Really new to financial Maths. I am currently having problems with the Carr-Madan Formula.

$$f(S_T)=f(F_t) + f'(F_t) (S_T - F_t) + \int_0^{F_t} f''(K) (K-S_T)^+ \ d K + \int_{F_t}^{\infty} f''(K) (S_T-K)^+ \ d K$$

I am struggling to understand what it is used for and I can't seem to find any good articles to explain what is going on. I was wondering could someone recommend any readings, given that I quite new to this.

• It's Carr-Madan not Mahan, and it's used for replicating and hedging exotics Jun 15 '16 at 15:03
• I appreciate your input, however I was just wondering do you know any articles where the carr-mahan formula is explained in laymans terms. Like I said I am really new to it. Jun 15 '16 at 15:20
• This is nothing special and more than the Taylor expansion formula.
– Hans
Jan 7 '20 at 2:30

For a sufficiently smooth function $f$, positive constant $a$, and $x>0$, Note that, \begin{align*} f(x) -f(a) &= \int_a^{x} f'(v) dv \\ &= \int_a^{x} \big[f'(v) -f'(a) + f'(a) \big] dv \\ &= f'(a) (x-a) + \int_a^{x}\!\! \int_a^v f''(u)du dv\\ &= f'(a) (x-a) + \int_a^{x}\!\! \int_u^{x} f''(u)dv du\\ &= f'(a) (x-a) + \int_a^{x}f''(u)(x-u)du. \end{align*} Then, \begin{align*} f(x) &= f(a) + f'(a) (x-a) + \int_a^{x}(x-u)f''(u)du \\ &= f(a) + f'(a) (x-a) + \int_a^{x}\big(\pmb{1}_{a \leq x} + \pmb{1}_{a > x} \big)(x-u)f''(u)du \\ &= f(a) + f'(a) (x-a) + \int_a^{x} \pmb{1}_{a \leq x}\,(x-u)f''(u)du + \int_x^{a} \pmb{1}_{a > x}\,(u-x)f''(u)du \\ &= f(a) + f'(a) (x-a) + \int_a^{x} \pmb{1}_{a \leq x}\,(x-u)^+f''(u)du + \int_x^{a} \pmb{1}_{a > x}\,(u-x)^+f''(u)du \\ &= f(a) + f'(a) (x-a) + \int_a^{\infty} \pmb{1}_{a \leq x}\, (x-u)^+f''(u)du + \int_{0}^a \pmb{1}_{a \geq x}\, (u - x)^+f''(u)du \\ &=f(a) + f'(a) (x-a)\\ &\qquad + \int_a^{\infty}(1- \pmb{1}_{x < a})\, (x-u)^+f''(u)du + \int_{0}^a (1-\pmb{1}_{x>a})\, (u - x)^+f''(u)du \\ &= f(a) + f'(a) (x-a) + \int_a^{\infty}(x-u)^+f''(u)du + \int_{0}^a(u - x)^+f''(u)du. \end{align*} This formula is used in the valuation of a variance swap, and, as an approximation, the constructuion of the VIX index; see https://www.cboe.com/micro/vix/vixwhite.pdf.

• Thanks @Siron. The delta function is nice, but I like to derive the formula using a rudimentary approach. Jun 16 '16 at 18:39
• Since vanilla option prices are readily observable, why doesn't this formula solve option pricing, period. Any European claim can be priced this way, doesn't it? Then why bother with modeling? Mar 28 '19 at 19:23
• It is the Taylor expansion. However, to reach the final form is not trivial. Moreover, the Taylor is not related to options. I will think it worth a new name. Jan 7 '20 at 3:07
• Does the this proof of Carr-Madan formula work also for every $x\in\mathbb{R}$ and every $a\in\mathbb{R}$?
– B_B
Aug 30 '20 at 9:58
• @B_B: By going through the proof, you will notice that the proof works for all $x, a \in \mathbb{R}$. Aug 31 '20 at 12:12

The main interest of the formula is that it allows you, at least theoretically, to replicate any European option with payoff $f(\cdot)$ using only Call and Put options. As simple examples, consider $f(S)=S$ and $f(S)=(S-K)^+$.

The formula also implies that knowing all Puts and Calls for all strikes for a given maturity gives you the price of any European option with the same maturity.

• we need $f$ to be smooth, however, $f(s)=(s-k)^+$ is not. Jun 15 '16 at 17:20
• One could understand the right hand side in the sense of distributions in that case Jun 15 '16 at 17:22
• Non-smooth payoffs can often be written $f(S)1_{S\in \mathcal{A}}$ where $\mathcal{A}$ is some set thus the formula can be applied to $f(S)$ and the whole result can then be multiplied by $1_{S\in \mathcal{A}}$. Mar 27 '19 at 13:44

If $$f\colon\mathbf{R}\to\mathbf{R}$$ has a piecewise continuous second derivatve, then \begin{align*} f(x) = f(a) + f'(a)(x-a) + \int_{-\infty}^a (k - x)^+ f''(k)\,dk + \int_a^\infty (x - k)^+ f''(k)\,dk. \end{align*} Note this formula holds for $$x = a$$. Taking a derivative with respect to $$x$$ yields \begin{align*} f'(x) &= f'(a) + \int_{-\infty}^a -1(x \le k) f''(k)\,dk + \int_a^\infty 1(x \ge k) f''(k)\,dk\\ &= f'(a) - \int_{\min\{x, a\}}^a f''(k)\,dk + \int_a^{\max\{x,a\}} f''(k)\,dk\\ \end{align*} Note this formula holds for $$x = a$$. Taking a derivative with respect to $$x$$ yields \begin{align*} f''(x) = f''(x)1(x < a) + f''(x)1(x > a)(k) \end{align*} for $$x\not= a$$. Note the left and right limits as $$x\to a$$ equal $$f''(a)$$. This proves the original formula is valid.