# Conditional Deep Levy Models for Exotic Derivatives: History-Aware Path Generation and P-Q Payoff Diagnostics

arXiv: https://arxiv.org/abs/2509.13374 · HTML: https://mlsanity.com/paper/2509.13374/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 31 references that are in the map.
- First posted on arXiv 16 Sep 2025.

## Builds on (most cited first, 11 of 11)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Chronos: Learning the Language of Time Series](https://mlsanity.com/paper/2403.07815/) (2024)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [The Rise of Diffusion Models in Time-Series Forecasting](https://mlsanity.com/paper/2401.03006/) (2024)
- [Robust pricing and hedging via neural SDEs](https://mlsanity.com/paper/2007.04154/) (2020)
- [Arbitrage-free neural-SDE market models](https://mlsanity.com/paper/2105.11053/) (2021)
- [Heavy-Tailed Diffusion with Denoising L'evy Probabilistic Models](https://mlsanity.com/paper/2407.18609/) (2024)

## Abstract (arXiv)

We develop and audit a history-aware financial path generator based on Denoising Levy Probabilistic Models (DLPMs) for conditional equity-index path generation. The model combines symmetric alpha-stable diffusion noise with a conditional U-Net observing the contract state, 60- and 252-day return histories, and pre-start trend, drawdown, and volatility state. A chronological protocol evaluates one frozen model on 6,824 untouched windows from eight Chinese and U.S. equity indices. The generator attains terminal CRPS 0.0485, path energy score 0.3368, and a generated-to-realized volatility ratio of 0.882, improving on contract-only DLPM and Gaussian diffusion controls. An unconditional historical block bootstrap remains competitive on pooled marginal metrics; against a stronger state-matched empirical control using the same index, tenor, volatility, trend, and drawdown information, DLPM lowers terminal CRPS from 0.0633 to 0.0485 (23.3%), with joint calendar-block confidence intervals below zero from 20 to 150 days. We then embed the learned physical-measure distribution in a transparent P-Q payoff diagnostic against an independently fitted, martingale-corrected Student-t GARCH(1,1) benchmark. At a pre-specified 0.05 dealer spread and a 1%-of-spot materiality threshold, the aligned unconditional diagnostic is 0.91% of initial spot for vanilla calls and 0.36% for Asian calls. Drift-neutral and one-time-delta counterfactuals decompose this gap: physical direction is a large component, leaving a 0.33% vanilla residual and a near-zero Asian residual. The P-Q framework is a belief-based payoff diagnostic, distinct from an option-surface-calibrated pricing engine.