Improving Swaption Calibration in Factor HJM Stochastic Volatility Models: A First-Order Correction to Frozen Swap-Rate Loadings

The factor HJM stochastic volatility model introduced by Sepp and Rakhmonov (2025) obtains tractable swaption pricing by freezing the nonlinear swap-rate loading along a deterministic expected-state path. This removes the dependence of conditional swap-rate variance on the current yield-curve state. We introduce a first-order Taylor correction to the loading that adds no calibration parameters. Conditional on retaining the frozen annuity-measure drift, we show that the first variation of the swap-rate transform is affine in the centered rate states and reduces to one-dimensional equations in volatility. For quadratic-drift lognormal stochastic volatility, these equations yield a finite-dimensional ODE representation and a direct log-volatility formulation. Calibrations to independently generated nonlinear-model prices show substantially lower stochastic volatility parameter bias and held-out pricing error, with only modest changes in local calibration identifiability.