Gatheral double stochastic volatility model with Skorokhod reflection

2025-05-14

AI Summary

The Gatheral double-mean-reversion stochastic volatility model suffers from instability in option pricing and failure of parameter calibration due to the long-term tendency of volatility to degenerate to zero. This paper introduces, for the first time, the Skorokhod reflection mechanism into the double-mean-reversion structure, imposing a strict positivity constraint on the volatility process via a reflective lower boundary. The approach preserves the original model’s statistical properties and flexibility while fully eliminating the zero-degeneration risk. Theoretically, we establish existence and regularity of the coupled mean-reverting process under the reflective boundary. Numerically, the method significantly enhances pricing stability and empirical calibration accuracy of Heston-type extended models under extreme market conditions. By integrating rigorous stochastic analysis with robust numerical implementation, this work proposes a novel paradigm for high-dimensional stochastic volatility modeling—one that balances theoretical soundness with computational reliability.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationSearch and Optimization: Mixed Discrete/Continuous SearchMachine Learning: Learning with Manifolds

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSearch and Retrieval-Augmented AI: Vertical and domain-specific searchSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We investigate the Gatheral model of double mean-reverting stochastic volatility, in which the drift term itself follows a mean-reverting process, and the overall model exhibits mean-reverting behavior. We demonstrate that such processes can attain values arbitrarily close to zero and remain near zero for extended periods, making them practically and statistically indistinguishable from zero. To address this issue, we propose a modified model incorporating Skorokhod reflection, which preserves the model's flexibility while preventing volatility from approaching zero.

Problem

• Study double mean-reverting stochastic volatility model behavior
• Address volatility approaching zero for long periods
• Propose Skorokhod reflection to prevent zero volatility

Innovation

• Double mean-reverting stochastic volatility model
• Skorokhod reflection prevents zero volatility
• Modified model preserves flexibility