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 – Kiwiakos 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. – user3223190 Jun 15 '16 at 15:20

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.

• So nice and I like it – user16651 Jun 15 '16 at 17:39
• Very nice proof! I only knew the proof that makes use of Dirac Delta functions. – Cavents Jun 16 '16 at 17:32
• Thanks @Siron. The delta function is nice, but I like to derive the formula using a rudimentary approach. – Gordon 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? – starovoitovs Mar 28 at 19:23
• You need models for path dependent or basket options. – Gordon Apr 4 at 0:28

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. – Gordon Jun 15 '16 at 17:20
• One could understand the right hand side in the sense of distributions in that case – user22171 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 $1_{S\in \mathcal{A}}$. – Daneel Olivaw Mar 27 at 13:44