Carr–Madan Payoff Decomposition for European Claims

Summary

The document explains how the Carr–Madan identity represents a sufficiently smooth terminal payoff using a fixed component, a linear exposure to the underlying, and a continuum of puts and calls across strikes. The terminal price is a random future value in the payoff identity; it is not an input that must be known at the initial date. The reference level κ can be chosen freely, with the expected terminal price offered as one possible choice.

The decomposition describes replication of a payoff, not its price by itself. To value the claim, take discounted expectations under the appropriate pricing framework: the linear component depends on the expected terminal price, and the option components depend on expected put and call payoffs at each strike. The answer illustrates this transformation algebraically but does not give a specific payoff, market data, or assumptions for computing the expectations. Practical replication and valuation therefore require additional modeling and market inputs.

Key ideas

• A smooth terminal payoff can be decomposed into a constant, a linear underlying exposure, and weighted put and call payoffs.
• The terminal underlying price remains a random variable when writing the payoff decomposition at the initial date.
• The reference level κ may be selected freely; the expected terminal price is one example.
• The decomposition identifies a replication portfolio but does not by itself determine the claim's value.
• Valuation requires discounted expectations of the underlying and option payoffs under a pricing framework.

Tags

# Carr-Madan european contingent claim payoff decomposition formula - application


# Carr-Madan european contingent claim payoff decomposition formula - application












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?

## Answer by Gordon (score 3, accepted)

https://quant.stackexchange.com/a/26127

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.