Yet another Bitcoin price model

Erwin Schrödinger
Aug 26, 2017 · 6 min read

A simple exponential model for the long term Bitcoin price dynamics including “to the moon” peaks.

TL;DR

Crazy predictions based on this model:

  • Next month: 1 BTC = $18 000
  • Mid 2018: 1 BTC = $12 000
  • Late 2019: 1 BTC = $20 000
  • Early 2020: 1 BTC = $200 000

Introduction

Well, I am not a financier, but I do data analysis quite often. Usually I work with data from quite well understood systems like atomic spectra and other physics stuff. In most cases this data can be described by simple enough models — exponentials, Gaussian or Cauchy distributions. Or a bit more complicated like their convolution or combination of any other kind.

I am looking at the Bitcoin price for a few years now and it looks like it can be described reasonably well by a pretty simple model. And there are plenty of similar processes in the world with the same behaviour. One example is a photomultiplier tube. I will mention this analogy at the end of the post.

Bitcoin price model

First thing I want to emphasize: it doesn’t make any sense to look at the Bitcoin price on the linear scale. Only logarithmic! Otherwise you will see only the latest price spike and nothing else. And don’t think about “why” for now. Let’s try to make a model based only on the data itself. And later we will try to connect it to the real world.

When building a model there is a very simple criterion to estimate how good is it. The less free parameters you have the better is your model.

Zero order approximation

The simplest model would be just an exponential, or linear fit on the log scale:

The result of this fit looks like this:

Exponential fit of the Bitcoin price.

Even this linear fit describes the overall trend pretty well, but we can try to extend it by including price spikes (appearing on the chart in 2011, 2013, 2014 and the current one in 2017) to the model.

Spikes model

Looking closer at the spikes we see that they have something in common. They all look like triangles on the log scale and time constants are also quite similar.

The triangular shape on the log scale can be described as two exponents. We will add two extra parameters, common for all price spikes: time constant of the rising and falling edges and add them to our base exponent.

Left: Spike model on top of the base exponent, Right: Bitcoin price spike

If you like equations, here is one for a spike:

Here the first part is from base exponent, H — Heaviside step function (basically, it is zero if the argument is less then zero and 1 if the argument is larger than zero), A is the amplitude of the peak, τ — time constants for rising and falling fronts of the spike.

Putting all the spikes together we will get:

Here we keep amplitude and time constants the same for all peaks and add just one free parameter per peak — its position.

Exponential model with spikes

Not perfect, but better than just an exponent. There are a few things that I don’t like here though.

Problems

  1. In the very beginning this model works quite poorly. I think it’s just because the market was very unstable at that time and no one knew what Bitcoin was actually worth. Let’s say it’s a high noise region.
  2. Number of peaks is too small right now, and number of free variables is too high, so the model could be completely wrong. We need to wait for another 10 years to see if it is valid.

Reducing number of free variables

In total there are N+5 free parameters, where N is number of spikes in the model (2 for the base exponent, 3 for the spike model and N for spike positions). Ideally I would prefer to include periodicity in the model, so spikes would appear every two years or so. In this situation the model will look much less attractive, but probably more useful for predictions.

Here I reduced number of free parameters to 7 for any number of peaks:

Periodic spikes model

Mapping to the real world

Now, when we have a model that somehow describes the data, we can try to explain why is it so.

I would say, Bitcoin is quite new and only a tiny fraction of people know its potential and why it is great. Some people only heard about it and just know that it’s “some kind of money in the Internet”, but most of the people don’t even know that it exists. This means, that Bitcoin adoption grows proportionally to number of people who know. And it is clearly exponential behaviour that describes our zero approximation. This exponential growth will stop when number of people that use Bitcoin will be comparable to total number of people on Earth. But we are quite far from that point.

Spikes are also quite understandable. They are described by people’s expectations. When people see Bitcoin growing slowly but steady (outside the spike) they start thinking to invest in it, and more it grows more people want to jump on this train. It explains the growing edge of the spike. When Bitcoin reaches x10 of it’s base price, people start thinking more about risks and that it’s too much already (bubble etc.). Then something bad happens (China, MtGox, whatever) and expectations switch the sign — spike switches to the falling phase.

As market is an open system, it can be influenced by external factors and spikes periodicity is not exact. For instance, peak in early 2014 is probably caused by the MtGox’s Willy bot, so it appeared much earlier than in the periodic model.

Photomultiplier analogy

As I wrote in the very beginning, there are other processes in the world with similar behaviour. And a photomultiplier tube (PMT) is an example.

PMT is a device that allows to detect single photons by electron avalanche effect. The idea is quite simple: when photon hits a metal plate, it kicks an electron out from the metal. This electron is accelerated by electric field and guided to the next metal plate. When accelerated electron hits the second plate, it kicks out several electrons. And now more electrons start accelerating and flying to the next plate. Overall number of electrons is multiplied on every state. Comparing to Bitcoin dynamics I would say that Satoshi and his whitepaper were a photon, and every spike that we see on the price chart is similar to electron multiplication on every metal plate. And in between the spikes (plates) we are losing some electrons (value) due to background collisions or whatever.

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