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FINANCIAL ECONOMICS

A Bitcoin price prediction model assuming oscillatory growth and lengthening cycles

ORCID Icon & ORCID Icon
Article: 2087287 | Received 01 Oct 2021, Accepted 05 Jun 2022, Published online: 14 Jun 2022

Abstract

This article’s motivation is to understand the volatile Bitcoin price increase. The objective is to develop price estimation methods. The methodology is to present five differential equation models estimated against the 23 July 2010–21 June 2021 Bitcoin data. The findings are that Gompertz growth fits the damped oscillations and lengthening cycles well, and tracks the early data better with the weighted least squares method. Gompertz growth combined with charged capacitor growth tracks the early data even better. Logistic growth is too slow to track the early data. Logistic growth combined with charged capacitor growth to some extent tracks the early data. Pure charged capacitor growth is unrealistic. The dates for the future bull market maxima depend to a low degree on the growth model carrying capacity approached asymptotically, assumed to match gold at $10 trillion, and to be 50 times higher. The implications for traders are to focus on the large standard deviations. Investors should understand the growth potential compared with other asset classes. Regulators should ensure financial stability by focusing on the fluctuations. Central banks should adjust the money supply while acknowledging. Bitcoin competition. Collective units should understand Bitcoin growth models to determine whether to accept Bitcoin transactions.

JEL Classification:

1. Introduction

1.1. Background

Since the genesis block was mined on 3 January 2009 at 18:15:05 UTC, the Bitcoin price has increased above 100% per year subject to fluctuations. Understanding the nature of the growth and fluctuations is of paramount importance. A variety of opinions emerge on how the Bitcoin price evolves into the future. Skeptics believe the Bitcoin price is in a bubble and will collapse. Others see Bitcoin, accompanied with layer 2 solutions for scaling (e.g., the Lightning Network) and layer 3 solutions for interoperability, as the future dominant means of payment, measure of value, medium of exchange, basis of credit, standard of postponed payment, store of value, and possibly unit of account. Other cryptocurrencies may contribute. The decentralized nature of Bitcoin, where anyone can run a node which stores the entire blockchain, emerges as a competitor to traditional media of exchange and stores of value which require an intermediary. Thus for example El Salvador on 7 September 2021 and the Central African Republic on 27 April 2022 accepted Bitcoin as legal tender. Cryptocurrencies and their underlying ledger technologies currently impact how most central banks develop digital currencies. These developments can be expected to reshape the financial system.

1.2. Contribution

This article’s motivation, objectives, research hypotheses, and research questions are as follows. First, the Bitcoin price has increased apparently unpredictably since 3 January 2009, which suggests a need both to understand the evolution so far and to predict the future evolution. Second, applying Bitcoin’s price data since 23 July 2010, methods are developed to estimate and understand this price evolution as accurately as possible. The Bitcoin empirics are such that the methods involve growth models, while accounting for oscillations and lengthening cycles. Third, two different Bitcoin carrying capacities are considered, assumed to express reasonable outer limits for what can be expected over the next decades. Fourth, five differential equation models are compared against each other to determine which is best, applying the least squares method and the weighted least squares method. Fifth, the methods are used to predict the future Bitcoin price and future bull marked maxima.

More specifically, first, a generalized logistic growth model is presented, which depicts the Bitcoin price’s growth with four characteristics: logistic growth, damped oscillation, retracement in bear markets, and lengthening cycles. Second, a generalized charged capacitor growth is introduced with damped oscillation, retracement in bear markets, and lengthening cycles. Third, the article introduces what the authors believe are two hitherto unknown theoretical combinations of growth models, i.e., logistic growth combined with charged capacitor growth, and Gompertz growth combined with charged capacitor growth. This gives five models which are solved analytically and analyzed numerically.

The least squares method is applied to estimate the models’ parameters. Supplementation is made with the weighted least squares method since the Bitcoin price variance increases over time, exhibiting heteroscedasticity. Based on the three bull market local maxima and three bear market local minima during the period 23 July 2010–21 June 2021, the scaling of the inverse of the cycle length of the sine oscillations, and the scaling of the inverse of the degree of lengthening of each subsequent cycle, are estimated. The amplitude of the oscillations, and the start time adjustment parameter for the sine oscillations, are estimated to predict the future bull market local maxima and bear market local minima.

The article outperforms other alternative approaches and adds to our knowledge in various ways. First, the dynamic nature of the Bitcoin price is such that differential time equations are especially well suited. Such differential equations do exist in the literature, but are perhaps in the minority. Second, the Bitcoin price is not only characterized by dynamics, but by dynamic growth. Hence this article focuses explicitly on growth models. Third, the Bitcoin price is characterized by dynamic oscillatory growth with lengthening cycles, which is explicitly incorporated into the analysis.

1.3. Literature

The existing literature predicts the Bitcoin price applying various methods, occasionally using differential equations, and more generally accounting for statistics, econometrics, machine learning, neural network, deep learning, etc. The literature is divided into five groups, i.e., 1. Differential equations, 2. Bitcoin price dynamics, 3. Gompertz growth and Metcalfe’s Law, 4. Machine learning, and 5. Neural network, deep learning and memory models. This article correlates most with the first two groups, while introducing growth with damped oscillatory and lengthening cycles. The last three groups are included for broader positioning.

1.3.1. Differential equations

Relatively few studies apply differential equations to predict the Bitcoin price. K. S. Chen and Huang (Citation2020) adopt a stochastic differential equation to capture the evolution of the Bitcoin price 2015–2018. Their differential equation considers the instantaneous expected return, the instantaneous volatility, and jumps focusing explicitly on the crash after the 17 December 2017 and Brownian motion. They focus on the jump risk distribution of the Bitcoin price and Bitcoin options pricing and hedging. Such a focus on jumps is implicitly present in the current article which determines moves back and forth between bull market maxima and bear market minima. Jalali and Heidari (Citation2020) adopt grey system theory and propose a first order differential equation to predict the Bitcoin price. The approach requires an appropriate time frame. They focus explicitly on five-day predictions. That differs from the current article which predicts over any future time horizon. Wang and Wang (Citation2020) introduce a partial differential equation model to predict the Bitcoin price January 1–31 December 2017. They incorporate the daily Bitcoin transaction volumes and google trends index, and the spatial heterogeneity of chainlet clusters, which proceeds beyond this article’s focus. This article differs from these other articles applying differential equations by focusing explicitly on the Bitcoin price growth patterns. That is, the differential equations consider the Bitcoin price, two different Bitcoin carrying capacities, damped oscillations, lengthening cycles, and bull market maxima and bear market minima for five different growth models.

1.3.2. Bitcoin price dynamics

The following articles pertain to Bitcoin price dynamics, but with a different focus and applying other models than in the current article, thus implicitly illustrating a gap in the literature. Statistics and econometrics are widely used methods to forecast the Bitcoin price. Begusic et al. (Citation2018) demonstrate slowly decaying tails in the distributions of Bitcoin returns, and a power law with 2 < α < 2.5, which means heavier tails than for stocks with alpha around 3. Such slowly decaying tails seem consistent with damped oscillations, and heavy tails seem consistent with the substantial fluctuations between maxima and minima, found in the current article.

Caporale et al. (Citation2019) apply statistical methods for 2013–2018. They find that the frequency of price overreactions is informative about Bitcoin price movements and the Bitcoin price exhibits no seasonality. Their approach constitutes an alternative way of assessing the drive towards bull market maxima and bear market minima.

Roy et al. (Citation2018) apply 2013–2017 data and present an autoregressive integrated moving average model which predicts the Bitcoin price volatility with 90% accuracy, thus also capturing fluctuations between maxima and minima.

Indera et al. (Citation2017) apply 2012–2017 data and develop a multi-layer perceptron-based non-linear autoregressive model to predict the Bitcoin price with good accuracy. They generate moving averages, account for input and output lags, and apply regression analysis, validation and fitting tests. They focus less on the timing and magnitudes of the maxima and minima than the current article.

Cretarola and Figa-Talamanca (Citation2021) apply a continuous time stochastic model to determine how bubbles in the Bitcoin price in 2012–2013 and in 2017 are linked to the correlation between the market attention to Bitcoin and the Bitcoin return being above a threshold, known as market exuberance. Such bubbles are yet another way of assessing bull market maxima. Jana et al. (Citation2021) apply 2013–2019 data to forecast the Bitcoin price through a differential evolution-based regression framework, shown to be superior to six advanced predictive modeling algorithms. Instead of differential equations, they apply polynomial regression on time series.

