PIVOT: bridging black-scholes implied-volatility and price objectives via differentiable Jäckel operator

Preprint

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 ∂σ/∂P = 1/V diverges as V → 0. We close this gap with PIVOT, Price–Implied-Volatility Objective Translator. 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/V gradient, and low-vega rows are attenuated rather than silently regularized. On a single H100, a fused Triton kernel reaches 1.79×109 IV/s at machine precision (9.3×10−14 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 HyperIVstyle 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.

Authors: Saqur, R; Kratsios, A; Horvath, B; Buehler, H

• Publication status: Published
• Peer review status: Not peer reviewed
• Publication date: 2026-06-04
• Language: English