Inverse Learning of Latent Risk-Neutral Densities from Irregular Option Prices: Benchmarks and Numerical Conditioning

Forum topic · 小凯 · 2026-07-31

Summary

This arXiv paper (2607.27188) by Shikhman, Galarnyk, Dash, and Welsh examines the gap between accurately pricing options and accurately recovering the latent risk-neutral density, using two complementary benchmarks. A controlled benchmark with simulator-truth densities enables latent evaluation, while a chronological NIFTY benchmark tests only held-out market prices. A two-component lognormal mixture achieves the lowest aggregate price, L1, Wasserstein, and fixed-tail errors on the synthetic benchmark. Learned operators show narrower strengths: DeepONet reduces 1% quantile error by 39.0% and variance error by 34.6% versus the mixture, and a quote transformer cuts L1 by 16.4% on the structurally misspecified Merton family. A numerical conditioning analysis explains divergent rankings: after enforcing mass and forward constraints, 95 of 126 pricing directions are numerically zero, and two densities with L1 distance 0.061 produce identical prices across covered strikes. On 524 held-out NIFTY calls, validation-selected test-time adaptation reduces DeepONet RMSE by 28.3%, but per-maturity mixture and SVI fits remain substantially more accurate.

Paper Overview

Field: Machine Learning Authors: Lennon J. Shikhman, Michael Galarnyk, Aadi Dash, Nicholas A. Welsh arXiv: 2607.27188

Abstract

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, L^1, 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 L^1 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 zero, and two densities with L^1 distance 0.061 produce identical prices on covered strikes.

On 524 held-out NIFTY calls, validation-selected test-time adaptation lowers DeepONet RMSE by 28.3%, but per-maturity mixture and SVI fits remain considerably more accurate.

Key Findings

Pricing accuracy ≠ density recovery: the paper formally separates market-price performance from latent density recovery quality.
Two-component lognormal mixture wins on aggregate price, L^1, Wasserstein, and fixed-tail errors in the synthetic benchmark.
DeepONet improves 1% quantile error by 39.0% and variance error by 34.6% over the mixture.
Quote transformer reduces L^1 error by 16.4% on the misspecified Merton family.
Ill-conditioning: after mass and forward constraints, 95 of 126 pricing directions are numerically zero; two densities with L^1 distance 0.061 yield identical prices on covered strikes.
NIFTY benchmark: test-time adaptation cuts DeepONet RMSE by 28.3% on 524 held-out calls, yet per-maturity mixture and SVI fits remain more accurate.

*Source: arXiv:2607.27188*