PIVOT: Bridging Black-Scholes Implied-Volatility and Price Objectives via Differentiable J\"ackel Operator
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Quantitative Finance > Computational Finance
arXiv:2606.17065 (q-fin)
[Submitted on 4 Jun 2026]
Title:PIVOT: Bridging Black-Scholes Implied-Volatility and Price Objectives via Differentiable Jäckel Operator
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Abstract: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 the sensitivity 1/vega diverges as vega -> 0.
We close this gap with PIVOT, the 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/vega gradient, and low-vega rows are attenuated rather than silently regularized. On a single H100, a fused Triton kernel reaches 1.79e9 IV/s at machine precision (9.3e-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.
| Comments: | 30 pages, 17 figures, 12 tables |
| Subjects: | Computational Finance (q-fin.CP); Artificial Intelligence (cs.AI); Machine Learning (cs.LG) |
| MSC classes: | 91G60, 91G20, 68T07, 65K10 |
| ACM classes: | I.2.6; G.4; G.1.0 |
| Cite as: | arXiv:2606.17065 [q-fin.CP] |
| (or arXiv:2606.17065v1 [q-fin.CP] for this version) | |
| https://doi.org/10.48550/arXiv.2606.17065
arXiv-issued DOI via DataCite
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