Understanding Risk-Neutral Density Functions

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

C(K)=e−rT∫K+∞(s−K)p(s)ds=e−rTlim⁡a→+∞∫Ka(s−K)p(s)ds∂C(K)∂K=e−rT∂(lim⁡a→+∞∫Ka(s−K)p(s)ds)∂K=e−rTlim⁡a→+∞∂(∫Ka(s−K)p(s)ds)∂K=e−rT[]0−(K−K)p(K)∗1+∫K+∞(−1)p(s)ds]=e−rT(∫0Kp(s)ds−1)∂C2(K)∂K2=e−rTp(s)P(K)=e−rT∫0K(K−s)p(s)ds∂P(K)∂K=e−rT∂(∫0K(K−s)p(s)ds)∂K=e−rT[(K−K)p(s)∗10−0+∫0Kp(s)ds]=e−rT∫0Kp(s)ds∂P2(K)∂K2=e−rTp(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 ∂C2(K)∂K2=∂P2(K)∂K2=e−rTp(s)\frac{\partial{C^{2}(K)}}{\partial{K^2}} = \frac{\partial{P^{2}(K)}}{\partial{K^2}} =e^{-rT}p(s)

Reference

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