Further studies consider market attention, market sentiment, active addresses, etc. for Bitcoin price prediction, which is a broader focus than in the current article. Sabalionis et al. (Citation2021) found that the amount of active addresses impacts the Bitcoin and Ethereum prices more than other factors such as google search interest and number of tweets. Haffar and Le Fur (Citation2021) applied a structural vector error correction model to determine that the Bitcoin price in the short run is influenced positively by Asian emerging countries and negatively by North America. In the long run, the influence is negative from all countries in Asia and the Pacific, and positive from Europe. This article adopts a wider range of Bitcoin price data than the above articles, applying growth models to explain and predict the Bitcoin price.

1.3.3. Gompertz growth and Metcalfe’s Law

The quick initial increase in Gompertz growth (commonly used for e.g. tumor growth; see Yorke et al. (Citation1993)) is found to be descriptive in the current article. Two other articles have also identified Gompertz growth as descriptive. Peterson (Citation2018) applies the Gompertz curve to capture the inflationary impacts of the creation of new Bitcoin, shown to follow Metcalfe’s Law. Patel et al. (Citation2020) found that the price of cryptocurrencies follows a Gompertz growth function, which links the traditional time-value-of-money concepts to Metcalfe’s law, and that the growth rate of users impacts the Bitcoin price. This article extends this focus to other growth models, i.e., logistic growth, charged capacitor growth, and combinations of growth models, accounting for damped oscillation and lengthening cycles.

1.3.4. Machine learning

Several studies apply machine learning methods to explain and predict the Bitcoin price. Chevallier et al. (Citation2021) applied six machine learning algorithms to parameterize and disentangle the non-stationary behavior of the Bitcoin price data, as an alternative to classical parameter models. They suggest that machine learning does not teach how to trade due to the substantial Bitcoin price variability, and that long term holding may be preferable. Such a suggestion seems compatible with the current article which determines overall 2010–2021 growth, interrupted by substantial downturns towards bear market minima. Dutta et al. (Citation2020) present a framework of machine learning forecasting methods to predict the Bitcoin’s price. They compare various approaches, arguing that the gated recurring unit model with a recurrent dropout performs best. Z. Chen et al. (Citation2020) predict the Bitcoin price with various frequencies data via machine learning techniques. They also incorporate high-dimension features like property and network, trading and market, attention, etc. They show that statistical methods perform better than machine learning algorithms, reaching accuracy of 66% and 65.3%, respectively.

Some articles combine machine learning and econometrics. Mudassir et al. (Citation2020) developed high-performance machine learning-based classification and regression models to predict the Bitcoin price. The models have accuracy of 65% and 64% for next-day forecast and seventh–ninetieth-day forecast, respectively. Gupta and Nain (Citation2021) use time series involving moving averages, autoregressive integrated moving averages, and multiple machine learning approaches including Support Vector Machine, Long Short Term Memory and Gated Recurrent Unit. They compare these models to determine their accuracy. The machine learning approach is challenging since it requires appropriate data input. Long term forecasting is challenging. Instead of machine learning, this article applies differential equations and least squares methods which more directly cause price explanation and prediction.

1.3.5. Neural network, deep learning and memory models

Recent articles adopt neural networks and deep learning to predict the Bitcoin price, which enables the analysis of instructed data including documents, images, and texts. Ji et al. (Citation2019) explored the performance of a deep neural network model, a Long Short-Term Memory model, and a Convolutional Neural Network model, to predict the Bitcoin price. They show that the deep neural network model predicts price increases and decreases nicely, and that classification models are more effective than regression models. Patel et al. (Citation2020) present a Long Short Term Memory and Gated Recurrent Unit based hybrid cryptocurrency prediction model to predict the price of Litecoin and Monero. They found that the model accurately predicts the prices.

Hua (Citation2020) compares the accuracy of predicting the Bitcoin price via Long Short Term Memory model and an Autoregressive Integrated Moving Average model. He finds that the former performs better, but requires more time to train the neural network. Cocco et al. (Citation2021) compared several approaches to predict the Bitcoin price. They show that two-stage frameworks usually outperform one-stage frameworks, except for one-stage Bayesian Neural Network. Jaquart et al. (Citation2021) proposed a stochastic neural network model based on random walk to predict the price of cryptocurrencies. The approach induces a layer-wise randomness into the neural networks to capture market volatility. Using multi-layer perceptron and Long Short Term memory models, they found that the proposed models perform well compared with deterministic models.

Chkili (Citation2021) applies a long memory model and a Markov switching model to determine the Bitcoin price volatility, which relates to the focus in the current article of assessing fluctuations between maxima and minima. A common challenge faced by the deep learning approach is finding the optimal network hyperparameters. That contrasts with the current article which applies least squares methods to estimate the parameters.

1.4 Article organization

Section 2 presents the materials and methods. Section 3 analyzes the model and presents the results. Section 4 discusses the results. Section 5 concludes.

2. Materials and methods

This section identifies and develops the differential equations believed to capture the Bitcoin price evolution most accurately.

2.1. Nomenclature

Parameters

kGrowth rate, kR

KCarrying capacity, K0

νParameter for generalized logistic growth impacting near which asymptote maximum growth occurs, ν0

λAdjustment parameter for combined generalized logistic and charged capacitor growth, λR

αOscillation amplitude, expressing strength of bull and bear markets, αR

ωScaling of the inverse of the cycle length of the sine oscillations, ωR

γScaling of the inverse of the degree of lengthening of each subsequent cycle, γR

δStart time adjustment parameter relative to time t=t0 for the oscillation of the Sin function, γR

t0Initial time t

TFinal time t

p0Initial price p at time t=t0

Independent variable

tTime

Dependent variable

pPrice

2.2. Generalized logistic growth

This section generalizes Richards’ (Citation1959) model for growth modeling to

(1) pt=k+αSinωtt0γ+δp1pKvp=K1+Kp01νek+Qνtt01/ν,limtp=K,Qiα2γeiδ1eiωtt0γqqγ1γdqeiδ1eiωtt0γqqγ1γdq,i1(1)

where means partial differentiation, t means time, t0 is the start time, kR is the growth rate which expresses how quickly the price p changes, and K0 is the carrying capacity, defined as the maximum sustainable price p. EquationEquation (1) expresses that the price p changes logistically from p0, p00, at the initial time t=t0 towards p=K as time t approaches infinity. The parameter ν, ν0, impacts near which asymptote maximum growth occurs.

Whereas Richards (Citation1959) assumes a constant growth rate, (1) supplements the growth rate k with a Sin function and four additional parameters. The Sin function oscillates between +1 and −1 to reflect bull markets with increased growth rate when the Sin function is positive, and bear markets with decreased growth rate when the Sin function is negative.

The parameter αR expresses the strength of the bull and bear markets, and thus the size of the positive and negative amplitudes in the oscillations. EquationEquation (1) simplifies to Richards' (Citation1959) model when α=0 which eliminates the sine oscillations causing Q=0.

The parameter ωR scales the inverse of the cycle length of the sine oscillations. Higher ω gives shorter cycle length, since ω is multiplied with time t, and higher ω means that each cycle with length 2π gets completed more quickly.

The parameter γR scales the inverse of the degree of lengthening of each subsequent cycle as time t progresses. Lower γ gives more lengthening of each subsequent cycle as time t progresses. Mathematically, if we consider γ=1 as a common benchmark giving linear oscillatory progression through time t, decreasing gamma below 1 causes ωtt0γ<ωtt0, and hence more time t is needed for each subsequent cycle with length 2π to be completed. In contrast, increasing gamma above 1 causes ωtt0γ>ωtt0, and hence less time t is needed for each subsequent cycle with length 2π to be completed. EquationEquation (1) also simplifies to Richards’ (Citation1959) model at the limit when limγQ=0, since each subsequent cycle gets completed immediately and thus the sine oscillations have no impact.

The parameter δR adjusts the start time at time t=0 for the oscillation of the Sin function. For example, if δ=0, Sinωtγ+δ=0 when t=0, which gives zero amplitude and thus no impact of the Sin function at time t=0.

2.2.1. Conventional logistic growth

Inserting ν=1 into (1) gives conventional logistic growth (Lotka, Citation1924; Verhulst, Citation1845) with oscillation, retracement in bear markets, and lengthening cycles, where both the initial value asymptote t=t0 and the future value asymptote t are approached symmetrically.

2.2.2 Gompertz growth

Inserting the limit ν>0+ into (1) gives conventional Gompertz (Citation1825, p. 518) logistic growth with oscillation, retracement in bear markets, and lengthening cycles, i.e.,

(2) pt=k+αSinωtt0γ+δpLnKpp=Kp0Kek+Rtt0,Riα2γtt0ω1γeiπ+2γδ2γitt0γωq1γ1eqdq1eiπ+2γδ2γitt0γωq1γ1eqdq1(2)

where Ln is the natural logarithm, and the initial value asymptote t=t0 is approached more quickly than the future value asymptote t. EquationEquation (2) simplifies to Richards’ (Citation1959) model when R=0.

