# Implied state price density (Question 1 - derivation of the formula)

I came upon the term "implied state price density" in a couple of papers. As far as I understand the concept one basically tries to extract the "pricing density" from the market data.

For the sake of simplicity we assume a constant interst rate $r$ and also don't make any assumptions on the model used to evolve $S_t$.

$C(t,S_t,K,r,T)=e^{-r(T-t)}\int_0^{\infty}(S_T-K)^+f(S_T|S_t)dS_T$

According to Douglas T. Breeden and Robert H. Litzenberger in their paper Prices of State-Contingent Claims Implicit in Option Prices one can recover the density via the formula:

$p(S_T|S_t)=e^{r(T-t)}\frac{\partial^2 C(t,S_t,K,r,T)}{\partial K^2}|_{K=S_T}$

How does one arrive at this formula? I tried to differentiate $C(t,S_t,K,r,T)$ but according to the rules for differentiating parameter integrals this is not how one can arrive at above formula (what am I missing?)

P.S. You can read the paper online for free at JSTOR after you register. Or just email me and I will sent you the pdf-file