幂律资产动态下凯利配置的渐近不变性：来自比特币的证据

中文摘要 AI 辅助

我们研究了当资产的长期价格遵循幂律轨迹$P(t)=At^\alpha$且其瞬时收益方差按$\sigma^2(t)=\sigma_0^2 t^{-2\gamma}$衰减时的对数最优投资组合配置。在零无风险利率基准下，连续时间凯利比例缩放为$K^*(t)=(\alpha/\sigma_0^2)t^{2\gamma-1}$。因此，当$\gamma=1/2$时出现精确的时间不变性，而偏离该值则会在配置中产生系统的年龄依赖性。我们提出了一个缩放假设，将网络参与度的增长、有效市场流动性和波动率下降联系起来。在一组特定的缩放假设下，该模型预测基准指数$\gamma=1/2$。利用历史每日比特币价格，我们估计了幂律价格指数，并考察了波动率指数对滚动窗口长度的敏感性。对于四到九年的窗口，估计的波动率指数的算术平均值为0.53，跨窗口标准差约为0.03。由于这些估计值来自重叠观测和相同的底层价格历史，该离散度被解释为模型敏感性的度量，而非正式的置信区间。最后，我们表明收益方差中随时间变化的乘性贡献通常会破坏精确的凯利不变性。我们通过一个场景说明了这一结果，其中比特币交易费用可变性影响有效方差过程。这些结果确定了在对数最优配置在非平稳幂律资产动态下保持稳定的条件，并阐明了将此结果应用于比特币时所需的假设。

英文摘要

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.

发表机构

• Department of Physics and Astronomy, University of Manchester（曼彻斯特大学物理与天文系）

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