A 0.3771 normal-vol basis-point smile fit can conceal a rate-vol correlation parameter that is wrong by a factor of 2.5. Brongers demonstrates this in the Moderate synthetic regime of the Sepp-Rakhmonov factor HJM stochastic volatility model. The true correlated volatility loading is -0.0500. Calibration with the frozen swap-rate loading returns -0.0199, even though the largest standard error among the reference quotes is 0.0061 bp. The discrepancy sits well above simulation noise.
The error varies sharply by regime. On the flat curve in Moderate, the true implied instantaneous rate-volatility correlation is -0.267644, while the fitted value is -0.121730. In Negative skew, it shifts from -0.554700 to -0.505109, a shrink of about 9%. Brongers builds a first-order correction for this distortion, and it works on synthetic data.
The loading being frozen
Sepp and Rakhmonov add a scalar stochastic volatility driver, correlated with the curve factors, to the factor Heath-Jarrow-Morton curve model. Their tractable swaption pricing relies on freezing the swap rate's diffusion loading under the annuity measure. They denote that loading by Lambda.
Lambda depends nonlinearly on the yield-curve state because bond prices appear through ratios and weighted sums of exponential-affine terms. The pricing approximation evaluates it along a deterministic expected-state path. The annuity-measure drift coefficient is treated similarly, and Brongers notes that his annuity-measure drift is also frozen unless stated otherwise.
Freezing Lambda removes the current yield-curve state from the swap-rate diffusion loading and, consequently, from the conditional swap-rate variance. The calibration can respond by moving the stochastic volatility parameters, the remaining free quantities able to bend the smile. Brongers states the implication directly: a good fit to option prices does not guarantee recovery of the dynamics that generated them.
His remedy expands the loading to first order around the same centering path. It adds no free parameters. The initial curve, swap cash-flow dates, factor basis and existing parameters determine the Jacobian.
Why the full linearization resists an affine solution
Squaring the state-linear loading inside the swap-rate variance produces terms in sigma-squared times xi-squared, sigma-squared times xi eta and sigma-squared times eta-squared. Brongers therefore says the complete state-linear model is not affine in the usual sense of Duffie, Filipović and Schachermayer. The exponential-affine transform available for an affine diffusion cannot be used.
The paper instead works with the first variation. A scalar gamma multiplies only the state-dependent part of the loading, and Brongers differentiates the transform at gamma equals zero. The resulting first variation is affine in the centered rate states. Since the annuity drift remains frozen, the state process has no gamma dependence, leaving the swap-rate increment pathwise affine in gamma.
This yields one-dimensional parabolic equations in the volatility state. A correction to a two-state-plus-volatility problem consequently requires three one-dimensional solves. Proposition 3.1 and Equation (3.25) contain the result.
Brongers gives two implementations. The first couples the correction to E1, the paper's quadratic expansion of the frozen-loading log-transform in shifted volatility v equals sigma minus theta. At degree six, this creates a finite block-triangular ODE system with 21 extra complex coefficients beyond E1's three. The second solves the same one-dimensional equations directly in log volatility, either through Crank-Nicolson finite differences or Chebyshev-Lobatto collocation. Brongers describes this route as removing the finite polynomial approximation while retaining the frozen annuity drift, linearized loading and first-order truncation. The stress results depend on that separation.
Price fit, holdout and parameter recovery
The calibration panel contains three payer smiles: 1Yx5Y, 3Yx7Y and 5Yx10Y. Each has five fit strikes and four interlaced holdout strikes, giving 15 fitted quotes and 12 held out. Reference prices come from the fully nonlinear loading and nonlinear annuity drift, independently of either calibration model. The setup uses a flat 2.5% initial curve and one-factor scalar Cheyette.
In Moderate, held-out RMSE drops from 0.3169 to 0.0118 bp, a factor of 26.8. Negative skew falls from 0.2622 to 0.0323, a factor of 8.1. The parameter result is more relevant for anyone using calibrated beta as a hedge input. Across time buckets, the largest absolute beta bias in Moderate declines from 0.03005 to 0.00006. For the two-bucket second interval, the true value is -0.2200, the frozen estimate -0.1566 and the corrected estimate -0.2136.
Brongers then reprices both fitted parameter vectors under the full nonlinear dynamics. In Moderate, parameters obtained from frozen calibration miss the nonlinear surface by 0.8688 bp fit RMSE and 0.7297 bp held out. Corrected parameters reduce those figures to 0.0170 and 0.0178. The largest quote-level Monte Carlo standard error in this comparison is 0.0070 bp. Brongers also reports that the paired confidence intervals for the RMSE reductions exclude zero. The calibration shifts change the nonlinear price map itself.
The frozen annuity drift has a much smaller effect. Across the five parameter regimes Brongers studies, it contributes between 0.02% and 2.09% of the state-loading price effect. In every case, it remains below the loading-linearization error. Both pricers leave that drift frozen, with its linearization deferred to future work.
The six-parameter Two buckets result warrants restraint. Held-out RMSE moves from 0.1102 to 0.0658, only a factor of 1.7. Its frozen fit also began from a better level, 0.1236 bp versus 0.3771 bp in Moderate, so the correction has less room to help. My own conjecture, which the paper does not test, is that the benefit contracts as the parameterization expands: 26.8x and 8.1x in the one-bucket cases, compared with 1.7x for six parameters. The multi-factor extension in the paper is theoretical. These results therefore leave open the behavior of a Nelson-Siegel specification with a term structure of parameters.
