Fourier Transform Sign Conventions in Option Pricing

Summary

The document asks why the Fourier transform in Carr and Madan’s option-pricing paper uses a positive sign in the exponential, while a commonly taught transform formula uses a negative sign. The answer explains that Fourier transforms have multiple sign conventions across fields. In this case, the convention used in the paper is described as the probabilist’s convention and is aligned with the definition of a characteristic function.

This distinction matters when implementing transform-based pricing formulas: the convention, variable definitions, and inverse transform must be kept consistent. The document offers no derivation, pricing example, or evidence beyond the explanation in the answer, and it does not spell out the implications for a specific implementation. It is therefore a concise clarification of notation rather than a full account of the Carr–Madan method or its numerical requirements.

Key ideas

• Fourier transform definitions can differ in the sign used in the exponential.
• The convention described in Carr and Madan’s paper uses a positive sign and matches a characteristic-function convention.
• Implementations must keep transform and inverse-transform conventions consistent.
• The document clarifies notation but does not provide a pricing derivation or implementation guidance.

Tags

# Carr and Madan Fourier Transform


# Carr and Madan Fourier Transform












I am bit confused by Carr and Madan's paper. In it the authors write that the Fourier transform $ c_T(k)$ is defined by

\begin{align} \psi_T(v) = \int_{ - \infty}^{\infty} e^{ivk} c_T(k)dk \end{align}

Yet traditionally it is know that the Fourier Transform formula is \begin{align} X( \omega) = \int_{- \infty}^{\infty} x(t) e^{ - i \omega t} dt \end{align}

The negative being the difference between the two.

## Answer by LocalVolatility (score 1)

https://quant.stackexchange.com/a/33253

Different fields use different conventions to define the Fourier transform. The one in Carr and Madan is often referred to as the "probabilist's Fourier transform". It coincides with the definition of the characteristic function.

See https://www.johndcook.com/blog/fourier-theorems/ for a nice overview and the link between the different definitions.

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