Bitcoin Bubbles and Crashes collectively behave as Power Law

True for minor bubbles as well as the major ones

Mini-Bubbles

Minor bubbles (mini-bubbles) can be selected over shorter time frames without considering the power law nature directly. However, somewhat incredibly (but not surprising on deeper reflection) it is implicitly emergent, as I will show below. In Figure 1 there are 31 mini-bubbles displayed by vertical bands and meeting the criteria of velocity greater than 0.003 in log10 daily price increase, and persisting for 15 or more days. The minimum bubble with these criteria yields an 11% increase in price.

Figure 1. These 31 mini-bubbles were chosen by showing a sufficiently steep rise in price (daily change greater than 0.003 in log10 of price) and sufficient duration (15 days or more). One notices that bubbles were much more frequent in the first half of the series. Two-thirds occurred prior to 2018.

This filter found 31 mini-bubbles, 2/3 of which occurred prior to 2018. Note the clustering, as many of these define the substructure of the major bubbles of 2011, 2013, 2017, and to a lesser degree, 2021. The rate of occurrence has fallen to about one per year with this selection method. This is only the upward bubbles, we will show the anti-mini-bubbles, or small crashes, below.

Figure 2 presents the bubble energies as a function of Bitcoin age, with the energy being defined as the sum of the log price changes over the bubble duration. The energy appears to drop by a factor of 3 or more, especially evident in the maximal values, and can be fit by a reciprocal relation with small offset. Note that a bubble energy of 0.5 is sufficient to triple the price (since 10 raised to the 0.5 power is 3) and a bubble energy of 0.3 is sufficient to double the price during its duration. The era of doubling and tripling or more in a single mini-bubble appears to have terminated, however, by the end of 2021.

Figure 2. Mini-bubble energy has declined by around a factor of 3 or so with Bitcoin’s age, roughly as a reciprocal relation (1/t) with some offset. The volatility of the energy itself has also declined considerably (shaded region represents a one standard deviation plus or minus band).

Anti-Bubbles

Anti-bubbles can be selected with the same filter, but in the downward direction. Over the same 14 plus year period 23 such small crashes or anti-bubbles were found, and are shown in Figure 3. There have been none since the beginning of 2023, although there is a more balanced number of events in the first half versus second half of the data set. Again we see clustering, especially earlier on with the crashes of 2011, 2014, and 2018, and to a lesser degree the 2022 decline. The crash of 2011 with 5 back to back mini-crashes brought the price down by a factor of over 10, but the 2022 group of three did damage of more like a factor of 2.5.

Figure 3. There were 23 small crashes or anti-bubbles isolated with the same filter as for mini-bubbles but as greater than 0.003 in log10 daily change in price downward.

In Figure 4 we show a fit of the anti-bubble energy vs. Bitcoin age. There is only weak evidence for overall decline but the upper boundary appears to have dropped by a factor close to 2. The volatility has dropped perhaps by a factor of over 2 as well, from its peak.

Figure 4. Anti-bubble energy with time graph for 23 anti-bubbles, there is little evidence of decline, the R^2 is quite low. The maximum energy level does appear to have declined by close to a factor of 2.

Collective behavior

Now collectively, these 54 mini-bubbles and anti-bubbles took up just 21.5% of the total duration of the data set history, spending about twice as much total time in the mini-bubbles as in the anti-bubbles. I have isolated just the effect of those, ignoring 78.5% of the daily price data to produce this stair step function.

Figure 5 shows how the price would have behaved due to the 54 mini-bubbles and crashes alone, apart from the overall trend, and very interestingly, it still would have increased by 5 orders of magnitude. This occurs since the mini-bubbles outweighed the anti-bubbles in average energy (0.25 vs. 0.15), and in aggregate duration (2:1).

One sees on this log-log chart that the overall nature is roughly a straight line, it can be fit by a power law of index 5.12 with a high

\(R^2 = 0.93.\)

Figure 5. Stairstep diagram of the cumulative energy in both 31 mini-bubbles and 23 mini-crashes. Each is located at its actual age position, but the trend outside these events is not considered. Although they only occupy 21.5% of the days in the price series they collectively result in a large price gain of over 5 orders of magnitude. The series can be fit with a power law of index 5.12 (4.97 with quantile regression) and with a high value of R^2 = 0.93.

It is quite remarkable that the net path of the series of mini-bubbles and anti-bubbles is a power law of index equal to or greater than 5, and not much lower than the 5.7 index seen for OLS regressions of the full Bitcoin price history. Since this series represents the tops we expect it to be lower, consistent with what is seen in quantile regressions for high levels of say 0.8 (appropriate to around 20% of the overall data series). One of my recent quantile regressions of Bitcoin’s full price history at the level 0.9 had a slope of 5.1 as well.

