When Kelly Allocation Stays Stable Under Power-Law Bitcoin Dynamics
Summary
The document derives how the continuous-time Kelly fraction changes when an asset’s long-run price follows a power law and its return variance declines over time. With zero risk-free return, the allocation depends on the growth exponent, the volatility scale, and the rate at which variance decays. It is time-invariant only when that decay rate takes a particular value; otherwise, the optimal fraction changes with asset age.
The authors estimate price and volatility exponents from Bitcoin’s daily history, examining volatility estimates across rolling windows of four to nine years. Their mean estimate is close to the theoretical benchmark, but the overlapping windows make the spread a measure of model sensitivity rather than a confidence interval. They also propose a scaling explanation involving network participation, liquidity, and volatility, and show that an additional time-varying variance contribution, illustrated with transaction-fee variability, can disrupt invariance. The Bitcoin estimates and scaling explanation depend on modeling assumptions and do not establish that Kelly allocation is stable in future markets.
Key ideas
- The Kelly fraction varies with time unless the volatility decay exponent reaches the model’s invariance condition.
- Bitcoin volatility estimates across rolling windows are presented as a sensitivity check, not a formal confidence interval.
- A proposed scaling relationship links network participation and effective liquidity to declining volatility.
- An additional time-varying source of return variance can break the predicted allocation stability.
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# Asymptotic Invariance of Kelly Allocation Under Power-Law Asset Dynamics: Evidence from Bitcoin
# Asymptotic Invariance of Kelly Allocation Under Power-Law Asset Dynamics: Evidence from Bitcoin
We examine log-optimal portfolio allocation when the long-run price of an asset follows a power-law trajectory, $P(t)=At^α$, and its instantaneous return variance decays as $σ^2(t)=σ_0^2 t^{-2γ}$. Under a zero risk-free-rate benchmark, the continuous-time Kelly fraction scales as $K^\ast(t)=(α/σ_0^2)t^{2γ-1}$. Exact temporal invariance therefore occurs when $γ=1/2$, whereas deviations from this value produce systematic age dependence in the allocation. We propose a scaling hypothesis connecting growth in network participation, effective market liquidity, and declining volatility. Under a specified set of scaling assumptions, this model predicts the benchmark exponent $γ=1/2$. Using historical daily Bitcoin prices, we estimate the power-law price exponent and examine the sensitivity of the volatility exponent to the length of the rolling window. For windows of four to nine years, the estimated volatility exponents have an arithmetic mean of 0.53 and a cross-window standard deviation of approximately 0.03. Because these estimates are obtained from overlapping observations and the same underlying price history, this spread is interpreted as a measure of model sensitivity rather than a formal confidence interval. Finally, we show that a time-dependent multiplicative contribution to return variance generally breaks exact Kelly invariance. We illustrate this result using a scenario in which Bitcoin transaction-fee variability affects the effective variance process. The results identify the conditions under which log-optimal allocation can remain stable under non-stationary power-law asset dynamics and clarify the assumptions required when applying this result to Bitcoin.Shown in full with attribution under the source's licence. Licence: abstract CC0
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