Asymmetric Nonlinear Return Extrapolation and Optimal Portfolio Choice under Stochastic Volatility

2026-06-09

AI Summary

This study addresses the limitations of traditional linear return extrapolation models, which neglect the saturation of investor belief updating and the asymmetry in responses to gains versus losses. The authors propose a smooth, nonlinear, and asymmetric extrapolation function embedded within the Heston stochastic volatility framework to derive the optimal portfolio for a CRRA investor. This approach endogenously generates sentiment-distorted myopic demand, variance hedging demand, and sentiment hedging demand. By incorporating belief saturation and gain–loss asymmetry into extrapolative expectations for the first time, the model reveals that saturation mechanisms intrinsically mitigate welfare losses. A semi-linear Hamilton–Jacobi–Bellman equation is solved via an alternating direction implicit (ADI) finite difference scheme combined with a deep learning-based iterative algorithm, successfully replicating four major behavioral anomalies and demonstrating that nonlinear extrapolation significantly outperforms its linear counterpart.

Abstract

We extend the return extrapolation framework of Atmaz (2022) to incorporate two behaviorally realistic features absent from the linear benchmark: saturation in belief updating and asymmetry between gains and losses. We introduce a smooth, nonlinear, asymmetric extrapolation function and characterize the optimal portfolio of a CRRA investor under Heston (1993) stochastic volatility as the sum of a sentiment-distorted myopic demand, a variance hedging demand, and a sentiment hedging demand. The resulting semilinear Hamilton-Jacobi-Bellman equation is solved by two independent numerical methods, a finite-difference ADI scheme with time-step policy iteration and a deep learning-driven iterative scheme. The model generates four investor-level behavioral anomalies: asymmetric responses to gains and losses, attenuated reactions at extremes, excess trading volume, and welfare loss rising with the strength of extrapolation, each of which maps onto documented empirical patterns. Its central finding is that saturation acts as an endogenous correction mechanism: at the same local slope at the origin, the asymmetric nonlinear extrapolator carries a smaller welfare loss than a linear one.

Problem

• return extrapolation
• asymmetry
• saturation
• stochastic volatility
• optimal portfolio choice

Innovation

• asymmetric nonlinear extrapolation
• stochastic volatility
• sentiment hedging
• semilinear HJB equation
• behavioral anomalies