Forward-looking physical tail risk: a deep learning approach

Jingyan Zhang, Cong Ma and Wim Schoutens

Quantitative Finance, 2026, vol. 26, issue 6, 981-992

Abstract: This study introduces a deep learning framework that integrates risk-neutral information extracted from options markets into physical measure estimates. Although previous research links risk-neutral and physical measures using a pricing kernel, its functional form remains unsolved. To address this challenge, we develop the forward-looking physical return variational autoencoder Wasserstein generative adversarial network (FPR-VAE-WGAN), which is a generative model that reconstructs the mapping between the two measures. This approach allows us to infer forward-looking physical returns exclusively from risk-neutral information. A numerical analysis based on S $ \& $ &P 500 option data demonstrates that forward-looking physical return densities have leptokurtic and heavy-tailed characteristics, as one believes that the real physical return distribution has. Furthermore, by leveraging the joint elicitability of value-at-risk (VaR) and expected shortfall (ES), we derive forward-looking tail risk estimates from the generated physical return distributions.

Date: 2026
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DOI: 10.1080/14697688.2026.2623897

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