Beta-Dependent Gamma Feedback and Endogenous Volatility Amplification in Option Markets

TL;DR

The paper develops a beta-dependent gamma-feedback framework that links delta-gamma hedging with stock-specific volatility scaling to explain endogenous volatility amplification during gamma squeezes. It derives a stability condition via the denominator $D_i = 1 - \lambda G \phi(x_i)$ and introduces a dynamic recursive model featuring decay and saturation, ensuring bounded responses. A beta-normalized shock variable $x_t = \frac{|\Delta S_t|/S_t}{\beta \sigma_m}$ and a saturating hedging function $I(y)=\tanh(cy)$ capture nonlinear amplification and self-limitation. Numerical simulations show that low-$\beta$, high gamma-exposure stocks are most susceptible to transient amplification, offering a structured framework for risk management and empirical calibration in option-driven markets.

Abstract

We develop a theoretical framework that aims to link micro-level option hedging and stock-specific factor exposure with macro-level market turbulence and explain endogenous volatility amplification during gamma-squeeze events. By explicitly modeling market-maker delta-neutral hedging and incorporating beta-dependent volatility normalization, we derive a stability condition that characterizes the onset of a gamma-squeeze event. The model captures a nonlinear recursive feedback loop between market-maker hedging and price movements and the resulting self-reinforcing dynamics. From a complex-systems perspective, the dynamics represent a bounded nonlinear response in which effective gain depends jointly on beta-normalized shock perception and gamma-scaled sensitivity. Our analysis highlights that low-beta stocks exhibit disproportionately strong feedback even for modest absolute price movements.

Figures (7)

• Figure 1: (a) Stability map of market-maker feedback dynamics. Color intensity represents the stability denominator $D_i$, with warmer colors corresponding to higher stability. The black dashed line indicates the critical threshold $D_i = 0$, where even small exogenous shocks can produce disproportionately large price movements. (b) Extreme-event amplification and stability contours. The red dash-dotted line indicates amplification $\ge 2$, while the black dashed line denotes the critical stability boundary $D_i = 0$. These contours reveal regions of pronounced gamma feedback and potential instability.
• Figure 2: Time-series response to initial $\mu_t$ shocks, using the first-step market impact. Stock price changes are computed according to $\Delta S_t = \mu S_t + \lambda G \Delta S_t \, \phi(x_i)$, applied iteratively to visualize the trajectory. Different curves correspond to different shock values $\mu_t$.
• Figure 3: Time-series response to a fixed initial shock $\mu = 0.025$ under varying levels of $\beta$. Lower-$\beta$ stocks perceive the same absolute price change as a larger normalized surprise, yielding stronger hedging demand and a larger initial price displacement. The trajectories reflect only the one-time delta-neutral hedge and therefore plateau after the initial jump.
• Figure 4: Simulated stock price trajectories under recursive feedback with cumulative position decay and saturating impact. Larger initial shocks produce steeper early-stage amplification, while cumulative decay of $N_t$ and saturation of $I(\cdot)$ gradually dampen recursive growth, leading to a stable plateau.
• Figure 5: Time-series trajectories of stock prices under recursive hedging feedback for varying $\beta$ values, given a fixed initial shock $\mu_0 = 0.025$. Lower-$\beta$ stocks exhibit a stronger initial displacement, as the same absolute shock corresponds to a larger normalized surprise $x_i$, triggering more aggressive hedging. The inset highlights the early-stage divergence, while the long-run trajectories converge as saturation and inventory decay dominate the dynamics.