Critical volatility threshold for log-normal to power-law transition

Abstract

Random walk models with log-normal outcomes fit local market observations remarkably well. Yet interconnected or recursive structures - layered derivatives, leveraged positions, iterative funding rounds - periodically produce power-law distributed events. We show that the transition from log-normal to power-law dynamics requires only three conditions: randomness in the underlying process, rectification of payouts, and iterative feed-forward of expected values. Using an infinite option-on-option chain as an illustrative model, we derive a critical volatility threshold at $\sigma^* = \sqrt{2\pi} \approx 250.66\%$ for the unconditional case. With selective survival - where participants require minimum returns to continue - the critical threshold drops discontinuously to $\sigma_{\text{th}}^{*} = \sqrt{\pi/2} \approx 125.3\%$, and can decrease further with higher survival thresholds. The resulting outcomes follow what we term the Critical Volatility ($V^*$) Distribution - a power-law whose exponent admits closed-form expression in terms of survival pressure and conditional expected growth. The result suggests that fat tails may be an emergent property of iterative log-normal processes with selection rather than an exogenous feature.