The fit is bought from the curve, not the volatilities.

The book is one instrument: a USD at-the-money cap struck at 2.712% 2.712% single ATM cap strike across all caplets cap.csv CapStrike column on $10M $10M cap notional cap.csv Notional column of notional, from a Bloomberg SWPM cashflow table dated April 16, 2019 — 119 119 USD ATM cap cashflow rows, Bloomberg SWPM, as of 2019-04-16 cap.csv, archived repo rows, 118 118 caplets entering the corrected calibration (row 0 has no preceding discount, matching the archived Black loop) export_web_json.py stage 3 priced caplets, payments out to 30.4y 30.4y longest caplet payment time (30/360) cap.csv T_i column . Two curves carry the result. The discount curve fixes f(0,0) = 2.53% 2.53% instantaneous forward rate at t=0 from the discount curve; the market-consistent value of r(0) export_web_json.py stage 1 , the market-consistent value of the model's initial short rate; the implied-volatility term structure rises off a low short end, peaks through the belly, and steps down once near 15.5 years — the shape a constant-σ one-factor model has to hold.

The archived pipeline — graduate coursework, 2024; the committed notebook reproduces bit-for-bit — built its calibration target with a units slip: strike and volatility, already decimals, were divided by 100 again. Priced at a strike near zero, the target collapses to pure forward value — double every input volatility and it moves by <0.0001% <0.0001% largest move in the archived calibration target when every input vol is DOUBLED -- the target carried effectively no volatility information export_web_json.py stage 2 . A separate floating-point truncation in the accrual grid halved the accrual on 49 49 caps whose arange sub-division silently dropped half the accrual period -- the sawtooth in the archived price plot export_web_json.py stage 1 of the 119 119 USD ATM cap cashflow rows, Bloomberg SWPM, as of 2019-04-16 cap.csv, archived repo caplets: the sawtooth in the figure below, drawn hatched.

That explains the archived numbers. The report's σ = 1.75% 1.75% parameters printed in the archived course report archived course report (2024), optimization section at an SSE of 8.97% 8.97% SSE printed in the archived course report; this pipeline evaluates the report parameters at 0.0890 on the archived target with tau/8 subdivision -- same problem, better optimum than the committed notebook run archived course report (2024), optimization section and the notebook's a = 0.812 0.812 archived notebook differential-evolution run (2024) notebook cell 65, commit 025f598 , σ = 9.13% 9.13% archived notebook run; implausibly large for a Gaussian short rate -- a symptom of the corrupted target notebook cell 65, commit 025f598 — implausibly large for a Gaussian short rate — are both genuine optima of the same corrupted target, whose flattened objective holds a long trough of near-equivalent minima. And the 22.2% 22.2% archived IV-error metric; equals the gap between the inverter's own starting guess and the mean market vol, exactly -- the inversion never left its starting point because the double-divided strike flattened the objective notebook cell 70 + stage 2 implied-vol error it reports is not a fit statistic: with the objective flat, the price inverter never leaves its starting guess, and the figure equals that guess measured against the market mean, exactly.

On a corrected target — strike restored, accrual grid repaired — the model calibrates, with r(0) free, to a = 0.150 0.150 mean-reversion speed, corrected-target recalibration, free r(0) export_web_json.py stage 3 , σ = 145 bp 145 bp Gaussian short-rate volatility (absolute, per year), corrected-target recalibration, free r(0) export_web_json.py stage 3 , r(0) = 0.17% 0.17% free r(0) as the archived setup treats it; compare fm00 -- the gap is the curve tilt the optimiser buys itself export_web_json.py stage 3 , and prices the caplet book to 1.4 vol pts 1.4 vol pts mean absolute model-vs-market implied-vol gap at the recalibrated optimum export_web_json.py stage 4 of implied volatility at a 12.1% 12.1% RMS relative cap-pricing error at the recalibrated optimum export_web_json.py stage 4 price RMSE. The accuracy has one legible source: r(0) sits well below the market-consistent f(0,0) = 2.53% 2.53% instantaneous forward rate at t=0 from the discount curve; the market-consistent value of r(0) export_web_json.py stage 1 , and the model's discount factor scales with exp(B(0,T)·(f(0,0) − r(0))) — a low r(0) lifts the model's long-maturity discounts above the market's, as the playground's “curve tilt at 30y” readout reports. Part of the fit re-prices the curve rather than describing the volatilities.

Pin r(0) to f(0,0) and the same model class collapses: a runs to its lower bound (0.000 0.000 mean-reversion speed with r(0) pinned to f(0,0) export_web_json.py stage 3 ), σ falls to 33 bp 33 bp sigma when r(0) is pinned to f(0,0) (market-consistent HW) export_web_json.py stage 3 , and the fit widens to 11.6 vol pts 11.6 vol pts IV MAE under the pinned-r0 restriction export_web_json.py stage 4 at a 56.5% 56.5% RMS relative pricing error under the pinned-r0 restriction export_web_json.py stage 4 price RMSE, almost every model volatility below the market's. The reason is structural. Market caplet prices imply a normal-volatility term structure that rises with maturity; a constant-σ one-factor model with a ≥ 0 can only produce one that is flat or falling. No parameter search fixes the wrong shape — the miss prices the next model class, a time-dependent σ(t) or a second factor as in G2++.

The closed forms are verified against Monte Carlo, and the estimators themselves are part of the result. The archived T-measure estimator for zero-coupon-bond puts omits the M^T drift — the T-forward change-of-measure term — and sits +0.08% +0.08% mean MC-vs-closed-form gap of the archived T-measure estimator (missing M^T drift); small but systematically positive across every maturity export_web_json.py stage 6 above the closed-form price: small, but positive at every maturity, a signature rather than sampling noise. With the drift restored it closes to within 2.6 SE 2.6 SE worst |MC - closed form| of the drift-corrected estimator, in MC standard errors export_web_json.py stage 6 of Monte-Carlo error. A second estimator, defined in the notebook but never invoked, draws the short rate independently at every step of its discount integral and so loses the convexity term: off by -14.8% -14.8% worst ZCB gap of the archived estimator (independent marginal draws kill the convexity term), at the committed parameters, 30y export_web_json.py stage 6 at the long end, where exact-transition path simulation stays within +0.47% +0.47% worst ZCB gap of the exact-transition path estimator (within MC error) export_web_json.py stage 6 .