Blogs & Articles: Modeling Bitcoin Value with Vibes 🔗 40 weeks ago

The Market Price of Bitcoin and the “Number Go Up” Model

visualization of the analysis presented herein by twitter user @wearehodl123

Introduction

Ever since Bitcoin has had a market price, and ever since scarcely imaginable Twitter clout has been up for grabs in attempting to predict it, such predictions have been readily forthcoming. We assume familiarity with a range of such prior models and will not recount here the spectrum of silliness on which they sit.

While Bitcoin has many intriguing properties worth considering as model inputs, in our opinion, all previous attempts overlook the only characteristic of Bitcoin that is truly novel: its revolutionary Number Go Up technology. Only NGU truly sets Bitcoin apart from altcoins, gold, and any other asset we might consider a competitor. It behooves us to explore this property more rigorously.

To start with, we must surely ask ourselves, with respect to what does the number go up? When does this happen?

In time, that’s when.[i]

We must also ask ourselves, is anything more important than Number Go Up? Is it possible, even in theory, that there are other factors involved?

Nah, I don’t think so.

Our task is therefore to build a price model for Bitcoin that uses as its independent variable time, and time alone. Let us work from first principles to build as rigorous a model as we possibly can, justifying every construction with logic, and backing up every data point with facts. Let’s sit down in our thinking chair and think. Let’s build the NGU Model.

Intuition

photo by Jonathan Cosens, via Unsplash

Since Bitcoin is a network and benefits from network effects in the form of positive feedback, it makes sense to assume the number will go up exponentially.[ii]

Therefore, let us take the entire price history of Bitcoin, regress it logarithmically, and find a growth rate at which we think the number is likely going up as the core of our thesis, per Figure 1:

Figure 1: Bitcoin’s price history charted logarithmically with a best fit line added and its equation shown

As the reader can see, the R² of the regression is 0.87, which is pretty high for having put no thought into this whatsoever. This suggests we are tiptoeing towards a profound and almost certainly causal relationship between Bitcoin’s value and our identification of the fundamental breakthrough of the underlying Number Go Up technology. If we keep tweaking it to better fit the data we might really be onto something. Let us continue.

To make our model more realistic, we should note that the problem with Number Go Up is that sometimes the number goes down. This is most mysterious given the issuance schedule is perfectly well known over literally every period of time. We might hypothesize that there are recurring phases of excitement, perhaps caused by misleading memes on social media that cause a wave of larping in on false hopes and, naturally, larping out again when these hopes fail to materialize. Animal spirits and whatnot, as Keynes rightly observed.

Starting from our assumption of exponential growth, we can modulate the path of growth by adding a sinusoidal component, as demonstrated in the figures below.

In Figure 2 we see an exponential growth curve in red and a sine wave that has been shifted entirely above the x-axis in blue:

Figure 2: exponential function (red) and shifted sine function (blue)

Consider the product of these two:

In Figure 3 we see this new function overlaid in green:

Figure 3: exponential function (red), shifted sine function (blue), and their product (green)

It is perhaps easier to conceptualize what is really happening here if we adjust the y-axis to a logarithmic scale, as below in Figure 4. Here the exponential increase is a straight line, and the sinusoidal modulation fluctuates around this line but demonstrates the same long-term rate of growth:

Figure 4: exponential function (red), shifted sine function (blue), and their product (green), all on log scale

This is also easier to then compare to Bitcoin given we arguably need a logarithmic y-axis for the price to adequately capture its history.[iii]

Another idea might be that when people get all excited by price memes, larp in and larp out, etc., that tends to happen pretty quickly, whereas the fallout tends to take up a longer stretch of the periodicity. Bear markets are for building, after all.[iv]

We can address this by “compressing” and “stretching” our sinusoidal modulation, so to speak, as follows. Rather than f(x) = sin(x), consider instead something like f(x) = sin(x — cos(x)).

