Carr–Madan Replication of Payoffs with Calls and Puts
Summary
The document explains the Carr–Madan formula as a way to represent a sufficiently smooth payoff as a linear position in the underlying, a cash amount, and integrals of European put and call payoffs across strikes. The decomposition expands the payoff around a reference level using its first derivative and expresses curvature through the second derivative. This gives an intuition for why a broad strip of options can reproduce nonlinear payoffs.
The replies describe applications to valuing European claims when option prices are known across strikes, as well as variance-swap valuation and, approximately, construction of the VIX index. The formula is presented as a theoretical replication result, with smoothness conditions on the payoff and suitable strike coverage implicit in its use. The document does not work through a numerical pricing example or explain practical issues such as discrete strike grids, market frictions, or hedging error. It is an explanatory discussion rather than a complete implementation guide.
Key ideas
- A sufficiently smooth payoff can be decomposed into cash, underlying exposure, and option payoffs across strikes.
- The first derivative determines the underlying position around a chosen reference level.
- The second derivative weights the put and call components that capture payoff curvature.
- A complete set of same-maturity option prices can, in theory, support valuation of other European payoffs.
- The document also connects the decomposition to variance swaps and approximate VIX construction.
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# Carr-Madan Formula
# Carr-Madan Formula
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.
## Answer by Gordon (score 27)
https://quant.stackexchange.com/a/27628
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.
## Answer by user22171 (score 6)
https://quant.stackexchange.com/a/27632
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.
## Answer by Keith A. Lewis (score 2)
https://quant.stackexchange.com/a/48909
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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