Critical Volatility Threshold for Log-Normal to Power-Law Transition: Iterated Options Model

Battle Count: 50

The paper provides a theoretical framework for understanding when iterative/compounding financial structures transition from log-normal to power-law (fat-tailed) behavior. This is relevant for: (1) tail risk assessment in derivative portfolios, (2) understanding when optionality can dominate underlying value during stress events, (3) identifying critical volatility regimes for high-vol instruments, (4) explaining power-law VC returns. However, it is primarily a theoretical/mathematical paper rather than a practical trading strategy paper. The critical thresholds (250.66% unconditional, 125.3% with survival) are directly actionable for risk management but less so for alpha generation. The time-to-criticality table is practically useful for identifying when instruments enter supercritical regimes.

The analytical framework involves nonlinear recursions, Gaussian CDF/PDF evaluations, and Mills ratio computations - mathematically sophisticated but well-defined. The Monte Carlo simulation is straightforward (numpy-based, ~15 lines of core logic). The main complexity lies in correctly implementing the survival threshold logic and interpreting the phase space. For practical risk management applications, computing β_eff and the power-law exponent α requires only standard normal distribution functions. The theoretical derivations are complex but the computational implementation is moderate.

Simulation code is available on GitHub (github.com/sci2sci-opensource/research). The analytical derivations are fully self-contained with explicit formulas. The Monte Carlo simulation parameters (N=10M, T=15, w0=$20,000, threshold_k=2.5) are clearly specified. However, the paper relies on idealized assumptions (constant volatility, perfect pricing, no volatility amplification) that limit direct empirical validation. No external datasets are required.