Sign up ×
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It's 100% free, no registration required.

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 $S_t$.


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

share|improve this question

1 Answer 1

up vote 1 down vote accepted

Look the first answer of this thread: How to derive the implied probability distribution from B-S volatilities?

Also many papers in Dupire volatility have your formula derivation. For example, look at (10) in

share|improve this answer
damn this was actually quite simple:( – Probilitator Mar 9 '14 at 6:27

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.