Derivation — The Heston Characteristic Function

European option prices under the Heston model are computed by Fourier inversion of the characteristic function $\phi(u, \tau) = \mathbb{E}^{\mathbb{Q}}[e^{iu\log S_T} \mid \mathcal{F}_t]$. This derivation sub-note carries the algebra behind that characteristic function: the PDE it satisfies (derived from the Heston SDE by Itô's lemma), the Riccati ODE system that the exponential-affine ansatz reduces it to, and the closed-form solution with the correct complex-logarithm branch.

This note is heavy algebra. The practical output — the closed-form $\phi$ — is used directly in the Fourier pricing lesson. Readers who want only the formula without the derivation can proceed directly there.