Gatheral double stochastic volatility model with Skorokhod reflection
Yuliya Mishura, Andrey Pilipenko, Kostiantyn Ralchenko
TL;DR
This work scrutinizes Gatheral's double mean-reverting stochastic volatility framework and identifies a practical issue: volatility paths can spend extended periods near zero, undermining model flexibility. It develops a Skorokhod-reflected extension to enforce a positive lower bound on volatility without sacrificing flexibility, and rigorously analyzes the internal mean-reverting process across CKLS/CIR regimes. The paper proves existence and uniqueness for the coupled external/internal diffusion, reveals that near-zero regions are revisitable with high probability, and introduces a reflected CKLS variant to guarantee strict positivity. Collectively, the results provide a mathematically robust pathwise foundation for stable, nonnegative stochastic volatility dynamics with controlled near-zero behavior, offering a practical tool for more flexible and realistic financial modeling.
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.