Overview
The Breden Litzenberger analysis provides a way to extract a risk neutral distribution from a set of options prices.Derivation
Starting with a call option (here labeled {% c %}), the risk neutral pricing framework shows that the call price can be obtained by calculating the following integral.
{% c = e^{- \int_0^T r_u du} \int_K ^{\infty} (S_T - K) \phi(S_T) dS_T %}
Taking the partial derivative with respect to the strike price, we have
{% \frac{\partial c}{\partial K} = e ^{-\int_0^T r_u du} \int_K^{\infty} -\phi(S_T) = e ^{-\int_0^T r_u du} (\Phi(K) - 1) %}
(recall the
Fundamental Theorem of Calculus)
{% \frac{\partial^2 c}{\partial K^2} = e ^{-\int_0^T r_u du} \phi(K) %}
Using the
second derivative approximation
we get
{% \phi(K) \approx e ^{-\int_0^T r_u du} \frac{c(K-h) -2c(K) + c(K+h)}{h^2} %}
(see Kelliher chpt 12)