A logistic model of Bitcoin's priceAuthor:
@gbianchi 1GbianchiJ6EeBU8ua77719Ur7qwLZVk3x
Revision 0.5 04/11/2024
PrefaceSeveral models of Bitcoin's price have circulated over the years, and recently two have become particularly well known: PlanB's S2F model
[1] and Santostasi's PowerLaw model
[2]Without going into the logic behind these models, two orders of problems immediately emerge, which irretrievably invalidate their foundations:
1) the models are referenced to a fiat currency, usually BTC price in USD.
2) The patterns tend to grow infinitely over time, so you already start with the certainty that they will be invalidated sooner or later.
I will try in this article to derive a model that removes these basic problems.
Choosing a correct reference value: goldIn setting up a Bitcoin pricing model that is valid over time, we need to choose a reference value that we can consider reasonably constant and stable.
Choosing a fiat currency as a reference value exposes the model over time to potential variations in that currency. Such variations can result from inflationary problems, political problems,
economic problems of the issuing country of the fiat currency.
In the case of the U.S. dollar, despite the fact that it is often considered a reference currency in international trade, all these problems become evident, thus both devaluation problems due to inflation,
volatility problems from political tensions, and huge problems of debt-to-GDP ratio sustainability. In this already difficult problem of finding a model for the price of Bitcoin,
we also introduce a whole host of other variables that are totally unrelated to the intrinsic dynamics of the model, but rather dependent on a number of situations contingent on the U.S.
For this reason, as suggested by Plutosky, we will follow the method of using as a reference value the value of a troy ounce of gold, i.e., the unit of measurement internationally used for trading in gold.
Gold is useful to be used as a reference sample of value because it is transnational, so its value does not depend on local political and economic situations in a single country,
it is not subject to particular inflationary pressures, that is, production is known and and relatively constant, furthermore it has been recognized as a value system for millennia in every part of the world.
The choice of this reference system, i.e., price of BTC/GOLD Oz t immediately takes out of the equation a whole series of external variables that could irreparably distort any kind of long-term valuation.
A sustainable mathematical model: the logistic curveAnother problem with the previous models realized arise from the fact that they use “eternally increasing” curves, which are therefore ontologically wrong. No physical system can grow indefinitely,
since physical resources are by definition finite, and therefore there must be “upper bounds” toward which the growth phenomenon must move asymptotically.
This is a general argument that obviously applies to any kind of physical resource: from the consumption of fossil fuels (which will run out at some point) to any phenomenon of expansion of living populations
(which will have to stop when all the resources necessary for survival are finished) and generically valid for any phenomenon of expansion with use of physical resources.
That kind of problem is well studied and summarized in a sigmoid curve called logistics. It dates back as far as 1838 to Pierre F. Verhulst's studies of logistic growths,
and since then a series of insights and refinements in the study of population growth.
It is obvious that in the progressive adoption of Bitcoin as a universally accepted means of payment, we face precisely to such a problem, namely the growth of the population that is using Bitcoin,
and that there will be an upper bound, that is, a moment when all possible users will have been reached, and therefore the system cannot grow any further.
From this phenomenon, which is commonly referred to as the “adoption of Bitcoin”, depends the value, since the more of people who will use Bitcoin as a means of payment,
the greater its value will be as a result of Metcalfe's law
[3], and also in view of the fact that the total number of Bitcoins is limited to 21,000,000 units.
Accordingly we can consider a direct dependence of Bitcoin's value on the population using it.
Examples of other resources growing according to logistic curves.
In the field of economics, the adoption of virtually every product follows a logistic curve: from the time of its release to the market, until the upper bound of users reachable by that product is reached.
Here are two graphs representing the adoption of some products and services according to a logistic curve:
and here's another one more detailed
[4] that also highlights the concept of “growth pulse”, which is the period of time of market penetration from 10 percent market to the time of 90 percent market penetration.
Locating the logistic equationMathematical analysis provides us with several forms of logistic equations, the simplest being totally symmetrical equations that are in the form
[5]In this case there is a lower bound that is 0 an upper bound that is 1 and an inflection point that is at the coordinate (0,0.5)
Obviously for our type of analysis we need a more malleable type of logistics, as it has to fit to already known constraints:
1) lower bound = 0
2) must follow with good approximation the path of the BTC/GOLD Oz t price so far
3) upper bound configurable.
These kinds of equations exist and are called “asymmetric logistic”; the need for asymmetric logistics comes from the fact that natural phenomena
do not grow following perfect logistics, but they obviously adapt to the environment, and the final shape of the sigmoid is somewhat distorted.
Among the various forms of asymmetric logistic I will use the following for the simplicity of modeling on the problem at hand:
[6]Five parameters can be set in this logistics, and they are:
LL = Lower Bound
LU = Upper Bound
Ix = X of the inflection point
S = Slope, slope of the inflection point
c = Asymmetry factor.
