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Looking for some clarification to the values of the parameters used in the Carr-Madan payoff decomposition formula.

$$f(S_T)=f(\kappa) + f'(\kappa) (S_T - \kappa) + \int_0^{\kappa} f''(K) (K-S_T)^+ \ d K + \int_{\kappa}^{\infty} f''(K) (S_T-K)^+ \ d K$$ which represents replication by investments in risk free bond, forward and put and call options.

what values would be used for:
1) $S_T$ ? the future price is not known at t=0, so what is used for $S_T$?
2) $\kappa$? is this the closest forward price to the strike price used for options?

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    $\begingroup$ As you've stated it, there are no bonds, forwards or call/put options. These instruments only appear when taking the risk-neutral expectation as illustrated in @Gordon's answer. From that answer it is also easy to see why picking $\kappa=E [S_T]$ (= forward price) is useful: the term $f'(\kappa) ... $ then disappears. You should look at this formula as a generic way of expanding any (sufficiently well behaved) function $f (x)$ of a variable $x$ (it so happens that here $x=S_T$) using an arbitrary parameter $\kappa$. Kind of a Taylor expansion but with quantities that have financial sense. $\endgroup$
    – Quantuple
    May 19, 2016 at 17:00
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    $\begingroup$ Yes that is correct. $\endgroup$
    – Quantuple
    May 19, 2016 at 20:10
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    $\begingroup$ The capital $K$ is a variable for integration, which you can change to anything else. For example, $\int_a^b f(x) dx = \int_a^b f(K)dK$. It has nothing to do with $\kappa$, which you can set to any positive value. $\endgroup$
    – Gordon
    May 19, 2016 at 20:20
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    $\begingroup$ You can use any symbol to represent a strike. We use $K$ for exactly the reason you have mentioned, that is, people like to use $K$ for strike. Some other sources may use $X$, for example, in John Hull's book. To compute the integrals on the right hand side, certain approximations has to be employed. for example, Gaussian or Hermite quadrature; see the book "numerical recipes in C". $\endgroup$
    – Gordon
    May 19, 2016 at 20:29
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    $\begingroup$ For replication, you have to use traded options. However, for computational of RHS, the strikes are selected based on our approximation (not based on what currently traded), and the prices of the options corresponding to our selected strikes are computed based on a given volatility surface, provided by data vendors, where interpolation or extrapolations are also needed. $\endgroup$
    – Gordon
    May 19, 2016 at 20:46

1 Answer 1

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This formula is used for replication of certain payoffs, for example, the log-payoff in Variance replication using options. The value of $\kappa$ can be set to any number, for example, $\kappa=E(S_T)$. This is a decomposition of the payoff, which is not a valuation of the payoff itself, and then further valuation is still needed. For example, based on the above decomposition, the value is given by \begin{align*} e^{-rT}E\big(f(S_T)\big) &= e^{-rT} E\bigg(f(\kappa) + f'(\kappa) (S_T - \kappa) + \int_0^{\kappa} f''(K) (K-S_T)^+ \ d K \\ & \qquad \qquad\qquad \qquad\qquad \qquad \qquad + \int_{\kappa}^{\infty} f''(K) (S_T-K)^+ \ d K \bigg)\\ &=e^{-rT}\bigg[f(\kappa) + f'(\kappa) \big(E(S_T) - \kappa\big) + \int_0^{\kappa} f''(K) E\big((K-S_T)^+\big) \ d K \\ & \qquad \qquad\qquad \qquad\qquad \qquad \qquad + \int_{\kappa}^{\infty} f''(K) E\big((S_T-K)^+\big) \ d K\bigg]. \end{align*}

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