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TL;DR

A transformation that decomposes a function into its frequency components: \( \hat{f}(\omega) = \int f(x) e^{-i\omega x} dx \). The inverse recovers the original function from its spectrum.

By Valenke Exam Prep Team·Last updated 2026-06-03

Fourier Transform

A transformation that decomposes a function into its frequency components: \( \hat{f}(\omega) = \int f(x) e^{-i\omega x} dx \). The inverse recovers the original function from its spectrum.

Why it matters for interviews

Fourier methods are used in option pricing (Carr-Madan formula), characteristic function-based pricing (Heston), and signal processing for time series. The FFT algorithm makes these computations fast.

Definition and Mathematical Foundation

A transformation that decomposes a function into its frequency components: \( \hat{f}(\omega) = \int f(x) e^{-i\omega x} dx \). The inverse recovers the original function from its spectrum.

Application in Quantitative Finance

Fourier methods are used in option pricing (Carr-Madan formula), characteristic function-based pricing (Heston), and signal processing for time series. The FFT algorithm makes these computations fast.

Related Terms

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Frequently Asked Questions

What is the Carr-Madan formula?▼
An option pricing formula that expresses the call price as a Fourier integral of a modified payoff function: \( C = \frac{e^{-\alpha k}}{\pi} \int_0^\infty e^{-i\omega k} \psi(\omega) d\omega \) where \( \psi \) depends on the characteristic function. The FFT computes prices for all strikes simultaneously.
What is the FFT and why does it matter?▼
The Fast Fourier Transform computes the discrete Fourier transform in \( O(n \log n) \) instead of \( O(n^2) \). This makes Fourier-based option pricing practical: computing implied volatilities for thousands of strikes in milliseconds.
How are Fourier methods used in time series analysis?▼
The power spectral density (Fourier transform of the autocorrelation) reveals periodicities and dominant frequencies in return data. This is used for seasonal pattern detection and noise filtering.