Stock returns revisited and variances hedged by machine learning

Dilip B. Madan and King Wang

Quantitative Finance, 2026, vol. 26, issue 3, 341-354

Abstract: The empirically supported property of absolute variations dominating quadratic variations are employed to motivate the construction of models with percentage returns bounded by unity in absolute value. Characteristic functions are developed for the log price relative for the new return models. The models are estimated on time series and option data and demonstrate improvements delivered by the inverse logistic transformation. Applications to pricing return variations and options on them are developed. Hedging strategies use Machine Learning methods on simulated sample spaces. It is observed that the log contract hedge introduces a high volatility dynamic hedge that theoretically may be compensated by the log contract leaving the variance contract as a residual. The Machine Learned hedge works with other functions estimated here by a Gaussian Process Regression that substantially reduces the volatility of the dynamic hedge and yet leaves, approximately, the variance payout as the residual.

Date: 2026
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DOI: 10.1080/14697688.2025.2609651

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