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

2026-07-29

AI Summary

Recovering the implied risk-neutral density from irregular option quotes is highly ill-posed, making stable estimation challenging even with accurate price data. This study systematically evaluates multiple methods on both synthetic and real-market (NIFTY) benchmarks, exposing the inherent numerical instability of the problem and introducing the concept of “target-dependent inductive bias” to tailor estimators to specific objectives. Combining a two-component lognormal mixture model, DeepONet, a Quote Transformer, SVI parametrization, and test-time adaptation strategies, experiments demonstrate that the mixture model achieves the lowest aggregate error on synthetic data, while DeepONet reduces tail and variance errors by over 34%. Incorporating test-time adaptation further decreases DeepONet’s RMSE by 28.3% on NIFTY data, substantially enhancing its generalization performance.

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 null, and two densities separated by $L^1 = 0.061$ 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.

Problem

• risk-neutral density
• inverse learning
• option pricing
• numerical conditioning
• latent inference

Innovation

• risk-neutral density
• inverse learning
• DeepONet
• numerical conditioning
• option pricing