2.3. Generalized charged capacitor growth

This section assumes generalized charged capacitor growth with damped oscillation, retracement in bear markets, and lengthening cycles, with growth rate k+αSinωtt0γ+δ, i.e.,

(3) pt=k+αSinωtt0γ+δ1pKνp=InverseFunction2F11,1v,1+1v,#Kν#&k+Qtt0p02F11,1v,1+1v,p0Kν,limtp=K(3)

where Q is defined in (1), which simplifies to

(4) pt=k+αSinωtt0γ+δ1pKp=KKp0ek+Qtt0/K,limtp=K(4)

when ν=1, and simplifies to pt=0p=p0, i.e., no growth, when ν=0. The function 2F1a,b,c,z is hypergeometric with power series assuming z<1. The Mathematica (www.wolfram.com) notation in (3) is as follows: InverseFunctionf is the inverse of the function f, defined so that InverseFunctionfy gives the value of x for which fx=y. The symbol # is the first argument supplied to a pure function, so that f#&x evaluates to fx. The symbol & expresses the end of the argument. Charged capacitor growth expresses a quick initial price increase, due to the high value of the right-hand side of the differential EquationEquation (3) when p0 is low.

2.4. Combined generalized logistic and charged capacitor growth

Combining (3) and (1) gives

(5) pt=k+αSinωtt0γ+δpλ1pKνp=InverseFunction#1λ1λ2F11,1λv,1+1λv,#Kν&k+Qtt0p01λ1λ2F11,1λv,1+1λv,p0Kν,limtp=K(5)

where the price p is raised to an exponent λ, λR, and 2F1a,b,c,z is defined in (3). The adjustment parameter λ can be thought of as weighing generalized logistic growth (conventional logistic growth and Gompertz growth) and charged capacitor growth against each other. EquationEquation (5) simplifies to generalized logistic growth in (1) when λ=1, simplifies to Gompertz growth in (2) when λ=1 at the limit v0+, and simplifies to generalized charged capacitor growth in (3) when λ=0. Hence 0<λ<1 enables growth intermediate between quick generalized charged capacitor growth when λ=0, and slower generalized logistic growth when λ=1. Compared with generalized logistic growth in (4) when λ>1, initial growth when the price p is low is damped since pλ in (5) is comparatively low, and eventual growth is amplified more when the price p is high since pλ in (5) is comparatively high.

2.4.1 Combined logistic and charged capacitor growth

Inserting ν=1 into (5) gives combined conventional logistic growth and generalized charged capacitor growth expressed as

(6) pt=k+αSinωtt0γ+δpλ1pKp=InverseFunction#1λK1λ2F11,1λ,2λ,#K&k+Qtt0K+Kλ0p0/Kqλ1qdq,limtp=K(6)

2.4.2 Combined Gompertz and charged capacitor growth

Inserting the limit v0+ into (5) gives combined Gompertz growth and generalized charged capacitor growth expressed as

(7) pt=k+αSinωtt0γ+δpλLnKpp=InverseFunctione1λLnK1λLnK#eqqdq&k+Qtt0+K1λ1λLnKp0eqqdq,limtp=K(7)

3. Results

The results are presented over seven subsections. Sections 3.1 and 3.2 assume no oscillation and estimate and predict the Bitcoin price with the various methods developed in the previous section. Sections 3.3 and 3.4 generate results needed to account for oscillation. These results pertain to Bitcoin price maxima and minima, cycle length and cycle lengthening. Sections 3.5 and 3.6 allow for oscillation and predict the Bitcoin price with the various methods. Section 3.7 estimates future bull market maxima.

More specifically, section 3.1 estimates the Bitcoin price assuming no oscillation amplitude α = 0 and the Bitcoin carrying capacity K = $476,190 which corresponds to Bitcoin eventually approaching the market capitalization of gold estimated at $10 trillion. Section 3.2 repeats the exercise for the 50 times higher Bitcoin carrying capacity K = $23,809,524 which corresponds to Bitcoin eventually approaching a market capitalization of $500 trillion. Section 3.3 determines the three bull market local maxima and the three bear market local minima which have been established at the writing of this article. Section 3.4 estimates the scaling ω of the inverse of the cycle length of the sine oscillations and the scaling γ of the inverse of the degree of lengthening of each subsequent cycle. Section 3.5 allows for oscillation amplitude α ≥ 0 as determined by the previous two subsections, and estimates the Bitcoin price assuming the carrying capacity K = $476,190. Section 3.6 repeats the exercise for the 50 times higher Bitcoin carrying capacity K = $23,809,524. Section 3.7 estimates bull market local maxima 4,5,6,7,8 assuming the carrying capacities K = $476,190 and K = $23,809,524.

3.1. Bitcoin carrying capacity k=$476,190 and no oscillation amplitude α=0

The Bitcoin carrying capacity K is estimated as the maximum sustainable market capitalization divided by the circulating supply. The Bitcoin circulating supply is capped at 21 million coins, expected to be mined by ca year 2140. Estimating Bitcoin’s maximum sustainable market capitalization is extremely uncertain. This section assumes that Bitcoin approaches a maximum sustainable market capitalization of $10 trillion, which is similar to the market capitalization of gold.Footnote1 The comparison with gold is made since it would constitute a major milestone if Bitcoin were to reach it. Dividing $10 trillion with 21 million coins gives the Bitcoin carrying capacity K=$476,190. Using daily midnight 11:59.99 pm UTC closing time Bitcoin data,Footnote2 the initial Bitcoin price at the initial time 23 July 2010 is p0=$0.04951. Figure shows the empirical price pE for the period 23 July 2010–21 June 2021, which increases overall, with intermittent decreases.Footnote3

Figure 1. Assuming no oscillation amplitude α=0, the empirical price pE, logistic growth pL, Gompertz growth pG, charged capacitor growth pC, combined logistic and charged capacitor growth pLC and pLCλ, and combined Gompertz and charged capacitor growth pGC and pGCλ, for 23 July 2010–21 June 2021 (panels a and c) and until 1 January 2100 (panels b and d), K=$476,190. Panels a and b apply the least squares method. Panels c and d apply the weighted least squares method.

Figure 1. Assuming no oscillation amplitude α=0, the empirical price pE, logistic growth pL, Gompertz growth pG, charged capacitor growth pC, combined logistic and charged capacitor growth pLC and pLCλ, and combined Gompertz and charged capacitor growth pGC and pGCλ, for 23 July 2010–21 June 2021 (panels a and c) and until 1 January 2100 (panels b and d), K=$476,190. Panels a and b apply the least squares method. Panels c and d apply the weighted least squares method.

The subsequent seven curves in each panel in assume α=0, K=$476,190 and p0=$0.04951 and estimate the growth rate k, and the two parameters λ and β, for the models in section 2.2. These seven curves increase strictly, in contrast to the empirical price pE, due to the nature of growth models. In sections 3.5 and 3.6 oscillatory growth is modeled.

3.1.1. Least squares method

Applying the least squares method, ) shows the historical estimates. ) predicts until 1 January 2100. Using data over ca 11 years to predict ca 79 more years into the future, i.e., a ratio 79/11 ≈ 7.2, entails some uncertainty for the more distant future.

The curve pL estimates the growth rate k=1.28 by assuming logistic growth in (1) for ν=1, approaching K=$476,190 more quickly than the other six curves. The curve pL appears almost linear on a logarithmic plot with base 10.

The curve pG estimates the growth rate k=0.16 by assuming Gompertz growth in (2). The curve pG is concave to reflect that the future value asymptote t is approached more gradually than the initial value asymptote t=t0. That is, initial growth is quick, and K=$476,190 is approached more slowly.

The curve pC estimates the growth rate k=1178 by assuming charged capacitor growth in (4). The curve pC is extremely concave. It initially increases more quickly than the other six curves, and eventually approaches K=$476,190 more slowly than the other six curves.

The curve pLC estimates the growth rate k=1.21 and adjustment parameter λ=1.02 by assuming combined logistic and charged capacitor growth in (6). Since λ>1, initial growth for the curve pLC is slower than for the curve pL for logistic growth, see section 2.4.

The curve pLCλ is intermediate between the curve pL for logistic growth and pC for charged capacitor growth. This is obtained by assuming λ=0.88 and using the least squares method to optimize the growth rate k which gives k=2.18. The curve pLCλ is similar to Gompertz growth pG.

The curve pGC estimates the growth rate k=0.08 and adjustment parameter λ=1.17 by assuming combined Gompertz and charged capacitor growth in (7). Since λ>1, initial growth for the curve pGC is slower than for the curve pG for Gompertz growth, see section 2.4.

