0.1 Carr-Madan Formula
The Carr-Madan formula provides an equation for the price of a European-style call option with any payoff function \(h(S_{T})\). Denote the price of this option at time \(0\) with payoff \(h\), time to maturity \(T\) and strike \(K\) by \(C_{h}(T,K)\). Denote \(C(T,K)\) as the price of European call option at time \(0\) with payoff \(h(S_{T})=(S_{T}-K)_{+}\), with time to maturity \(T\) and strike \(K\).
The price of a European-style call option with any payoff function \(h(S_T)\), denoted by \(C_{h}(T,K)\) is given by \[ C_{h}(T,K)=h(0)e^{-rT}+h'(0)S_{0}+\int _{0}^\infty h''(x)\cdot C(T,x) \, dx . \]
Proof: If we assume that the density of \(S_T\) under \(\mathbb{Q}\) is assume to exist then we have \[ \begin{align} C_{h}(T,K) & =e^{-rT}\mathbb{E}_{\mathbb{Q}}[h(S_{T})] \\ & = e^{-rT}\int _{0}^\infty h(x)\cdot p(T,x) \, dx. \end{align} \] Plugging in the Breeden-Litzenberger Formula for the distribution \(p\) and we get \[ \begin{align} C_{h}(T,K) & =\int _{0}^\infty h(x)\cdot \frac{{\partial^2C}}{\partial K^2}(T,x) \, d \\ & =h(x) \frac{{\partial C}}{\partial K}(T,x)\big|_{x=0}^\infty +\int _{0}^\infty h'(x)\cdot \frac{{\partial C}}{\partial K}(T,x) \, dx. \end{align} \] Note that \(\frac{{\partial C}}{\partial K}(T,K)=-e^{-rT}\int _{K}^\infty p(T,x) \, dx\) from the proof of the Breeden-Litzenberger formula \[ \begin{align} C_{h}(T,K) & =h(0)e^{-rT}-\int _{0}^\infty h'(x)\cdot\frac{{\partial C}}{\partial K} \, dx \\ & = h(0)e^{-rT}-h'(x)C(T,x)\big|_{x=0}^\infty +\int _{0}^\infty h''(x)\cdot C(T,x) \, dx . \end{align} \] Finally, we note that \(C(T,\infty)=0\) since the strike is too high to ever be exercised, and \(C(T,0)=S_{0}\) since the call option with zero strike has payoff \(S_{T}\), the same as a forward contract, which gives the result \[ C_{h}(T,K)=h(0)e^{-rT}+h'(0)S_{0}+\int _{0}^\infty h''(x)\cdot C(T,x) \, dx \] ### Put-Call Parity
The Carr-Madan formula implies [[put-call-parity|put-call parity]]. To see this, take \(h(x)=(K-x)_{+}\) so that \(C_{h}(T,K)=P(T,K)\) is the price of a European put. Plug into Carr-Madan formula, \(h(0)=K\), \(h'(x)=-\mathbb{1}_{{x<K}}\) and \(h''(x)=\delta_{\{K\}}\) where \(h', h''\) are in the sense of weak derivative and \(h''\) is a Dirac point mass. The result \[ P(T,K)=Ke^{-rT}-S_{0}+C(T,K) \] is exactly put-call-parity.
0.2 Results
For any \(x,x_{0}>0\) the following equation holds \[ G(x)=G(x_{0})+G'(x_{0})+\int _{x_{0}}^\infty G''(K)(x-K)_{+} \, dK + \int _{0}^{x_{0}}G''(K)(K-x)_{+} \, dK. \]
\begin{proof} Splitting into cases \(x>x_{0}\), \(x=x_{0}\) and \(x<x_{0}\) we prove for \(x>x_{0}\) since the proof for other cases is similar. Starting with the RHS we have that \[
\begin{align}
& G(x_{0})+G'(x-x_{0})+\int _{x_{0}}^x(x-K)G''(K) \, dK \\
& = G(x_{0})+G'(x_{0})(x-x_{0})+x[G'(x)-G'(x_{0})]-\int _{x_{0}}^xKG''(K) \, dK \\
& = G(x_{0})-x_{0}G'(x_{0})+xG'(x)-xG'(x)+x_{0}G'(x_{0})+\int _{x_{0}}^xG'(K) \, dK \\
& =G(x_{0})+\int _{x_{0}}^x G'(K)\, dK \\
& =G(x).
\end{align}
\] \end{proof} The point here is not only introduce the European call but also introduce the European put. Applying the Lemma with \(x=S_{T}\), \(G(x)=\log x\) and \(x_{0}=S_{0}\) specified \[
\log S_{T}=\log S_{0}+\frac{1}{S_{0}}(S_{T}-S_{0})+\int _{S_{0}}^\infty \left( - \frac{1}{K^2} \right)(S_{T}-K)_{+} \, dK+\int _{0}^{S_{0}}\left( - \frac{1}{K^2} \right)(K-S_{T})_{+} \, dK,
\] and taking expectation under \(\mathbb{Q}\) and discounting both sides gives the Carr-Madan extension \[
\begin{align}
e^{-rT}\mathbb{E}_{\mathbb{Q}}\log S_{T} & =e^{-rT}\log S_{0}+e^{-rT} \frac{{\mathbb{E}_{\mathbb{Q}}S_{T}-S_{0}}}{S_{0}}+\int _{S_{0}}^\infty\left( - \frac{1}{K^2} \right)C(T,K) \, dK \\
& \quad\quad\quad+\int _{0}^{S_{0}}\left( - \frac{1}{K^2} \right)P(T,K) \, dK,
\end{align}
\] that connects the BS formula with the price of a European option with log payoff.
The importance of Carr-Madan is that t tells us that any European style option with twice differentiable payoff function can be replicated using European call and put.