Figure 6. A randomization of the mini-bubble energies shown as a log price vs. log age graph. We use the same dates of occurrence but shuffled the energy values (up or down) for the 54 events for 100 realizations. While there was a wide variation in the individual power law indices (the first 25 are seen in the figure) the collective behavior can be fit by a power law of index 5.59, with a standard deviation of 1.05. This is remarkably close to the overall Bitcoin power law regression (OLS) of 5.7 index (5.8 for QR).

In Figure 6 we present the results of shuffling the sequence of bubbles up and down while preserving their timing. There are 100 simulated outcomes in the figure, and each has been measured for its own power law slope. While there is a broad range of slopes from 2.8 to 8.4, the mean value for the collective behavior is 5.59, with a standard deviation of 1.05.

That is remarkably close to the overall Bitcoin power law regression index of 5.7 using ordinary least squares (OLS). So somehow, the mini-bubbles and the crashes manifest the overall power law nature of Bitcoin.

We expect this, because after each major bubble Bitcoin has reverted to its core power law behavior that characterizes its long term rise. The core power law, resulting from network adoption squared, acts as a price attractor.

Power law aware bubble definition

For long-term analysis it is better to incorporate the fundamental power law trend in analyzing bubbles. It is their movement relative to the power law core that matters, as we have demonstrated with multiple techniques.

In order to do that we first fit the power law long term trend, one can use either an OLS or QR (quantile regression) method; here I have used the QR median (0.5 level) and computed log price residuals from that long term trend. One then defines a residual threshold, e.g. 0.2 or 0.3 etc. and identified bubbles with a certain duration. I used 10 days as the minimum and if two bubbles were separated by only 3 days from each other they were merged into a single event.

Here’s what it looks like on a log-linear plot with a threshold of 0.2. That is appropriate to the volatility that has been experience during the past four or five years, although the volatility long-term average is more like a 0.3 threshold (one standard deviation).

Figure 7. Power law aware bubbles, looking at some threshold above or below trend, in this case 0.2 in log10 price terms. The bubbles identified are represented by triangles, and the energy counted is only the energy above the threshold, not from the mean power law trend. There is much more energy in the bubbles than in the anti-bubbles. There are also many fewer than found with the first method that is not power law aware.

Somewhat surprisingly the number of bubbles varies little with threshold, because as the threshold is lowered, smaller bubbles can merge together. Here we have 10 bubbles and 14 anti-bubbles. The total energy across all bubbles or anti-bubbles is much higher in the bubbles: 467 vs. 44. In fact the mean bubble energy is 47, as large as the sum of all the anti-bubbles.

For these power law residual defined bubbles, the duration is longer, and the number is smaller, than one finds with the method that ignores the power law trend.

Figure 8. Residuals after the power law is subtracted out. The triangles represent the bubble peak and duration, and only represent the energy above the threshold, which is the same for both bubbles and anti-bubbles except with the sign reversed. There are only 4 major bubbles lasting at least 169 days in this case and with energies above 70. The right tail skew is very clear, with smaller excursions below the anti-bubble threshold. The mean bubble energy is 47 but the little bubbles have energies close to only 1 in these units, so are difficult to see. All anti-bubbles have energies less than 10 in these units, the mean is only 3.

Table 1. The number of bubbles and anti-bubbles and total bubble energy and anti-bubble (minor crashes) energy as a function of the log10 price threshold to define a bubble. In all cases the minimum duration is 10 days and two adjacent bubbles are merged if they are within 3 days of each other.

The Table 1 shows how the number of bubbles changes as the threshold is lowered, and how the aggregate energy increases. The number of bubbles is roughly constant due to the merging phenomenon, while the average energy per bubble of course rises, from about 35 to about 73. The overall bubble energy is much higher, even at the lowest threshold it remains over 3 times larger than that in anti-bubbles.

In the case of the anti-bubbles, there are only 3 at the highest 0.3 threshold but then it quickly rises to 14 in number and remains flat as the threshold is lowered toward 0.1. The average energy rises from just about 2 to about 13.

Where are we now?

There has been no bubble of this second type meeting the 0.2 threshold except for two very minor ones at the beginning of 2022. That is somewhat less than the prior cycle’s average volatility (0.24 single standard deviation). There were 5 anti-bubbles in 2022 and 2023. While we are approaching the threshold it has yet to be surpassed. In linear terms 0.2 corresponds to a factor of 1.58 and the QR median trend value is about $94,000 at present. A significant bubble would require us to reach $149,000 for Bitcoin.

We do have an upward trend since the beginning of 2023 and the liquidity outlook is favorable. But bubbles of large intensity driven or initiated by the halving cycle (as opposed to global liquidity and herding) do not look to be our current environment.