The curve pGCλ is intermediate between the curve pG for Gompertz growth and pC for charged capacitor growth. This is obtained by assuming λ=0.88 and using the least squares method to optimize the growth rate k which gives k=0.32. The curve pGCλ initially increases more quickly than all the other curves except the curve pC for charged capacitor growth.

3.1.2. Weighted least squares method

Applying the weighted least squares method, ) shows the historical estimates. ) predicts until 1 January 2100. The Bitcoin data exhibits heteroscedasticity so that the variance increases over time. That is, the Bitcoin price was $0.04951 on 23 July, 2010, with a few cents variation over the subsequent months until $1 was exceeded on 17 February 2011. In contrast, the Bitcoin price was $32,950 on 21 June 2021, with several thousand US$ variation over the preceding months until $1 was exceeded 17 February 2011. Hence the least squares method is more influenced by recent data than early data. This section assigns more weight to the earlier data by dividing each squared difference (between the model prediction and the data) at each time t with the 20-week moving variance in the data, i.e., the variance over 140 days from time t to time t+139. The variance calculation is constrained by the final time T so that at time T1 the variance over only the two final days at T1 and T is determined.

The two curves pL for logistic growth and pG for Gompertz growth have the same and slightly lower growth rates k=1.28 and k=0.15 as ) in section 3.1.1.

The curve pC for charged capacitor growth has the much lower growth rate k=4.32. That is because the early data is weighed more heavily, and more recent data is discounted. Hence the model prediction is worse for the more recent data, and the curve pC needs more time to approach the carrying capacity K=$476,190.

The curve pLC for combined logistic and charged capacitor growth estimates the higher growth rate k=2.33 and lower adjustment parameter λ=0.85, compared with Figure ). Weighing the early data more heavily causes more rapid initial growth.

The curve pLCλ for combined logistic and charged capacitor growth when λ=0.88 has the lower growth rate k=2.10 compared with ), since it becomes less important to adjust to the recent data.

The curve pGC for combined Gompertz and charged capacitor growth estimates the higher growth rate k=0.17 and lower adjustment parameter λ=0.97, compared with . Weighing the early data more heavily causes more rapid initial growth.

The curve pGCλ for combined Gompertz and charged capacitor growth when λ=0.88 has the lower growth rate k=0.22 compared with ), since it becomes less important to adjust to the recent data.

3.2. Bitcoin carrying capacity K=$23,809,524 and no oscillation amplitude α=0

As an alternative, assume that Bitcoin in the future eradicates all or most other digital currencies, overtakes gold, bonds, and most other assets except physical real estate and various other physical assets. That may suggest a maximum sustainable market capitalization of $500 trillion, which may account for future inflation of the US$. Dividing $500 trillion with 21 million coins gives the Bitcoin carrying capacity K=$23,809,524. Figure replicates Figure for K=$23,809,524.

Figure 2. Assuming no oscillation amplitude α=0, the empirical price pE, logistic growth pL, Gompertz growth pG, charged capacitor growth pC, combined logistic and charged capacitor growth pLC and pLCλ, and combined Gompertz and charged capacitor growth pGC and pGCλ, for 23 July 2010–21 June 2021 (panels a and c) and until 1 January 2100 (panels b and d), K=$23,809,524. Panels a and b apply the least squares method. Panels c and d apply the weighted least squares method.

Figure 2. Assuming no oscillation amplitude α=0, the empirical price pE, logistic growth pL, Gompertz growth pG, charged capacitor growth pC, combined logistic and charged capacitor growth pLC and pLCλ, and combined Gompertz and charged capacitor growth pGC and pGCλ, for 23 July 2010–21 June 2021 (panels a and c) and until 1 January 2100 (panels b and d), K=$23,809,524. Panels a and b apply the least squares method. Panels c and d apply the weighted least squares method.

The subsequent seven curves in each panel in assume α=0, K=$23,809,524 and p0=$0.04951 and estimate the growth rate k, and the two parameters λ and β, for the models in section 2.2, using the least squares method. ) shows the historical estimates. Figure ) predicts until 1 January 2100.

3.2.1. Least squares method

Applying the least squares method, ) shows the historical estimates. ) predicts until 1 January 2100. gives similar parameter estimates to those in .

The curve pL estimates slightly lower growth rate k=1.27 compared with for logistic growth in (1) for ν=1.

The curve pG estimates lower growth rate k=0.10 compared with for Gompertz growth in (2).

The curve pC estimates slightly lower growth rate k=1171 compared with for charged capacitor growth in (4).

The curve pLC estimates slightly higher growth rate k=1.23 and slightly lower adjustment parameter λ=1.01, compared with for combined logistic and charged capacitor growth in (6).

The curve pLCλ estimates slightly lower growth rate k=2.17 when assuming the same adjustment parameter λ=0.88, compared with for combined logistic and charged capacitor growth in (6).

The curve pGC estimates higher growth rate k=0.14 and lower adjustment parameter λ=0.94 compared with for combined Gompertz and charged capacitor growth in (7).

The curve pGCλ estimates lower growth rate k=0.20 compared with Figure when assuming adjustment parameter λ=0.88, for combined Gompertz and charged capacitor growth in (7).

3.2.2. Weighted least squares method

Applying the same weighted least squares method as in section 3.1.2, ) shows the historical estimates. ) predicts until 1 January 2100.

The two curves pL for logistic growth and pG for Gompertz growth have the same and slightly higher growth rates k=1.28 and k=0.11 compared with Figure ) in section 3.2.1.

The curve pC for charged capacitor growth has the much lower growth rate k=4.32. That is because the early data is weighed more heavily, and more recent data is discounted. Hence the model prediction is worse for the more recent data, and the curve pC needs more time to approach the carrying capacity K=$23,809,524.

The curve pLC for combined logistic and charged capacitor growth estimates the higher growth rate k=2.33 and lower adjustment parameter λ=0.85, compared with ). Weighing the early data more heavily causes more rapid initial growth.

The curve pLCλ for combined logistic and charged capacitor growth when λ=0.88 has the lower growth rate k=2.10 compared with ), since it becomes less important to adjust to the recent data.

The curve pGC for combined Gompertz and charged capacitor growth estimates slightly lower growth rate k=0.13 and the same adjustment parameter λ=0.94, compared with ).

The curve pGCλ for combined Gompertz and charged capacitor growth when λ=0.88 has the lower growth rate k=0.15 compared with ), since it becomes less important to adjust to the recent data.

3.3. Determining the three bull market local maxima and the three bear market local minima

The three bull market local maxima since 23 July 2010, are as follows:

$29.6 on 14 June, 2011, expressed as t1max=2011.449315, i.e., 327 days after the start date 23 July 2010 which is day 1.

$1131.992853 on 29 November, 2013 expressed as t2max=2013.909589, i.e., 1226 days after the start date 23 July 2010 which is day 1.

$19,378.35059 on 16 December, 2017 expressed as t3max=2017.956164, i.e., 2704 days after the start date 23 July 2010 which is day 1.

This gives 1226–327 = 899 days, i.e., 2.46027 years, from bull market local maximum 1 to bull market local maximum 2, and 2704–1226 = 1478 days, i.e., 4.04657 years, from bull market local maximum 2 to bull market local maximum 3.

The three bear market local minima since 23 July 2010 are as follows:

$2.2 on 20 November, 2011, expressed as t1min=2011.884932, i.e., 486 days after the start date 23 July 2010 which is day 1.

$178.712075 on 14 January, 2015 expressed as t2min=2015.035616, i.e., 1637 days after the start date 23 July 2010 which is day 1.

$3226.92952 on 14 December, 2018 expressed as t3min=2018.950685, i.e., 3067 days after the start date 23 July 2010 which is day 1.

This gives 1637–486 = 1151 days, i.e., 3.15068 years, from bear market local minimum 1 to bear market local minimum 2, and 3067–1637 = 1430 days, i.e., 3.915069 years, from bear market local minimum 2 to bear market local minimum 3.

The modeling assumes oscillations in the sense that a maximum is followed by a minimum, then a new maximum, etc.

3.4. Estimating the scaling ω of the inverse of the cycle length of the sine oscillations and the scaling γ of the inverse of the degree of lengthening of each subsequent cycle

This section estimates the scaling ω of the inverse of the cycle length of the sine oscillations, and the scaling γ of the inverse of the degree of lengthening of each subsequent cycle. The oscillatory growth rate with damped oscillation, retracement in bear markets, and lengthening cycles in all the equations in section 2 contain the sine of ωtt0γ+δ. One cycle has time length 2π. Hence the two equations

(8) ωt2maxt0γ+δωt1maxt0γ+δ=2π,ωt3maxt0γ+δωt2maxt0γ+δ=2π(8)

express the time length from bull market local maximum 1 to bull market local maximum 2, and the time length from bull market local maximum 2 to bull market local maximum 3, respectively. Solving (8) by using t1max, t2max, t3max from section 3.3 gives ωmax=7.05885 and γmax=0.499872.

