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

Research Question and Approach

This paper investigates the challenge of recovering the latent risk-neutral density (RND) from sparse, irregular option quotes. The authors distinguish between two goals: reconstructing observable market prices and recovering the underlying latent distribution. Because market RNDs are not directly observable, the authors employ a dual-benchmark approach: a controlled synthetic study where the true density is known, and a real-market study using NIFTY option data to test held-out price prediction. They compare four learned operator architectures (DeepONet, FNO, Quote Transformer, and Set Decoder) against classical baselines, including parametric lognormal mixtures and regularized discrete estimators.

Key Findings

The study demonstrates that the choice of estimator depends heavily on the intended application. A two-component lognormal mixture remains the strongest performer for aggregate metrics like $L^1$ error and Wasserstein distance, especially when the underlying data matches its parametric form. However, learned operators offer distinct advantages in specific regimes. For instance, DeepONet significantly reduces error in tail functionals and variance, while the quote transformer improves performance under structural misspecification (e.g., Merton jump diffusion). The authors also perform a numerical conditioning analysis, showing that the pricing map is highly ill-posed; they identify a subspace of "numerical-null" directions where different densities yield identical prices, explaining why different models can achieve similar pricing accuracy while producing vastly different latent densities.

This research clarifies that accurate option pricing does not guarantee accurate latent density recovery. For practitioners, this means that selecting a model requires aligning the inductive bias of the architecture with the specific downstream task—such as risk management or tail-risk assessment—rather than seeking a single "best" model. The results caution against over-reliance on aggregate pricing metrics when the objective is to infer the underlying market-implied distribution.