+}\) we are able to recover the density of the stock price using the information on the option price \(C(T,K)\), which is defined as the price of the European call option at time 0 with maturity \(T\) and strike \(K\).
The Breeden-Litzenberger Formula relates the option price \(C(T,K)\) to the density of the underlying stock price \(S_t\) under risk neutral measure \(\mathbb{Q}\), denoted as \(p(t,\cdot)\), as follows \[ p(T,x)=e^{-rT} \frac{{\partial^2C}}{\partial K^2}(T,x). \]
\begin{proof}
Assume that the density of stock price \(S_{t}\) under risk-neutral measure \(\mathbb{Q}\) is denoted as \(p(t,\cdot)\), then from risk-neutral pricing \[
C(T,K)=e^{-rT}\mathbb{E}^\mathbb{Q}(S_{T}-K)_{+}=e^{-rT}\int _{K}^\infty (X-K)\cdot p(T,x) \, dx .
\] Taking the derivative with respect to \(K\) (assuming its admissible) we find \[
\begin{align}
\frac{{\partial C}}{\partial K} & =e^{-rT}\left[ -Kp(T,K)-\int _{K}^\infty p(T,x)\, dx +Kp(T,K) \right] \\
& -e^{-rT}\int _{K}^\infty p(T,x) \, dx .
\end{align}
\] Taking the derivative again with respect to \(K\) gives the B-L formula. \end{proof} Note: This formula is restrictive in practice since it requires us to know a continuum of European call prices with respect to different strikes \(K\). However, it is not model specific and will help us build up more useful results.