Analogously, the two equations

(9) ωt2mint0γ+δωt1mint0γ+δ=2π,ωt3mint0γ+δωt2mint0γ+δ=2π(9)

express the time length from bear market local minimum 1 to bear market local minimum 2, and the time length from bear market local minimum 2 to bear market local minimum 3, respectively. Solving (9) by using t1min, t2min, t3min from section 3.3 gives ωmin=3.45348 and γmin=0.744082. The average of ωmax and ωmin is ω=5.25616. The average of γmax and γmin is γ=0.621977.

3.5. Bitcoin carrying capacity K=$476,190 and oscillation amplitude α>0

This section assumes positive oscillation amplitude α0 and assumes the same k, γ, and β as when α=0. Since ω=5.25616 and γ=0.621977 are estimated in the previous section, we only have to estimate α and δ. shows the empirical price pE for the period 23 July 2010–21 June 2021.

Figure 3. Assuming oscillation amplitude α0, the empirical price pE, logistic growth pL, Gompertz growth pG, charged capacitor growth pC, combined logistic and charged capacitor growth pLC and pLCλ, and combined Gompertz and charged capacitor growth pGC and pGCλ, for 23 July 2010–21 June 2021 (panels a and c) and until 1 January 2040 (panels b and d), K=$476,190. Panels a and b apply the least squares method. Panels c and d apply the weighted least squares method.

Figure 3. Assuming oscillation amplitude α≥0, the empirical price pE, logistic growth pL, Gompertz growth pG, charged capacitor growth pC, combined logistic and charged capacitor growth pLC and pLCλ, and combined Gompertz and charged capacitor growth pGC and pGCλ, for 23 July 2010–21 June 2021 (panels a and c) and until 1 January 2040 (panels b and d), K=$476,190. Panels a and b apply the least squares method. Panels c and d apply the weighted least squares method.

3.5.1. Least squares method

Applying the least squares method, ) shows the historical estimates. ) predicts until 1 January 2040.

The curve pL estimates the oscillation amplitude α=0.48 and start time adjustment parameter δ=3.90 for logistic growth in (1) for ν=1 and k=1.28. The oscillation amplitude α is moderate relative to the almost linear curve in .

The curve pG estimates the oscillation amplitude α=0.37 and start time adjustment parameter δ=4.29 for Gompertz growth in (2) when k=0.16. The oscillation amplitude α is higher than for the curve pL, and conforms with the bull and bear markets.

The curve pC estimates the oscillation amplitude α=7209 and start time adjustment parameter δ=5.03 for charged capacitor growth in (4) when k=1178. The curve pC starts with extreme concavity, thereafter oscillates according to the bull and bear markets, and eventually approaches K=$476,190 slowly.

The curve pLC estimates the oscillation amplitude α=0.39 and start time adjustment parameter δ=3.05 for combined logistic and charged capacitor growth in (6) when k=1.21 and λ=1.02. The curve pLC oscillates similarly to Gompertz growth pG.

The curve pLCλ estimates the oscillation amplitude α=3.93 and start time adjustment parameter δ=4.12 for combined logistic and charged capacitor growth in (6) when k=2.18 and λ=0.88. The curve pLCλ oscillates similarly to Gompertz growth pG.

The curve pGC estimates the oscillation amplitude α=0.04 and start time adjustment parameter δ=5.05 for combined logistic and charged capacitor growth in (6) when k=0.08 and λ=1.17 . The curve pGC initially grows slower than all the other curves.

The curve pGCλ estimates the oscillation amplitude α=0.91 and start time adjustment parameter δ=4.55 for combined Gompertz and charged capacitor growth in (7) when k=0.32 and λ=0.88. The curve is intermediate between Gompertz growth pG and combined logistic and charged capacitor growth pLC on the one hand, and charged capacitor growth pC on the other hand. The curve pGC conforms with the bull and bear markets.

3.5.2. Weighted least squares method

Applying the weighted least squares method, ) shows the historical estimates. ) predicts until 1 January 2040.

The curve pL estimates the oscillation amplitude α=1.42 and start time adjustment parameter δ=2.31 for logistic growth in (1) for ν=1 and k=1.28. The oscillation amplitude α is higher and the start time adjustment parameter δ is lower compared with ).

The curve pG estimates the oscillation amplitude α=0.08 and start time adjustment parameter δ=2.48 for Gompertz growth in (2) when k=0.15. Both the oscillation amplitude α and the start time adjustment parameter δ are lower compared with ).

The curve pC estimates the oscillation amplitude α=3.13 and start time adjustment parameter δ=4.28 for charged capacitor growth in (4) when k=4.32. The oscillation amplitude α is substantially lower, impacted by the much lower growth rate k=4.32, and the start time adjustment parameter δ is lower, compared with ). The curve pC eventually approaches K=$476,190 slowly.

The curve pLC estimates the oscillation amplitude α=1.88 and start time adjustment parameter δ=3.24 for combined logistic and charged capacitor growth in (6) when k=2.33 and λ=0.85. Both the oscillation amplitude α and the start time adjustment parameter δ are higher compared with ). The curve pLC oscillates similarly to Gompertz growth pG.

The curve pLCλ estimates the oscillation amplitude α=1.65 and start time adjustment parameter δ=3.02 for combined logistic and charged capacitor growth in (6) when k=2.10 and λ=0.88. Both the oscillation amplitude α and the start time adjustment parameter δ are lower compared with ). The curve pLCλ also oscillates similarly to Gompertz growth pG.

The curve pGC estimates the oscillation amplitude α=0.15 and start time adjustment parameter δ=3.26 for combined logistic and charged capacitor growth in (6) when k=0.17 and λ=0.97 . The oscillation amplitude α is higher and the start time adjustment parameter δ is lower compared with ). The curve pGC also oscillates similarly to Gompertz growth pG.

The curve pGCλ estimates the oscillation amplitude α=0.20 and start time adjustment parameter δ=3.93 for combined Gompertz and charged capacitor growth in (7) when k=0.22 and λ=0.88. Both the oscillation amplitude α and the start time adjustment parameter δ are lower compared with ). The curve pGCλ initially oscillates around higher values than the other curves except charged capacitor growth pC.

3.6. Bitcoin carrying capacity K=$23,809,524 and oscillation amplitude α>0

This section replicates the previous section with the higher Bitcoin carrying capacity K=$23,809,524. replicates for K=$23,809,524.

Figure 4. Assuming oscillations α0, the empirical price pE, logistic growth pL, Gompertz growth pG, charged capacitor growth pC, combined logistic and charged capacitor growth pLC and pLCλ, and combined Gompertz and charged capacitor growth pGC and pGCλ, for 23 July 2010–21 June 2021 (panels a and c) and until 1 January 2040 (panels b and d), K=$23,809,524. Panels a and b apply the least squares method. Panels c and d apply the weighted least squares method.

Figure 4. Assuming oscillations α≥0, the empirical price pE, logistic growth pL, Gompertz growth pG, charged capacitor growth pC, combined logistic and charged capacitor growth pLC and pLCλ, and combined Gompertz and charged capacitor growth pGC and pGCλ, for 23 July 2010–21 June 2021 (panels a and c) and until 1 January 2040 (panels b and d), K=$23,809,524. Panels a and b apply the least squares method. Panels c and d apply the weighted least squares method.

3.6.1. Least squares method

Applying the least squares method, ) shows the historical estimates. ) predicts until 1 January 2040. ) gives similar parameter estimates to those in .

The curve pL estimates lower oscillation amplitude α=0.34 and slightly higher start time adjustment parameter δ=4.00 for logistic growth compared with , assuming ν=1 and k=1.27.

The curve pG estimates lower oscillation amplitude α=0.17 and slightly higher start time adjustment parameter δ=4.34 for Gompertz growth compared with , assuming k=0.10.

The curve pC estimates slightly lower oscillation amplitude α=7200 and slightly lower start time adjustment parameter δ=5.00 for charged capacitor growth compared with , assuming k=1171.

The curve pLC estimates lower oscillation amplitude α=0.22 and higher start time adjustment parameter δ=3.69 for combined logistic and charged capacitor growth compared with , assuming k=1.23 and λ=1.01.

The curve pLCλ estimates lower oscillation amplitude α=3.58 and lower start time adjustment parameter δ=4.08 for combined logistic and charged capacitor growth compared with , assuming k=2.17 and λ=0.88.

