Power-law price models of bitcoin

Published in

Quantodian: Tracking Bitcoin

6 min readSep 17, 2019

Introduction

Power law relationships do a good job in describing bitcoin data quite accurately, as several publications have shown. A fundamental driver of this might well be the scarcity of bitcoin in relation to gold and silver as described so well in PlanB’s paper. This article supplements my previous article on price action. It presents some examples and background on the power law, lays down the formulas and charts for bitcoin and concludes with some key publications that have applied power law relationships to bitcoin so far.

Power law relationships

A power law relationship means that one variable (y) moves in relation with another variable (x) according to:

y = 10^B * (x + S)^A

or equivalent:

log(y) = A * log(x + S) + B

A, B and S are the three parameters of this model. The model fits log(y) vs. log(y) with a straight line, with A being the slope and B the intercept. S is a shift parameter working on x. Typically S=0, but if the origin of x is not or not accurately known (as in bitcoin’s case), S allows to estimate the x origin from the data.

Here are some examples of strict or strong power law relationships between two variables.

The length of two different sides (x and y) of a square (these are equal). B=0, S=0, A=1.
The circumference of a circle (y) and its radius (x) (these are proportional). B=log(2π), S=0, A=1
A square’s side length (x) and its area (y). B=0, S=0, A=2
A cube’s side length (x) and its volume (y). B=0, S=0, A=3
The distance of a planet to its sun (x) and its orbital period around its sun (y). B=?, S=0, A=2/3
The diameter of a crater (y) on Pluto (formerly known as a planet) and the number of similarly sized craters on the whole of Pluto (y). B=?, S=0, A=-2.5
Time (x) and the price of bitcoin (y) B= -13.4528, S=314.054, A=4.86822 (based on price data from 17 July 2010 to 16 September 2019)

Examples 1, 2, 3 and 4 are strictly deterministic, meaning that the y value is known with ultimate precision if the x value is known. These true relationships have no uncertainty.

Example 5 has an element of uncertainty due to measurement errors, but it holds for every observation made on stars and planets so it has become Keppler’s third law in physics.

Example 6 is stochastic with quite some uncertainty in it, because of limited measurement precision of crater size, but also of counting limitations e.g. a big crater will destroy small craters underneath it. The power law seems to hold across a very wide range of crater sizes, but it is not a strict law of physics.

The bitcoin case

The bitcoin example 7 is stochastic with lots of uncertainty in it. In finance this uncertainty is called volatility. Bitcoin only exists for about 10 years and so the dataset is limited. It is not sure at all if the power law will hold. Only in the far future we will know if our current extrapolations really worked, but at the moment the fit seems remarkably good in the overall sense, and by the lack of other plausible models and with some support of the fundamental characteristics of bitcoin I (and others) chose this model as a good option, at least to track the actual price, but also to rationally estimate future prices.

Per today, the parameter values for price vs. time were found by fitting the non-linear equation directly to the data, resulting in:

price = 10^(-13.4528) * (time + 314.054)^(4.88682)

where time is the number of days from 17 July 2010. Note that the parameter values have slightly changed to those published in my previous article, because 1. the data set has been extended by 15 more days and 2. a higher precision (6 significant digits) for these values is used here.

The following charts plot the realized price of bitcoin in blue vs. time together with the modeled price of bitcoin in red vs. time in four different ways.

Chart B: Logarithmic Price vs. Linear Time

Chart C: Linear Price vs. Logarithmic Time

Chart D: Logarithmic Price vs. Logarithmic Time

Data is identical in these four charts, just the way the axes are plotted is different. As you may note in charts A and C the red line looks like quite an insufficient description of the bitcoin price, while charts B and D seem to do a much better job in describing the data. The reason for this visual discrepancy is that the dynamic price range of bitcoin is so large that, when plotted on a linear scale, the small prices of the past become heavily compressed, switching the focus entirely on the highest price peak surge of 2017-2018.

If the power law model applies, we’re still left with the significant noise in the price data around the model line but we can estimate bitcoin’s price in the future with much more confidence than without the model. Also we then better make our investments in bitcoin earlier than later, because our lives proceed in linear time, and the steepness of the curve slowly but steadily diminishes (chart B). Also the model provides a good case to hodl our bitcoin as much as possible. This does not mean that we should never sell any of it, but if we do because bitcoin approaches new highs, we should buy back later, at new accumulation levels. The band around the fit as shown in my original article gives a hint how to make such tactical long-term trades, and this might be the subject in a next article.

Bitcoin is still relatively young, and the dataset is small. I’m privileged to have 3348 datapoints to fit my model. But the first power law model was already applied to bitcoin data as early as 2014, which is quite remarkable. Starting with that, here are a few key publications in chronological order. I have copied the leading chart in each article, because it is not certain that these articles will be maintained and/or still present on the web in the future.

Trolololo 14 Oct 2014: https://bitcointalk.org/index.php?topic=831547.msg9293359#msg9293359 (bitcoin price vs. date)

EconoPhysicist, 3 Sep 2018: https://www.reddit.com/r/Bitcoin/comments/9cqi0k/bitcoin_power_law_over_10_year_period_all_the_way/ to Genesis block (bitcoin price vs. date)

Dave The Wave, 12 Dec 2018: https://medium.com/max-exchange/bitcoin-the-logarithmic-growth-curve-by-dave-the-wave-a2ef93b02df1 (bitcoin price vs. date)

PlanB, 22 Mar 2019: https://medium.com/@100trillionUSD/modeling-bitcoins-value-with-scarcity-91fa0fc03e25 (Bitcoin stock to flow ratio vs Market cap)

Spencer Wheatley, Didier Sornette, Tobias Huber, Max Reppen and Robert N. Gantner 5 Jun 2019 https://royalsocietypublishing.org/doi/10.1098/rsos.180538 Are Bitcoin bubbles predictable? Combining a generalized Metcalfe’s Law and the Log-Periodic Power Law Singularity model. (Bitcoin market cap vs. number of active addresses)