X = independent variable.
Locating the Upper Bound.Identifying an upper bound is obviously the most difficult problem.
There are different estimates of gold mined and gold yet to be mined. To avoid wasting too much time on this topic, lets use a very rough estimate of 200,000 Tons of gold mined + 50,000 Tons yet to be mined.
[7]250,000/31.1035*1000000 is about 8,000,000 Oz t of total gold. Since there are 21,000,000 BTC, to have a parity of capitalization 1 BTC will have to be worth the same as 380 Oz t of gold.
We now set 4 upper bounds, which define the following scenarios:
1) A “pessimistic” upper bound called the Arulbero upper bound, where each BTC will be worth 100 Oz t of gold at the end of the adoption process, that is, the Bitcoin network will capitalize about a quarter of the gold, essentially a failure of the Bitcoin project, though a good result in absolute value.
2) A “conservative” upper bound called the Gbianchi upper bound, where each BTC will be worth at the end of the adoption process 600 Oz t of gold, that is, the Bitcoin network will capitalize 50 percent more than gold.
3) A “moderately optimistic” upper Bound defined as the Plutosky upper bound we set at 1100 Oz t of gold per BTC, i.e., the Bitcoin network will capitalize 200% more than gold.
4) An “optimistic” upper Bound defined upper bound Fillippone we set it at 2200 Oz t gold per BTC, double the previous one, a possible scenario that can safely be called hyperbitcoinization
[8]Final equationsWith the upper bound identified, we have all the elements to solve the parameters of the final logistic equations.
The other parameters are a lower bound = 0 and an optimal fitting with the BTC/GOLD Oz t value to date.
We derive the trend of the BTC/GOLD Oz t price as an optimal fitting constraint, deriving it from the BTC/USD price relative to the Fxing LBMA GOLD Oz t/USD
[9]
So we get these values for the parameters of the equations: the independent variable x represents in all models the days elapsed since 03/01/2009 (Genesis Block)
Arulbero "pessimistic" Upper bound
L_L = 0;
L_U = 100;
I_x = 5700;
S = 0.029;
c = -1;
Gbianchi "conservative" Upper bound
L_L = 0;
L_U = 600;
I_x = 8900;
S= 0.068;
c = -20;
Plutosky "moderately optimistic" Upper bound
L_L = 0;
L_U = 1100;
I_x = 10100;
S = 0.11;
c = -20;
Fillippone Optimistic "hyperbitcoinization" Upper bound
L_L = 0;
L_U = 2200;
I_x = 11400;
S = 0.197;
c = -20;
Moreover I also reparameterized Santostasi's Power Law on BTC Gold Oz t, to have a comparison of the “eternally increasing” trend of the Power Law with sigmoid trends; here is the equation:
val=(10**-18.76)*(x**5.42)
val= BTC price in Gold oz t
here is the graph on a logarithmic scale to 2026, as you can see on this scale all the patterns are practically coincident.
extrapolated graph to 2036 on a linear scale:
Instead, here is the same graph on a 70-year perspective, on a linear scale:
Analysis of inflection pointsOn each curve I have indicated the inflection point with a small rectangle of corresponding color. Note that being asymmetric sigmoids, the inflection point is off-center with respect to the interval of the growth pulse.
In the following table I report dates and values at the inflection points, and a range of the inflection points from -182 days to +182 days i.e. one year, to calculate absolute and percentage price increase in that interval.
| Scenario name | | | Date inflection | | | Value | | | Date inflection- 182 days | | | Value | | | Date inflection+ 182 days | | | Valore | | | Absolute difference | | | % difference |
| ___________________________________ | | | ______________ | | | ________ | | | ______________ | | | ________ | | | ______________ | | | ________ | | | ______________ | | | ______________ |
| Arulbero Upper Bound 100 BTC/GOLD Oz t | | | 2024-08-12 | | | 42.67 | | | 2024-02-12 | | | 37.41 | | | 2025-02-10 | | | 47.93 | | | 10.51 | | | 28.11 |
| Gbianchi Upper Bound 600 BTC/GOLD Oz t | | | 2033-05-17 | | | 220.72 | | | 2032-11-16 | | | 208.35 | | | 2033-11-15 | | | 233.09 | | | 24.73 | | | 11.87 |
| Plutosky Upper Bound 1100 BTC/GOLD Oz t | | | 2036-08-29 | | | 404.66 | | | 2036-02-29 | | | 384.65 | | | 2037-02-27 | | | 424.67 | | | 40.02 | | | 10.40 |
| Fillippone Upper Bound 2200 BTC/GOLD Oz t | | | 2040-03-21 | | | 809.33 | | | 2039-09-21 | | | 773.49 | | | 2040-09-19 | | | 845.17 | | | 71.68 | | | 9.26 |
Final observationsFrom the observation of these patterns, a number of considerations immediately emerge:
1) under
all scenarios (from the pessimistic Arulbero scenario to the reparameterized power law included), there would be a 1 BTC = 50 GOLD Oz t value reached in early 2026, practically a certainty.
