Carr-Madan Formula

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\mathbf{R}\to\mathbf{R} 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.