PIVOT: Bridging Price and Implied-Volatility Objectives via Differentiable Black–Scholes Inversion

Modern option-learning systems operate in two coordinates: *price space*, where markets quote and no-arbitrage constraints are most naturally enforced, and *implied volatility (IV) space*, where volatility surfaces are smoothed, regularized, and evaluated. The bottleneck is interface, not approximation: Jäckel's seminal *Let's Be Rational* (LBR) solver already inverts the Black–Scholes price to machine precision efficiently. What is missing is a differentiable layer that preserves LBR in the forward pass and avoids backpropagating through its branch logic. Such a layer must also confront the unavoidable singularity of the inverse map in the low-vega regime, where $\partial \sigma /\partial P=1/\mathcal{V}$ diverges as $\mathcal{V}\to 0$. We close this gap with PIVOT, **P**rice–**I**mplied-**V**olatility **O**bjective **T**ranslator. PIVOT keeps the LBR forward pass intact and supplies the backward pass by implicit differentiation through the smooth Black–Scholes/Black-76 price map, with an explicit gating contract: invalid domains return NaN, well-conditioned rows receive the exact $1/\mathcal{V}$ gradient, and low-vega rows are attenuated rather than silently regularized. On a single H100, a fused Triton kernel reaches $1.79\times 10^9$ IV/s at machine precision ($9.3\times 10^{-14}$ max relative error vs. the reference C solver); end-to-end label generation sustains 48.9M/s on synthetic chains and 16.6M/s on SPX OptionMetrics. In a HyperIV-style one-day reproduction on SPX, PIVOT-augmented objectives Pareto-dominate the baselines, reducing held-out price MAE by up to 43.4% and the strongest three-seed gated objective improving price MAE by 38.8% and IV MAE by 21.3% jointly; cross-asset results on RUT, VIX, and NDX show directional price-MAE gains of 40.1%, 24.2%, and 16.7%, while an ungated IV-roundtrip control collapses to a degenerate near-zero surface, confirming the gate as a correctness contract rather than a tuning knob. Code: https://anonymous.4open.science/r/pivot-anon.