Characteristic function
Quantitative Finance glossary · Quantitative Finance
Definition
varphi_X(u) = mathbbE[e^iuX] — the Fourier transform of the density. Exists for every distribution (unlike the MGF) and determines it uniquely (Lévy's inversion theorem). For Lévy processes the Lévy-Khintchine representation gives varphi in closed form, which is why Fourier pricing (Carr-Madan FFT) is the workhorse for affine models (Heston, VG, CGMY, Merton jump-diffusion) where the density has no closed form but varphi does.
Characteristic function in plain English
A characteristic function is a mathematical 'fingerprint' of a probability distribution — a compact code that captures everything about how likely each outcome is, and no two different distributions share the same one. Why bother, instead of just using the probabilities directly? Because for many realistic models of jumpy, unpredictable prices, there's no neat formula for the probabilities themselves, yet this fingerprint does have a neat formula. So quants compute the fingerprint and then run it through a fast standard math transform to get option prices in a flash. It's like not having a photo of someone but holding their fingerprint — from that alone a good system can still identify them exactly.
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