The curve pGC estimates substantially higher oscillation amplitude α=0.29 and lower start time adjustment parameter δ=4.22 for combined logistic and charged capacitor growth compared with , assuming k=0.14 and λ=0.94.

The curve pGCλ estimates the same oscillation amplitude α=0.91 and the same start time adjustment parameter δ=4.55 for combined Gompertz and charged capacitor growth compared with , assuming k=0.2 and λ=0.88.

3.6.2. Weighted least squares method

Applying the weighted least squares method, ) shows the historical estimates. ) predicts until 1 January 2040.

The curve pL estimates substantially higher oscillation amplitude α=1.53 and lower start time adjustment parameter δ=2.38 compared with ) for logistic growth, assuming ν=1 and k=1.27 as in .

The curve pG estimates lower oscillation amplitude α=0.11 and lower start time adjustment parameter δ=3.40 compared with ) for Gompertz growth, assuming k=0.11 as in Figure .

The curve pC estimates substantially lower oscillation amplitude α=3.13 and lower start time adjustment parameter δ=4.28 compared with ) for charged capacitor growth, assuming k=4.32 as in . The difference between ) and ) is similar to the difference between ) and ).

The curve pLC estimates substantially higher oscillation amplitude α=1.66 and lower start time adjustment parameter δ=3.23 compared with Figure ) for combined logistic and charged capacitor growth, assuming k=2.33 and λ=0.85 as in ).

The curve pLCλ estimates lower oscillation amplitude α=1.65 and lower start time adjustment parameter δ=3.00 compared with ) for combined logistic and charged capacitor growth, assuming k=2.10 and λ=0.88 as in .

The curve pGC estimates lower oscillation amplitude α=0.15 and start time adjustment parameter δ=3.5 compared with ) for combined logistic and charged capacitor growth, assuming k=0.13 and λ=0.94 as in .

The curve pGCλ estimates the much lower oscillation amplitude α=0.20 and lower start time adjustment parameter δ=3.93 compared with ) for combined Gompertz and charged capacitor growth, assuming k=0.15 and λ=0.88 as in .

3.7. Estimating bull market local maxima 4,5,6,7,8 when K=$476,190 and K=$23,809,524

predicts the dates and magnitudes of the five future Bitcoin bull market local price maxima, assuming K=$476,190 and K=$23,809,524, and assuming logistic growth pL, Gompertz growth pG, charged capacitor growth pC, combined logistic and charged capacitor growth pLC and pLCλ, and combined Gompertz and charged capacitor growth pGC and pGCλ.

Table 1. Dates and magnitudes of bull market local maxima 4,5,6,7,8 when K=$476,190 and K=$23,809,524, predicted until year 2050

analogously predicts the dates and magnitudes of the five future Bitcoin bear market local price minima, assuming K=$476,190 and K=$23,809,524, and assuming logistic growth pL, Gompertz growth pG, charged capacitor growth pC, combined logistic and charged capacitor growth pLC and pLCλ, and combined Gompertz and charged capacitor growth pGC and pGCλ.

Table 2. Dates and magnitudes of bear market local minima 4,5,6,7,8 when K=$476,190 and K=$23,809,524, predicted until year 2050

3.7.1. Bitcoin carrying capacity K=$476,190

Assuming the Bitcoin carrying capacity K=$476,190, for logistic growth pL no local maxima exist with the least squares method, which means that the Bitcoin price p increases monotonically and asymptotically towards K=$476,190 with no local maxima. The absence of local maxima is due to logistic growth pL approaching K=$476,190 more quickly than the other six curves (see Figure ), and also due to logistic growth pL exhibiting limited oscillation, thus not tracking the empirics very well. Two local maxima exist with the weighted least squares method, after which the Bitcoin price p increases monotonically towards K=$476,190. The presence of two local maxima is intermediate between no local maxima and five local maxima, reflecting more oscillation due to more weight being assigned to the early data, and better tracking of the early empirics. Since logistic growth pL approaches K=$476,190 more quickly than the other six curves, both local maxima are above $431,785, at 7/6/24 and 1/28/30.

For Gompertz growth pG, five local maxima exist with the least squares method, starting with $75,506 at 5/22/22, and ending with $462,526 at 6/11/47. These lower local maxima, compared with logistic growth pL, arise since Gompertz growth pG approaches the Bitcoin carrying capacity K=$476,190 more slowly. No local maxima exist with the weighted least squares method, reflecting less oscillation than with the least squares method.

For charged capacitor growth pC five local maxima exist with the least squares method. Since charged capacitor growth pC approaches the Bitcoin carrying capacity K=$476,190 slowly (see ), the local maxima are low, ranging from $18,032 at 8/8/21 to $47,827 at 3/25/46. No local maxima exist with the weighted least squares method, reflecting the much lower growth rate k=4.32, and inability to track the empirical oscillations.

For combined logistic and charged capacitor growth pLC, no local maxima exist with the least squares method and the weighted least squares method. This result is influenced by no local maxima existing for logistic growth pL with the least squares method, and no local maxima existing for charged capacitor growth pC with the weighted least squares method.

For combined logistic and charged capacitor growth pLCλ, five local maxima exist with the least squares method, starting with $113,012 at 8/19/22, and ending with $476,180 at 10/26/47. No local maxima exist with the weighted least squares method, reflecting less oscillation.

For combined Gompertz and charged capacitor growth pGC, no local maxima exist with the least squares method and the weighted least squares method, due to less oscillation.

For combined Gompertz and charged capacitor growth pGCλ, five local maxima exist with the least squares method, starting with $46,652 at 2/12/22, and ending with $342,930 at 1/10/47. No local maxima exist with the weighted least squares method, reflecting less oscillation.

3.7.2. Bitcoin carrying capacity K=$23,809,524

Assuming the Bitcoin carrying capacity K=$23,809,524, the results are remarkably similar to when K=$476,190. The local maxima mostly occur at the similar time t, but are naturally higher. For logistic growth pL, no local maxima exist with the least squares method, while three local maxima exist with the weighted least squares method. These range from $3,360,314 at 5/5/24 to $23,809,485 at 2/6/36.

For Gompertz growth pG the five local maxima with the least squares method range from $107,064 at 6/29/22 to $15,732,787 at 8/9/47, i.e., lower than for logistic growth pL. No local maxima exist with the weighted least squares method.

For charged capacitor growth pC the five local maxima with the least squares method only slightly exceed the local maxima when K=$476,190, and almost at the same time, ranging from $18,317 at 8/15/21 to $50,127 at 4/4/46. No local maxima exist with the weighted least squares method.

For combined logistic and charged capacitor growth pLC, no local maxima exist with the least squares method and the weighted least squares method.

For combined logistic and charged capacitor growth pLCλ, five local maxima exist with the least squares method, starting with $133,710 at 9/16/22, and ending with $23,181,174 at 12/8/47. No local maxima exist with the weighted least squares method.

For combined Gompertz and charged capacitor growth pGC, local maxima exist with both the least squares method and the weighted least squares method, contrary to when K=$476,190. With the least squares method the five local maxima range from $94,049 at 6/26/22 to $7,301,533 at 8/4/47. With the weighted least squares method the five local maxima range from the lower $65,062 at 6/26/22 to the lower $6,030,620 at 2/12/49.

For combined Gompertz and charged capacitor growth pGCλ, five local maxima exist with the least squares method, starting with $68,116 at 4/28/22, and ending with $2,960,784 at 5/6/47. No local maxima exist with the weighted least squares method.

4. Discussion

Ten results in the previous section are noteworthy. First, based on the current market capitalization of gold at approximately $10 trillion, the Bitcoin carrying capacity is estimated as K=$476,190. Using the least squares method, without modeling oscillations, logistic growth pL appears nearly linear with growth rate k=1.28 on a logarithmic plot with base 10. It initially increases more slowly, and eventually approaches K=$476,190 more quickly. Gompertz growth pG with growth rate k=0.16 is fast at the beginning and approaches K=$476,190 slowly. Charged capacitor growth pC initially grows much faster than the other six curves, with growth rate k=1178, and approaches K=$476,190 much slower than the other six curves. Combined logistic and charged capacitor growth pLC is slower than logistic growth pL but faster than combined Gompertz and charged capacitor growth pGC. The curve pLC estimates the growth rate k=1.21 and adjustment parameter λ=1.02. Assuming λ=0.88, the curve pLCλ with growth rate k=2.18 is intermediate between pL for logistic growth and pC for charged capacitor growth. Combined Gompertz and charged capacitor growth pGC displays initial slow growth rate k=0.08, but eventually approaches K=$476,190 quickly. Assuming λ=0.88, the curve pGCλ with growth rate k=0.32 is intermediate between pG for Gompertz growth and pC for charged capacitor growth, and grows as the second fastest among the six curves. Summing up impressionistically, the curves pGCλ, pLCλ and pG in ) fit the data relatively well.

