cm.md

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title: Carr-Madan Formula author: Keith A. Lewis institute: KALX, LLC classoption: fleqn fleqn: true abstract: Payoffs can be replicated with cash, a forward, and a portfolio of puts, and calls ...

\newcommand\RR{\mathbf{R}}

The Carr-Madan formula is $$ f(x) = f(a) + f'(a)(x - a) + \int_{-\infty}^a (k - x)^+ f''(k),dk + \int_a^\infty (x - k)^+ f''(k),dk, $$ if $f\colon\RR\to\RR$ is twice differentiable. The import is any sufficiently smooth payoff can be replicated with cash, a forward, and a portfolio of puts and calls.

This follows from applying the fundamental theorem of calculus twice $$ \begin{aligned} f(x) &= f(a) + \int_a^x f'(y),dy \ f(x) &= f(a) + \int_a^x (f'(a) + \int_a^y f''(z),dz),dy \ f(x) &= f(a) + f'(a)(x - a) + \int_a^x \int_a^y f''(z),dz,dy \ f(x) &= f(a) + f'(a)(x - a) + \int_a^x \int_z^x f''(z),dy,dz \ f(x) &= f(a) + f'(a)(x - a) + \int_a^x (x - z) f''(z),dz \ \end{aligned} $$

If $x > a$ then $\int_a^x (x - z) f''(z),dz = \int_a^\infty (x - z)^+ f''(z),dz$.

If $x < a$ then $\int_a^x (x - z) f''(z),dz = -\int_x^a (x - z) f''(z),dz = \int_x^a (z - x) f''(z),dz = \int_{-\infty}^a (z - x)^+ f''(z),dz$

If $x > a$ then $(z - x)^+ = 0$ for $z < a$ and if $x < a$ then $(x - z)^+ = 0$ for $z > a$, hence $$ \int_a^x (z - x) f''(z),dz = \int_{-\infty}^a (z - x)^+ f''(z),dz + \int_a^\infty (x - z)^+ f''(z),dz. $$

If $f$ is piecewise linear and continuous then $f'$ is piecewise constant and $f''$ is a linear combination of delta functions.