Paper: arXiv 2601.01269

Authors: Valerii Kremnev

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 $σ^* = \sqrt{2π} \approx 250.66%$ for the unconditional case. With selective survival - where participants require minimum returns to continue - the critical threshold drops discontinuously to $σ_{\text{th}}^{} = \sqrt{π/2} \approx 125.3%$, 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.

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces a novel theoretical framework showing how power-law distributions and fat tails emerge endogenously from iterated option-like payoffs with a critical volatility threshold at ~250%. However, it is entirely analytical/conceptual with no empirical market data backtesting or real-world validation, relying purely on mathematical abstractions and numerical toy simulations.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Explain transition from log-normal<br>to power-law (fat-tail) dynamics"] --> B["3 Core Conditions:<br>Randomness, Rectification, &<br>Iterative Feed-Forward"]
    
    B --> C["Model Input:<br>Infinite Option-on-Option Chain<br>(Iterative Log-Normal Processes)"]
    
    C --> D["Derive Volatility Thresholds<br>under different conditions"]
    
    D --> E1["Unconditional Case:<br>σ* = √(2π) ≈ 250.66%"]
    D --> E2["With Selective Survival:<br>σ_th* = √(π/2) ≈ 125.3%"]
    
    E1 --> F["Key Outcome:<br>Critical Volatility (V*) Distribution"]
    E2 --> F
    
    F --> G["Finding:<br>Fat tails emerge endogenously<br>from iteration and selection"]