2) On the concept of “growth pulse” to visualize the concept better, I have indicated with circles on the sigmoids the point where the curve reaches 10 percent of expansion (ramp start) and the one where it almost reaches saturation, that is, 90 percent of growth.
In the pessimistic arulbero scenario alone, we have already passed the ramp start point at the end of 2021, and we have also passed the inflection point in 2024.
Basically, this scenario tells us that we now have a few years left for the final stabilization to occur around 2031.
In the other three sigmoid scenarios
we have not even reached the ramp start point, i.e. Bitcoin is still in an extremely young adoption and thus growth state.
In the “fillippone” scenario, the growth pulse would reach the 10% ramp start around 2031, reaching near saturation of 90% around 2066, i.e., a period of 35 years of growth pulse.
In the “plutosky” scenario, the growth pulse would reach the 10% ramp start around 2028,and saturation of 90% around 2060, i.e., a period of more than 30 years of growth pulse.
Even the “gbianchi” scenario would have a period of growth pulse starting around 2025 and ending around 2054, or just under 30 years.
All quite different projections from the smartphone growth models of just over 10 years of growth pulse and the internet itself which is just over 20 years of growth pulse
3) The reparameterized “power Law” on gold shows all its weakness in the medium term, as it would exceed hyperbitcoinization in 2042, to continue in addition to grow infinitely.
4) One of the important effects of such modeling is learning to deal with a Bitcoin price based on a comparison with gold and not a fiat currency.
I consider this a very important mental step, for the better understanding of which phenomena are endogenous and which are derived from totally external causes, first of all inflation of the reference fiat.
References[1]
Modeling Bitcoin Value with Scarcity[2]
The Bitcoin Power Law Theory[3]
Metcalfe 's law[4]
Seeing What's Next[5]
Logistic function[6]
Reparametrization of asymmetric logistic function[7]
Chart: How Much Gold is in the World?[8]
Hyperbitcoinization[9]
LBMA: The Independent Precious Metals Authority
AcknowledgementsI would like to thank:
bitbollo who kindly read early drafts of the article and did the english translation,
plutosky who gave me the impetus to revive an old project I had shelved,
fillippone who suggested the inclusion of a “hyperbitcoinization” scenario,
arulbero who suggested the inclusion of a pessimistic scenario and the graphical and tabular highlighting of the inflection points.
Of course errors are my responsibility only.
FAQQ: This is not a model, it's an infinite number of models.A: The logic of the model is based on two fundamentals arguments.
1) using gold as a stable unit of value measurement.
2) to apply Asymmetric Logistics in studying the price of bitcoin, so as to have an “upper bound” and not an eternally increasing result.
Within this logic, 4 “upper bound” scenarios (from the most pessimistic to the most optimistic) were created to study what happens in different cases,
e.g., the duration of the growth pulse , the positions inflection points, the values in the surrounds of the inflection points, all very useful information
for studying the level of plausibility of each scenario.
Q: Using the logistic function is a no-brainer.A: I'm also convinced that mathematically it's an obviousness, I even wrote it down.
But the power law mathematically is also a truism, in fact a logistic initially evolves practically exponentially,
the only difference is that an exponential law grows infinitely, logistics slows growth until it asymptotically approaches an upper bound.
Thank you, but in the end I just applied the kind of curve, logistics, that needs to be applied in such a problem, so mathematically it's a truism.
I consider the principle of starting to think in BTC/GOLD by totally (and mentally) snubbing fiat the true quantum leap.
Q: Using gold or dollars as a unit of measure changes nothing.
A: The dollar is a FIAT currency issued by a state. It is subject to inflationary, political and economic pressures directly dependent on the situation in the U.S. meanwhile totally unrelated to what will happen to bitcoin.
Gold has been used as a unit of value for millennia, and is far less exposed to variations due to local and state phenomena.
To think of using the dollar or any other FIAT as a unit of value is equivalent to thinking that a surveyor should use a rubber band as a yardstick, and doing a medium/long-term model using a FIAT currency is conceptually wrong as an approach.
Q: You have no idea where the saturation, i.e., the upper bound, will be.A: That's true, which is why I've proposed different scenarios with various levels of plausibility.
With my model and the different scenarios I derive the tools (flexes, growth pulse) to test the plausibility of each scenario.
Whereas the plausibility of a model with infinite upper bound is practically zero.
Q: Your model is more complicated than PowerLaw and qundi Occam's razor favors power law.A: The model is not at all more complicated than the PowerLaw , it's a very simple sigmoid calculated always based on the days since the genesis block, exactly the same level of complication as the PowerLaw.
It is true that it introduces the concept of Upper Bound, but it introduces it not to make unnecessary additional assumptions, but to eliminate the infinite growth problem present in the power law.
