Stage 2–3 notes — Heston CF and the Fourier pricer

The convention (memorize this, everything hangs on it)

heston_cf(u, T, p) is the characteristic function of the de-drifted log return XT = log(ST/S0) − (r−q)T, i.e. ψ(u) = E[e^{iuXT}]. Market inputs (S, r, q) never touch the model CF; the pricer reattaches them through the forward F = S·e^{(r−q)T}. Two free identities become test gates:

ψ(0) = 1 — any CF at 0.
ψ(−i) = E[e^{XT}] = E[ST]/F = 1 — the martingale property of the discounted stock. If your CF gets this wrong, your drift or your Riccati solution is wrong, full stop.

This matches El Euch–Rosenbaum's L(a, T) (CF of log(ST/S0) at r = 0), which is why the rough extension later is a CF swap and nothing else.

The Little Heston Trap (the discriminating interview question)

Heston's 1993 closed form contains a complex logarithm, log[(1 − g₁e^{dT}) / (1 − g₁)] with g₁ = (β+d)/(β−d). The log of a complex number is multivalued; software returns the principal branch, with a cut along the negative real axis. As maturity T grows, the argument of that log spirals in the complex plane and crosses the cut, so the principal-branch value jumps by 2πi. The CF then has spurious discontinuities in u, and Fourier prices are silently, plausibly wrong — the classic failure is long-dated options at certain parameter sets.

The fix (Albrecher–Mayer–Schoutens–Tistaert 2007) is pure algebra: rewrite with g₂ = 1/g₁ = (β−d)/(β+d) and e^{−dT}. Mathematically identical; but with the decaying exponential the log's argument stays near 1, away from the cut, so the principal branch is continuous for typical parameters. The lesson worth saying in an interview: a formula and its floating-point evaluation are different objects — choosing the branch-safe algebraic form is part of the implementation, not an optimization.

Guard in the tests: at T = 10, sweep u finely and assert the CF curve has no jumps.

Structural gates instead of memorized numbers

The CF tests pin identities a wrong implementation cannot fake:

ψ(0) = 1, ψ(−i) = 1 (normalization, martingale).
ψ(−u) = conj(ψ(u)), |ψ(u)| ≤ 1 — X is a real random variable.
ξ → 0 collapse: with ρ = 0 and tiny ξ, variance follows the ODE path v(t) = θ + (v0−θ)e^{−κt}, so X_T is exactly Gaussian with variance w = θT + (v0−θ)(1−e^{−κT})/κ and ψ(u) = exp(−(iu+u²)w/2). This exercises κ, θ, v0 in closed form — the same role the α→1 gate plays for rough Heston later. (Tested at ξ = 1e-3, not 0: the CF has a 0/0 at ξ = 0, and the κθ/ξ² prefactor makes tiny-ξ floating-point cancellation-prone.)

Literature price values (Heston 1993 / Albrecher et al. tables) still need to be pinned before Stage 6 — the rough spec requires "published-value tests" as its entry gate.

Gil-Pelaez pricing, and why the pricer is model-agnostic

With m = log(F/K):

P₂ = 1/2 + (1/π)∫₀^∞ Re[e^{ium} ψ(u)/(iu)] du — risk-neutral P(S_T > K). P₁ = same integral with ψ(u−i) — exercise probability under the stock numeraire. call = e^{−rT}(F·P₁ − K·P₂).

Read that against Black-Scholes: C = S e^{−qT} Φ(d₁) − K e^{−rT} Φ(d₂) is the special case where the two probabilities have closed forms. The complex shift u ↦ u − i is the d₂ → d₁ measure change, one abstraction level up. That's also why ψ must accept complex arguments.

The pricer takes ψ as a function argument. It contains zero Heston-specific code — this is the binding architectural constraint from the rough-Heston spec, honored now so Stage 7 is u -> rough_heston_cf(u, T, p) and done.

AD note

ForwardDiff.gradient flows through CF → quadrature → price (tested against central finite differences for all five parameters). Everything downstream of HestonParams is type-generic; Complex{Dual} arithmetic does the rest. When calibration arrives: differentiate prices, and map to implied-vol space via the implicit-function shortcut dσ/dC = 1/vega rather than pushing Duals through the iterative inverter.