Second, applying the weighted least squares method when K=$476,190, early data (with low price fluctuations measured in US$) is weighed more heavily than late data (with high price fluctuations measured in US$), which eliminates or ameliorates the impact of heteroscedasticity since more equal weight is assigned over the period 23 July 2010–21 June 2021, causing some similar and some different results. Logistic growth pL estimates the same growth rate k=1.28. Gompertz growth pG has the higher growth rate k=0.15. Charged capacitor growth pC has the much lower growth rate k=4.32, and approaches K=$476,190 more slowly than with the least squares method. Combined logistic and charged capacitor growth pLC has higher growth rate k=2.33 and lower adjustment parameter λ=0.85. Assuming λ=0.88, the curve pLCλ estimates the lower growth rate k=2.10. Combined Gompertz and charged capacitor growth pGC has higher growth rate k=0.17 and lower adjustment parameter λ=0.97. Assuming λ=0.88, the curve pGCλ has lower growth rate k=0.22. Summing up, the curves pG, pGC, pGCλ, pLC and pLCλ in ) are relatively similar with seemingly good fit to the data.

Third, based on a Bitcoin market capitalization of $500 trillion, the Bitcoin carrying capacity is estimated as K=$23,809,524, i.e., 50 times higher than K=$476,190. The results and especially the dates of the local maxima are similar, but the local maxima are higher. Using the least squares method, without modeling oscillations, logistic growth pL is slightly lower at k=1.27. Gompertz growth pG is lower at k=0.10. Charged capacitor growth pC is slightly lower at k=1171. Combined logistic and charged capacitor growth pLC and pLCλ are similar at k=1.23 and k=2.17. Combined Gompertz and charged capacitor growth pGC and pGCλ (assuming λ=0.88) are higher at k=0.14 and lower at k=0.20, respectively. Summing up, the curves pG, pGC, pLCλ, pGCλ in ) seem to fit the data well.

Fourth, applying the weighted least squares method when K=$23,809,524, logistic growth pL and Gompertz growth pG are similar at k=1.27 and k=0.11. Charged capacitor growth pC is much lower than with the least squares method, at k=4.32 (due to weighing early data more heavily). Combined logistic and charged capacitor growth pLC and pLCλ are higher at k=2.33 and lower at k=2.10, respectively. Combined Gompertz and charged capacitor growth pGC and pGCλ are slightly lower at k=0.13 and lower at k=0.15, respectively. Summing up, the curves pG, pGC, pLC, pLCλ, pGCλ in ) seem to fit the data well.

Fifth, the three bull market local maxima during the period 23 July 2010–21 June 2021 are used to estimate the scaling of the inverse of the cycle length of the sine oscillations as ωmax=7.05885, and the inverse of the degree of lengthening of each subsequent cycle as γmax=0.499872. The three bear market local minima during the period 23 July 2010–21 June 2021 are analogously used to estimate ωmin=3.45348 and γmin=0.744082. Taking the average, ω=5.25616 as the scaling of the inverse of the cycle length of the sine oscillations, and γ=0.621977 as the inverse of the degree of lengthening of each subsequent cycle, are used in the remainder of the article.

Sixth, applying the same growth rate k and adjustment parameter λ as estimated without assuming oscillations (i.e., when α=0), and applying ω=5.25616 and γ=0.621977, the oscillation amplitude α and the start time adjustment parameter δ are estimated to model oscillatory growth for the models. With Bitcoin carrying capacity K=$476,190 and applying the least squares method, logistic growth pL oscillates minimally at the amplitude α=0.48. Gompertz growth pG oscillates more at α=0.37. Charged capacitor growth pC oscillates moderately at α=7209. Combined logistic and charged capacitor growth pLC oscillates similarly to logistic growth pL at α=0.39. The curve pLCλ oscillates similarly to Gompertz growth pG at α=3.93. Combined Gompertz and charged capacitor growth pGC oscillates minimally at α=0.04. The curve pGCλ oscillates at α=0.91. Summing up, the curves pG, pGC, pLCλ, pGCλ in ) seemingly oscillate nicely according to the data.

Seventh, applying the weighted least squares method when K=$476,190, logistic growth pL oscillates more at α=1.42. Gompertz growth pG oscillates less at α=0.08. Charged capacitor growth pC oscillates minimally at α=3.13. Combined logistic and charged capacitor growth pLC and pLCλ oscillate similarly to Gompertz growth pG at α=1.88 and α=1.65. Combined Gompertz and charged capacitor growth pGC oscillates at α=0.15. The curve pGCλ oscillates at α=0.20. Summing up, the curves pG, pGC, pLC, pLCλ, pGCλ in ) seem to oscillate according to the data.

Eighth, with Bitcoin carrying capacity K=$23,809,524 and applying the least squares method, logistic growth pL and combined logistic and charged capacitor growth pLC oscillate similarly and minimally at α=0.34 and α=0.22. Gompertz growth pG oscillates at α=0.17. Charged capacitor growth pC oscillates moderately at α=7200. The curve pLCλ oscillates similarly to Gompertz growth pG at α=3.58. Combined Gompertz and charged capacitor pGC oscillates at α=0.29. The curve pGCλ oscillates at α=0.91. Summing up, the curves pG, pGC, pLCλ, pGCλ in ) seemingly oscillate according to the data.

Ninth, applying the weighted least squares method when K=$23,809,524, logistic growth pL again oscillates more at α=1.53. Gompertz growth pG and combined logistic and charged capacitor growth pLC and pLCλ oscillate similarly at α=0.11, α=1.66, and α=1.65. Charged capacitor growth pC oscillates at α=3.13. Combined Gompertz and charged capacitor growth pGC and pGCλ oscillate at α=0.15 and α=0.20. Summing up, the curves pG, pGC, pLC, pLCλ, pGCλ in ) seem to oscillate according to the data.

Tenth, applying the two Bitcoin carrying capacities K=$476,190 and K=$23,809,524, the bull market local maxima 4,5,6,7,8 and bear market local minima 4,5,6,7,8 are estimated until 2050. These dates depend to a low degree on the growth model carrying capacity K. The magnitudes of the local maxima and local minima of course depend on K, assumed to vary broadly to assess the implications.

If the Bitcoin price evolves until the year 2100 as predicted in this article, that has substantial implications for today’s financial system. First, Bitcoin may become a more dominant investment class competing with today’s classes, i.e., stocks, bonds, real estate, money market instruments, non-inflationary instruments (minerals, art, etc.), etc. Second, if Bitcoin layer 2 solutions become common, as in El Salvador, such solutions may spread to more countries, and likely first to the world’s countries with the weakest currencies or countries without their own currency. The insights in this article may be useful for all humans, i.e., consumers choosing between Bitcoin layer 2 solutions and alternative payment rails, investors, politicians and regulators choosing how to regulate Bitcoin, regulators and developers of asset classes competing with Bitcoin, financial institutions competing with Bitcoin or developing Bitcoin-based instruments, and central banks developing digital currencies.

5. Conclusion

The motivation for this article is the apparently unpredictable Bitcoin price evolution since 3 January 2009, and the need for methods to understand the evolution so far and predict the future evolution. The methods are differential equation growth models incorporating oscillation and lengthening cycles. The analysis is interesting for traders (with time horizons from microseconds to months or years) exchanging Bitcoin with other cryptocurrencies, fiat currencies and asset classes, and savers and investors choosing Bitcoin as a mid term or long term store of value. The article is also relevant for regulators, central banks administering and developing competing currencies with specifically designed characteristics, banks offering competing financial products, collective units assessing whether to offer Bitcoin transactions, and countries assessing whether to accept/reject Bitcoin mining and trading, and whether to accept/reject Bitcoin as legal tender. Regulators want to understand Bitcoin to determine where and how Bitcoin trading and investing can occur, which Bitcoin-related financial products can be developed, how Bitcoin can interact and operate within the existing financial system, and which risk factors are involved. The study is unique in that a minority of other studies account for the dynamics of the Bitcoin price evolution with differential time equations. Further uniqueness consists in incorporating oscillation and lengthening cycles into growth models.

