# Inverse Learning of Latent Risk-Neutral Densities from Irregular Option Quotes

arXiv: https://arxiv.org/abs/2607.27188 · HTML: https://mlsanity.com/paper/2607.27188/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 40 references that are in the map.
- First posted on arXiv 29 Jul 2026.

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Latent Neural Operator for Solving Forward and Inverse PDE Problems](https://mlsanity.com/paper/2406.03923/) (2024)
- [Neural networks for option pricing and hedging: a literature review](https://mlsanity.com/paper/1911.05620/) (2019)
- [Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning](https://mlsanity.com/paper/2402.15734/) (2024)
- [Deep Smoothing of the Implied Volatility Surface](https://mlsanity.com/paper/1906.05065/) (2019)
- [Learning Physical Constraints with Neural Projections](https://mlsanity.com/paper/2006.12745/) (2020)
- [Invert to Learn to Invert](https://mlsanity.com/paper/1911.10914/) (2019)
- [One Operator to Rule Them All? On Boundary-Indexed Operator Families in Neural PDE Solvers](https://mlsanity.com/paper/2603.01406/) (2026)
- [Finance-Informed Neural Network: Learning the Geometry of Option Pricing](https://mlsanity.com/paper/2412.12213/) (2024)

## Abstract (arXiv)

Accurate option prices do not imply accurate recovery of the latent risk-neutral density. We study this distinction with two complementary benchmarks. A controlled benchmark exposes simulator-truth densities for latent evaluation, while a chronological NIFTY benchmark tests only held-out market prices. A two-component lognormal mixture has the lowest aggregate price, L1, Wasserstein, and fixed-tail errors on the synthetic benchmark. Learned operators retain narrower strengths: DeepONet reduces 1% quantile and variance error by 39.0% and 34.6% relative to the mixture, and a quote transformer reduces L1 by 16.4% on the structurally misspecified Merton family. A numerical conditioning analysis explains why these rankings can differ: after enforcing mass and forward constraints, 95 of 126 pricing directions are numerically null, and two densities separated by L1 = 0.061 produce identical prices on the covered strikes. On 524 held-out NIFTY calls, validation-selected test-time adaptation reduces DeepONet RMSE by 28.3%, but per-expiry mixture and SVI fits remain much more accurate. The evidence supports target-dependent inductive bias, not a universal winner.