Conditioning changes little. Brongers gives weighted calibration Jacobian condition numbers of 326.4 frozen and 271.4 corrected in Moderate, 61.1 and 59.9 in Negative skew, then 344.2 and 315.8 in Two buckets. Table 8 reports corrected-to-frozen ratios for the smallest singular value in the QMC replicates of 1.204, 1.020 and 1.091. The paper describes the 20.4% Moderate increase in prose and calls the other two changes smaller.
Cosines between the frozen and corrected weakest right-singular vectors exceed 0.999 throughout. Those directions are still dominated by log epsilon and, for Two buckets, contrasts between the epsilon buckets. The improvement in parameter recovery comes mainly from lower approximation-induced bias, with little change to the weakly identified directions in the calibration.
Where the finite E1 representation breaks
Wide volatility ranges defeat the finite E1 monomial projection in v equals sigma minus theta. In the E1 transform stress, simulated terminal-volatility quantiles span 0.16 at the 0.1% level to 3.68 at the 99.9% level, with theta equal to 1. Brongers calls a monomial expansion around v equals zero poorly conditioned over that range.
At Fourier frequency 5, degree six differs from the exact one-dimensional PDE by 105.4% of the PDE correction. The discrepancy falls to 61.0%, 19.1% and 1.2% at 15, 30 and 60. Higher degree can make matters worse. At frequency 15, the error runs 1.66e-3, 1.31e-3, 5.04e-3, 2.75e-2 and 4.21e-1 as N moves through 3, 4, 6, 8 and 10, as recorded in Equation (5.2).
The separate Broad-volatility calibration stress is harsher. Adding the finite E1 correction raises held-out RMSE from 1.7466 to 7.5129 bp. Refining degree and time changes individual prices by as much as 2844.85 bp.
Brongers attributes this failure to the finite E1 representation and carefully separates it from the first-variation result. His words: the affine first-variation reduction within the auxiliary state-linear family remains applicable and its one-dimensional equations can be solved accurately, while the finite E1 representation is numerically unreliable. The log-volatility PDE and Chebyshev collocation approaches solve the same one-dimensional system and remain stable. Final quadrature refinement changes corrected normal-volatility values by no more than 0.000351 bp.
With that pricer, the State-dependence stress reaches a fit RMSE of 0.1155 bp and a holdout RMSE of 0.1043 bp. The Broad-volatility stress reaches 0.0882 and 0.0809 bp. Both solutions are interior and insensitive to initialization.
Yet both miss the accuracy threshold Brongers set before examining the stress cases. The maximum quote-level total first-order residual is 0.0981 bp in the State-dependence stress and 0.0738 bp in the Broad-volatility stress, versus a 0.05 bp tolerance. For State-dependence, the error decomposition assigns 3.77% of the state-loading price effect to the loading Taylor term, 2.09% to frozen drift, 1.31% to first-order truncation and 1.35% to the total residual. Loading linearization is the dominant error. Brongers gives the warning plainly: numerical convergence of these equations can therefore persist after higher-order loading effects have become material.
The correction has a bounded domain. Brongers traces its edge along one interpolation path from Negative skew to the State-dependence stress. The maximum quote-level first-order residual is 0.0486 plus or minus 0.0051 bp at u equals 0.55, rising to 0.0529 plus or minus 0.0055 bp at u equals 0.60. He writes that this interpolation identifies a transition along one parameter path and does not define a validity region across the full parameter space.
Synthetic evidence only
The 12 holdout strikes are interlaced with the fitted strikes inside the same three smiles. The paper presents them as a within-smile test of interpolation at strikes excluded from the calibration objective. There are no swaption cubes, no historical dates and no market data of any kind. The evidence comes from one flat curve plus a single upward-sloping case.
The upward-sloping case repeats the result. Held-out RMSE falls from 0.3174 to 0.0104 bp, while implied correlation moves from a true -0.268 to -0.123 frozen and -0.269 corrected. Brongers states the limitation himself, writing that the study establishes parameter-recovery and pricing improvements in these controlled settings, not empirical performance on observed swaption markets, and naming historical swaption cubes as the future-work test.
We could not test any of this ourselves. The mechanism requires swaption prices and swap schedules alongside discount curves, and we have no interest-rate instrument data. Coupon-bond discounting creates the nonlinearity under correction, leaving nothing transferable to listed equity options.
Evidence that could change the verdict
Brongers identifies the test I would want: repeat the comparison on historical swaption cubes using a multi-factor Nelson-Siegel specification with a term structure of parameters, then examine whether calibrated beta paths become more stable across dates. The attenuation in Two buckets is the warning, 1.7x versus 26.8x and 8.1x. The paper also reports no runtime figures comparing the corrected pricer with frozen E1, a relevant omission for anyone recalibrating a cube every morning.
Anyone running the frozen-loading factor HJM stochastic volatility pricer should be wary of treating calibrated rate-vol correlation as a statement about the world. On the non-flat curve in the paper's Moderate case, it came out at -0.123 against a true -0.268, despite a frozen fit RMSE of 0.3779 bp.