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=dℚ/dℙ 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 ∼+100%), whereas XiNet's single self-consistent measure keeps the bias to +0.3%, beating maximum-entropy WMC (−24%), because ξ=fθ(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.
Related
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BibTeX
@misc{zhang2026neural,
title = {{Neural Networks Learning the Radon--Nikodym Derivative: Empirical Option Pricing in Incomplete Markets}},
author = {Ziyuan Zhang and Kiseop Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01941},
eprint = {2609.01941},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01941},
}
RIS
TY - UNPB
TI - Neural Networks Learning the Radon--Nikodym Derivative: Empirical Option Pricing in Incomplete Markets
AU - Ziyuan Zhang
AU - Kiseop Lee
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01941
UR - https://arxiv.org/abs/2609.01941
AB - 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=dℚ/dℙ 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 ∼+100%), whereas XiNet's single self-consistent measure keeps the bias to +0.3%, beating maximum-entropy WMC (−24%), because ξ=f_θ(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.
ER -