Understanding Risk-Neutral Density Functions

Given $p(s)$ is the Risk-Neutral Probability Density Function of underlying spot price s,

$$\begin{aligned} C(K) &= e^{-rT}\int_{K}^{+\infty}(s-K)p(s)ds \\ &= e^{-rT} \lim_{a \rightarrow +\infty} \int_{K}^{a}(s-K)p(s)ds \\ \frac{\partial{C(K)}}{\partial{K}} &= e^{-rT} \frac{\partial{(\lim_{a \rightarrow +\infty} \int_{K}^{a}(s-K)p(s)ds)}}{\partial{K}} \\ &= e^{-rT} \lim_{a \rightarrow+\infty}\frac{\partial{\left(\int_{K}^{a}(s-K)p(s)ds\right )}}{\partial{K}} \\ &= e^{-rT}\left [] 0 - (K-K)p(K)*1 +\int_{K}^{+\infty}(-1)p(s)ds \right ] \\ &=e^{-rT}\left ( \int_{0}^{K}p(s)ds -1 \right ) \\ \frac{\partial{C^{2}(K)}}{\partial{K^2}} &=e^{-rT}p(s) \\ \\ \\ P(K) &= e^{-rT}\int_{0}^{K}(K-s)p(s)ds \\ \frac{\partial{P(K)}}{\partial{K}} &= e^{-rT} \frac{\partial{(\int_{0}^{K}(K-s)p(s)ds)}}{\partial{K}} \\ &= e^{-rT}\left [ (K-K)p(s)*10 - 0 +\int_{0}^{K}p(s)ds \right ] \\ &=e^{-rT}\int_{0}^{K}p(s)ds\\ \frac{\partial{P^{2}(K)}}{\partial{K^2}} &=e^{-rT}p(s) \end{aligned}$$

Thus $\frac{\partial{C^{2}(K)}}{\partial{K^2}} = \frac{\partial{P^{2}(K)}}{\partial{K^2}} =e^{-rT}p(s)$

Reference

• Leibniz integral rule

• Leibniz rule for improper integral

• Chain rule

• The Risk Neutral Returns Distribution for the U.S. Stock Market

• Deriving option-implied probability densities for foreign exchange markets