This alters the sine wave, with the original in red, and the “quick up and down then long pause” version in blue in Figure 6 below :

Figure 6: regular sine function (red) and “compressed” sine function (blue) (per formula just above)

If we now go back to the logarithmic view, we can compare exponential growth (in blue), exponential growth modulated by a regular sinusoidal wave (in red) to exponential growth modulated by our “quick up and down” wave (in green) in Figure 7:

Figure 7: exponential function (blue), product of exponential and sine function(red), and product of exponential and “compressed” sine function (green), all on log scale

This seems sensible. We might also think that the underlying exponential growth, and possibly the amplitudes of the sine waves, should be dampened. This could conceivably be because silly memes that encourage larpiness have a decaying effectiveness across periodic cycles as more and more people get used to them. Let us introduce a decay term as follows:

Where f(t) increases monotonically in t and simply slows down the growth rate without ever turning it negative (number goes up, remember — this is very important). Hence, we might compare our above “quick up and down” modulated exponential growth (in green) to one that also dampens over time (in blue), in Figure 8:

Figure 8: product of exponential and “compressed” sine function (green), ditto with dampened growth (blue), both on log scale

There is a final element of market prices we ought to try to capture as well: they wibble wobble. When the number sometimes goes down, this rarely happens in so smooth a way as the preceding discussion suggests. This is readily explained by psychology.

We propose to introduce a wibble wobble factor in the form of another sinusoidal modulation with a faster integer period, as demonstrated below in Figure 9, in the simple case as compared to a sine wave.

Figure 9: a sine wave (orange) and a smaller, higher frequency sine wave (purple)

Such that their product is a wibbly wobbly bigger sinusoidal function, per Figure 10 below:

Figure 10: the product of the two waves in Figure 9, a wibbly wobbly sine wave.

With these simple, logical, and reasonable assumptions in mind, let’s put together a model that captures all of the above. This will take the form:

Where r is the regression coefficient derived from Figure 1, the f(t)’s are the monotonic dampening of this coefficient and the sinusoidal modulations, as captured in Figure 8, g(t) parameterizes how we input “time” such that the amplitude, frequency, shift, and so on, of the sinusoidal functions are suitable for this domain, and h(t) makes it all wibbly wobbly.

Figure 11 compares our model to the historical data from December 2011 to the present.[v]

Figure 11: NGU Model vs Historical Prices

This model has an R² of 0.97. You would probably have to run the universe three or four times in order to encounter such a correlation by chance. This suggests a relationship between Bitcoin’s value and time so profound as to be barely comprehensible.

Figure 12 extends this incomprehensibility into the far-flung future such that we might at least attempt to comprehend what is coming (apologies in advance to the reader if this is too much to handle):

Figure 12: all prices forever

Alternatively, here is the price on Jan 1st at 3-year intervals for the next 27 years. Given the astounding accuracy of the model, it is safe to say that only 3% of the actual price on these dates will not be explained by these predictions:

The full numerical specification is as follows:

Or:

Although this may look complicated, note that, as promised, there is only one independent variable:

time.

Let’s not draw too much attention to all these numbers that, okay, admittedly, define the model. Instead, let us propose the perfectly reasonable interpretation that:

The above analysis essentially proves that the price of Bitcoin is — and only is — a function of time, exactly as the Number Go Up thesis predicted. It’s unclear why we would care about any other potential input. We will move straight to questions the reader may naturally have.

Questions

photo by Towfiqu barbhuiya, via Unsplash

What is the physical significance of the model’s parameters? What are their dimensions? Can they be derived analytically? Do the physical inputs have numerical components that bear any relation?

These are natural questions to want to ask, but that’s not how Science works. These parameters were chosen because they make the backfitted R² very high, which in turn means we have to assume we have stumbled into a fundamental principle of economics. The onus is on us to work with these constants to derive further insights into marginal analysis, methodological individualism, and human action, not the other way around. This isn’t some Proof-of-Stake circle jerk, guys; we have to do the work.

Are we happy with the numeraire?

