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

TL;DR

This paper identifies a critical volatility threshold at approximately 250.66% for the transition from log-normal to power-law distributions in recursive financial models, explaining fat tails as emergent properties of iterative processes with selection.

Contribution

It derives a precise volatility threshold for the log-normal to power-law transition and introduces the Critical Volatility Distribution as a new framework for understanding fat tails.

Findings

Critical volatility threshold at ~250.66% for unconditional case

Threshold drops to ~125.3% with survival-based selection

Outcomes follow a power-law distribution with a closed-form exponent

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.