One of the main contributions of this article is to explain the Bitcoin price and predict its future evolution. The past evolution has been embedded within a structure of growth subject to damped oscillations and lengthening cycles. Future bull market maxima and bear market minima are predicted. Earlier studies mostly apply other methods to predict the Bitcoin price, or compare the accuracy of different prediction models, see e.g., Jana et al. (Citation2021); Roy et al. (Citation2018). This article develops differential equations which is uncommon in the literature. Differential time equations enable a different kind of dynamic understanding and explanation, which furthermore enable prediction. The differential equations assume Bitcoin price growth towards two different carrying capacities, subject to damped oscillations and lengthening cycles. Existing studies, e.g., K. S. Chen and Huang (Citation2020); Wang and Wang (Citation2020), capture some aspects of differential equations such as volatility, Bitcoin trading volume, market sentiment, etc. This article additionally incorporates oscillations which express the strength of past and future bull and bear markets, overall approaching one of two different Bitcoin carrying capacities. The authors believe that past studies unsatisfactorily, or at least differently, predict the Bitcoin price in future bull and bear markets. Acknowledging that the Bitcoin price, according to the best models developed in this article, is more influenced by recent data than early data, this article also adopts the weighted least squares method to estimate the parameters. Other studies incorporate the volatility in the models, see e.g., K. S. Chen and Huang (Citation2020); Jaquart et al. (Citation2021). This article furthermore uses a wider time range of the past Bitcoin prices to identify the optimum model parameters, i.e., 23 July 2010–21 June 2021, than has been common elsewhere in the literature, benefiting from more time having elapsed since Bitcoin’s emergence. Earlier studies mostly apply shorter data time ranges, see e.g., Caporale et al. (Citation2019); Cocco et al. (Citation2021); Cretarola and Figa-Talamanca (Citation2021); Gupta and Nain (Citation2021).

The parameters in the differential equation growth models are estimated with the least squares method against the 23 July 2010–21 June 2021 empirical data. The weighted least squares method is applied to account for heteroscedasticity. Logistic growth, Gompertz growth, charged capacitor growth, and two hitherto unknown combinations of these are merged with oscillation and damped lengthening cycles for increased realism.

For each of the five models the growth rate is estimated. Logistic growth is initially slow and eventually quick towards the asymptote. Gompertz growth is initially quick and thereafter slow. Charged capacitor growth is initially too quick and thereafter too slow. As theoretically novel contributions, logistic and Gompertz growth combined with charged capacitor growth exhibit intermediate growth rates, depending on an additional adjustment parameter which weighs the combination. This parameter is determined optimally (using the least squares method and the weighted least squares method) and by assumption, yielding seven growth curves in addition to the empirical curve.

The three bull market local maxima and the three bear market local minima in the available empirics are used to estimate the scaling of the inverse of the cycle length of the sine oscillations, and the scaling of the inverse of the degree of lengthening of each subsequent cycle. Two additional parameters are estimated, i.e., the oscillation amplitude, which expresses the strength of the bull and bear markets, and the start time adjustment parameter for the sine oscillations.

Gompertz growth tracks the growth and oscillations in the empirical data quite well, and tracks the early data better with the weighted least squares method which weighs the early data more heavily. Gompertz growth combined with charged capacitor growth tracks the early data even better since initial growth is quicker. Logistic growth is too slow to track the early empirical data, even when applying the weighted least squares method. Logistic growth combined with charged capacitor growth to some extent tracks the early data. Pure charged capacitor growth is judged to be least realistic.

Six of the curves (abandoning pure charged capacitor growth) are used to estimate the expected value ± the standard deviation of the dates of the future bull market local maxima and bear market local minima. These dates depend to a low degree on the growth model carrying capacities, approached asymptotically. The magnitudes of the bull market local maxima depend indeed on the two carrying capacities. When the carrying capacity is $476,190 to reflect the current market capitalization $10 trillion of gold, the future bull market local maxima and bear market local minima are lower than when the carrying capacity is $23,809,524 to reflect a $500 trillion market capitalization. The large standard deviations in the estimates are common for new assets in their early stages, and reflect the different predictions of the various models.

Modeling the Bitcoin price as oscillatory growth does not mean that the Bitcoin price can be expected to eventually stabilize towards a horizontal asymptote in the long run. The authors expect growth models to describe the Bitcoin price over the next few bull market local maxima towards various hypothetical carrying capacities. As cryptocurrency markets mature, at some point growth models will become less descriptive. Then alternative models may come into play. Examples of other kinds of evolution are the price fluctuations of more mature asset classes such as gold, stocks, bonds and real estate over the last centuries. Competition with other asset classes and means of exchange, and governmental policies, may increasingly impact the future Bitcoin price.

The implications of the study for all market participants are to be especially cognizant of Gompertz growth combined with charged capacitor growth of the Bitcoin price, and to realize that no growth is unlimited forever. Short term traders should focus on the large standard deviations which may indicate where to impose stop loss orders. Long term investors can focus less on the standard deviations and more on the Bitcoin price Gompertz growth compared with the potential growth of competing asset classes. Regulators focus on the stability and legality of the financial system which suggests a focus on the standard deviations and the fluctuations between the bull market maxima and bear market minima. Central banks focus on financial stability, which relates to inflation, unemployment, interest rates, and exchange rates. They should adjust the money supply of a fiat currency or a specifically designed central bank digital currency while acknowledging potential competition from a fixed supply and highly volatile cryptocurrency. Banks should adjust their competing financial products to account for the volatility and potential growth of the Bitcoin price. Collective units such as firms, institutions, governmental units (e.g., tax authorities), and countries need to account for the standard deviations and fluctuations of the Bitcoin price in order to determine whether to accept or reject Bitcoin transactions. For example, El Salvador addresses this by pricing goods and services in US$ while accepting Bitcoin transactions.

Future research may extend the analysis to other cryptocurrencies (e.g., Ethereum, Cardano, Polkadot, Chainlink) or other phenomena exhibiting growth. Other aspects to include are Bitcoin’s hash rate, mining difficulty, network value to transactions, active addresses and new addresses, on chain transaction volume, Bitcoin’s electricity consumption, renewable energy consumption, institutional investors, and other assets such bonds and stocks. The five models in this article may be generalized to include more parameters, and may be merged with other models, e.g., the stock-to-flow model, machine learning, neural networks, deep learning, and econometrics. The models may incorporate regulatory intervention, the policies and attitudes of various countries, and competition with other currencies and asset classes. Further extensions can be made to extreme value theory and stochastic analysis with probability distributions.

Acknowledgements

We thank three anonymous referees for useful comments, and John F. Moxnes for useful discussions before writing this article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The article contains no associated data. All data generated or analyzed during this study is included in this published article.

Additional information

Notes on contributors

Guizhou Wang

Guizhou Wang is a PhD student at the University of Stavanger, Norway, since 2020-06. His working PhD thesis title is “Game Theoretic Modeling of Economic Systems Involving Digital Currencies.” He has published 10 articles in peer reviewed journals. His research fields are digital currencies, game theory, risk analysis, cryptocurrencies, central bank digital currencies, econometrics. He holds a MSc degree in financial economics from the University of Chinese Academy of Sciences (Beijing, China), 2016-09 – 2019-06, focusing on mathematical finance, econometrics, venture capital, and cryptocurrency. He holds a BSc degree in finance from the Jinan University (Guangzhou, China), 2010-09 – 2014-06, focusing on finance, derivatives, and mathematical modeling. Email: [email protected].

Kjell Hausken

Kjell Hausken is a professor of economics and societal safety at the University of Stavanger, Norway, since 1999. His research fields are terrorism, societal safety, economics, economic risk management, economics and safety, political economy, information security, public choice, conflict, game theory, reliability, war, crime, risk analysis, disaster prevention, stochastic theory, dynamics, petroleum economics, resilience management. He holds a PhD from the University of Chicago (1990-1994), was a postdoc at the Max Planck Institute for the Studies of Societies (Cologne) 1995-1998, and a visiting scholar at Yale School of Management 1989-1990. He holds a Doctorate Program Degree (HAE) (“Philosophical, Behavioral, and Gametheoretic Negotiation Theory”) in Administration from the Norwegian School of Economics and Business Administration (NHH), a MSc degree in electrical engineering, cybernetics, from the Norwegian Institute of Technology (NTNU), focusing on mathematics and statistics, and a minor in Public Law from the University of Oslo. He has published 270 articles in peer reviewed journals, one book, edited two books, is/was on the Editorial Board for Theory and Decision (May 20, 2007 –), Reliability Engineering & System Safety (January 17, 2012 –), and Defence and Peace Economics (December 4, 2007 – December 31, 2015), has refereed 400 submissions for 85 journals, and advises and has advised seven PhD students. Email: [email protected].

Notes

1. https://8marketcap.com/metals.  Retrieved February 20, 2022.

2. https://messari.io/asset/bitcoin/historical.  Retrieved February 20, 2022.

3. The Mathematica 13 software package (www.wolfram.com) was used. The codes used for the simulations are available from the authors upon request.

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