Carr-Madan Formula

The Carr-Madan formula is f(x) = f(a) + f'(a)(x - a) + \int_0^a (k - x)^+ f''(k)\,dk + \int_a^\infty (x - k)^+ f''(k)\,dk for f\colon[0,\infty)\to\mathbf{R} when f'' is integrable and x,a\ge0.

A European option paying f(x) when x is the price of the underlying at expiration can be replicated using cash, a forward, puts with strikes below a and calls with strikes above a.

The cash position is f(a), the forward with strike a position is f'(a), the put positions are {f''(k)\,dk} for 0 \le k < a, and the call positions are {f''(k)\,dk} for k > a.

By the fundamental theorem of calculus applied twice and Fubini’s theorem \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''(k)\,dk)\,dy \\ f(x) &= f(a) + f'(a)(x - a) + \int_a^x \int_a^y f''(k)\,dk\,dy \\ f(x) &= f(a) + f'(a)(x - a) + \int_a^x \int_k^x f''(k)\,dy\,dk \\ f(x) &= f(a) + f'(a)(x - a) + \int_a^x (x - k) f''(k)\,dk \\ \end{aligned} If x > a then (x - k)^+ = 0 for k > x so \int_a^x (x - k) f''(k)\,dk = \int_a^\infty\ (x - k)+ f''(k)\,dk If 0\le x < a then (k - x)^+ = 0 for k < x so \begin{aligned} \int_a^x (x - k) f''(k)\,dk &= \int_x^a (k - x) f''(k)\,dk \\ &= \int_0^a (k - x)^+ f''(k)\,dk. \end{aligned}

If f is piecewise linear and continuous then f' is piecewise constant and f'' is a linear combination of delta functions. If f(x_i) = y_i, 0\le i\le n then {f'(x) = (y_{i+1} - y_i)/(x_{i + 1} - x_i) = m_i} is the right derivative on the interval [x_i, x_{i+1}), and the left derivative on (x_i, x_{i+1}], 0\le i < n. In both cases f'(x) = m_0 if x < x_0 and f'(x) = m_{n-1} if x > x_n. The second derivative is {f'' = \sum_{i=1}^{n-1} (m_i - m_{i-1}) \delta_{x_i}} where \delta_x is a point mass at x.

For example, if (x_0,y_0) = (0, 0), (x_1,y_1) = (1,0), (x_2,y_2) = (2,1) (x_3,x_3) = (3,0) and (x_4,y_4) = (4,0) then we have a butterfly spread. In this case m_0 = 0, m_1 = 1, m_2 = -1 and m_3 = 0. Taking a = 0 we have f(a) = 0 and f'(a) = 0 so there is no cash or forward position. The Carr-Madan formula indicates to buy m_1 - m_0 = 1 call at strike 1, m_2 - m_1 = -1 - 1 = -2 calls at strike 2, and m_3 - m_2 = 1 call at strike 3.

Exercise. Find the butterfly spread for a = 2.

Hint: Split (x_2,y_2) into (2 - \epsilon, 1 - \epsilon) and (2 + \epsilon, 1 - \epsilon) so f'(2) = 0.