This is an excellent question (well done!) given it is core to our thesis that the number will continue to go up until Bitcoin is the only money in the world, and possibly the only asset with any value whatsoever. This means we ought to at least be suspicious of non-Bitcoiny ways of measuring value lest the model begin to reflexively feed on itself in ways that, while not impossible to mathematically capture, are probably really hard.[vi] Thankfully, this is not a concern in this case because we have chosen an extremely robust measure of consistent purchasing power: the US Dollar.

It’s very impressive that you created such a model with only one independent variable, but have you considered any confounding variables?

Firstly, thank you. As to your question, no, time is an SI Unit, meaning it literally can’t be explained in terms of anything else. Typically, yes, you should always rack your brain to consider what other, simpler phenomena - potentially with fewer or zero parameters - might explain roughly similar behavior to what you have observed in the dependent variable. But we are not going to do that.

Does this have anything to do with fractals?

No.

The model clearly works astonishingly — almost scarily — well, but doesn’t it bother you that we don’t really know why it’s so accurate? Can we push our analysis further? What does it mean?

This is a relatable urge, but it betrays a lack of understanding of the fundamental purpose of economic analysis. The goal of the enterprise is to paramaterize the macroscopic outcome of economic activity in the past and extrapolate it to the future, not to try to understand why anybody is doing anything.

Besides, how would you even measure that? It is intrinsic to our analysis that people are kinda stupid (see the discussion preceding Figures 2 through 7) and may or may not believe any old meme rather than acting strictly rationally and in light of perfect information perfectly available to them as to Bitcoin’s properties in the future.[vii]

We see, therefore, that Number Go Up doesn’t really have a meaning,as such. It transcends such petty mathemetizations. What is the meaning of a baby’s laugh? What is the meaning of the smell of morning dew? Number Go Up is more of a vibe …

Conclusion

photo by Dynamic Wang, via Unsplash

Bitcoin is the first manifestation of Number Go Up technology the world has ever seen. Surely this relationship with time has value? In this article, I quantify Number Go Up and use Number Go Up to model Bitcoin’s value.

A statistically significant relationship between time and market value exists. The likelihood that the relationship between Number Go Up and market value is caused by chance is close to zero.

Adding confidence in the model:

• It was derived from first principles, not spurious real-world observations.
• It has one variable.
• That variable is “time.”
• Time is the only truly scarce asset.
• Did we just model scarcity also?
• 🤔

Regardless, the model clearly proves that for the price of Bitcoin at a given point in the future, the number will probably have gone up.

follow me on nostr for more mind-blowing market analysis: #npub1sfhflz2msx45rfzjyf5tyj0x35pv4qtq3hh4v2jf8nhrtl79cavsl2ymqt

and twitter @allenf32

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ENDNOTES:

[i]: Gigi’s Bitcoin Is Time already proved this rigorously, but it is good to go from first principles and not rely on the work of others. Interesting areas for further research beyond the scope of this study might be to explore Bitcoin’s value as a function of Hope, Venice, and Potatoes.

[ii]: negative feedback, on the other hand, will lead to being blocked.

[iii]: one of the more exciting intrinsic properties of Bitcoin is that not only does the number go up, it goes up fast.

[iv]: Building models, that is.

[v]: Technically, price data is available since July of 2010, but it messes up the model, seemingly no matter how we parametrize it. It’s important not to hamstring ourselves by taking these things too literally. The data on which a model is built and the rhetoric we use to discuss it needn’t be isomorphic.

[vi]: The mathematical term for what is being conceptually approached here is that the system may become “nonlinear.” While I have a first in mathematics from the highest-ranked undergraduate department in the United Kingdom, we only covered linear systems at that level of instruction. Nonlinear systems are waaaaay harder and their study typically begins at postgraduate level. Unfortunately, I got a job in finance, stumbled into Bitcoin, then stumbled into the present analysis.

[vii]: We note the distinct possibility that, having published this mind-blowingly accurate model, the entirety of the behavior of market participants will change. It is further possible that such an occurrence would refute many of the assumptions that went into building the model, but it’s also possible that this won’t happen.