Neural Networks Learning the Radon--Nikodym Derivative: Empirical Option Pricing in Incomplete Markets

Abstract

In incomplete markets, no-arbitrage (NFLVR) guarantees the existence, not the uniqueness, of an equivalent local martingale measure (ELMM): unhedgeable risks (jumps, stochastic volatility) admit a whole family of equivalent measures, and asset dynamics alone cannot pin down the one the market selects. We characterize the identifiability of $Q$ from option data and propose a measure that is identifiable from data yet prices any claim consistently. The key boundary is an ``identification wall'': European options identify only the terminal marginal, while out-of-sample tails and path/joint structure require instruments matched to the priced risk (variance or higher-moment swaps, path-dependent claims). Within this view, minimum-relative-entropy weighted Monte Carlo (WMC) is the optimal baseline for the marginal; we generalize it to a full path-space measure change parameterized by a neural network on physical scenarios---XiNet---which learns $ξ=\mathrm{d}\mathbb{Q}/\mathrm{d}\mathbb{P}$ directly from 8 model-free path features, with option prices as a soft constraint. On European (marginal) pricing XiNet matches but does not surpass WMC, and both are bound by the identification wall out-of-sample. On path-dependent claims the picture reverses: calibrated on the same European surface, per-marginal methods fail structurally (an ATM forward-start is mispriced by $\sim+100\%$), whereas XiNet's single self-consistent measure keeps the bias to $+0.3\%$, beating maximum-entropy WMC ($-24\%$), because $ξ=f_θ(\text{path features})$ captures joint structure Europeans cannot constrain. Identification is thus risk-specific, and XiNet is a single measure that absorbs available instruments and prices all claims consistently.