Critical volatility threshold for log-normal to power-law transition
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912801578024960 |
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| author | Kremnev, Valerii |
| author_facet | Kremnev, Valerii |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_01269 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Critical volatility threshold for log-normal to power-law transition Kremnev, Valerii Mathematical Finance Statistical Mechanics Theoretical Economics Probability 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. |
| title | Critical volatility threshold for log-normal to power-law transition |
| topic | Mathematical Finance Statistical Mechanics Theoretical Economics Probability |
| url | https://arxiv.org/abs/2601.01269 |