Article abstract
Article 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.
Keywords
Keywords
Bitcoin; logistic growth; oscillation; lengthening cycles; price prediction; differential equations; least squares method
1. Introduction
1. Introduction
Understanding the nature of the growth and fluctuations is of paramount importance. First, the Bitcoin price has increased apparently unpredictably since January 3, 2009, which suggests a need both to understand the evolution so far and predict the future evolution. Second, applying Bitcoin’s price data since July 23, 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 oscillation 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,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 July 23, 2010–June 21, 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.
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,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 July 23, 2010–June 21, 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.
2 Materials and methods
2 Materials and methods
This section identifies and develops the differential equations believed to capture the Bitcoin price evolution most accurately.
2.1 Nomenclature
2.1 Nomenclature
Parameters
Growth rate,
Carrying capacity,
Parameter for generalized logistic growth impacting near which asymptote maximum growth occurs,
Adjustment parameter for combined generalized logistic and charged capacitor growth,
Oscillation amplitude,expressing strength of bull and bear markets,
Scaling of the inverse of the cycle length of the sine oscillations,
Scaling of the inverse of the degree of lengthening of each subsequent cycle,
Start time adjustment parameter relative to time t=t_ 0 for the oscillation of the Sin function,
t0 Initial time t
T Final time t
p0 Initial price p at time t=t0
Independent variable
t Time
Dependent variable
p Price
k
k∈R
K
K>=0
ν
ν>=0
λ
λ∈R
α
α∈R
ω
ω∈R
γ
γ∈R
δ
γ∈R
t0 Initial time t
T Final time t
p0 Initial price p at time t=t0
Independent variable
t Time
Dependent variable
p Price
2.2 Generalized logistic growth
2.2 Generalized logistic growth
In[]:=
ν=.γ=.FullSimplifyDSolvep'[t](k+αSin[ω+δ])p[t]1-,p[t0]==p0,p[t],t
γ
(t-t0)
ν
p[t]
K
Out[]=
p[t]InverseFunction(Log[]-Log[-])(-)&k(-t+t0)-+(-)(Log[]-Log[-])
-ν
K
ν
#1
ν
K
ν
#1
ν
K
ν
#1
ν-1+
ν
#1
K
(t-t0)αExpIntegralE,-ω-ExpIntegralE,ω
-δ
2δ
-1+γ
γ
γ
(t-t0)
-1+γ
γ
γ
(t-t0)
2γ
-ν
K
ν
K
ν
p0
ν
p0
ν
K
ν
p0
-1+ν
ν
p0
K
2.2 .1 Conventional logistic growth
2.2 .1 Conventional logistic growth
In[]:=
γ=.FullSimplifyDSolvep'[t](k+αSin[ω+δ])p[t]1-,p[t0]==p0,p[t],t
γ
(t-t0)
p[t]
K
2.2 .2 Gompertz growth
2.2 .2 Gompertz growth
In[]:=
γ=.FullSimplifyDSolvep'[t](k+αSin[ω+δ])p[t]Log,p[t0]==p0,p[t],t
γ
(t-t0)
K
p[t]
Out[]=
p[t]K
-
k(-t+t0)+
α-Gamma,ω+Gamma,-ω-Gamma,-ω+Gamma,ω
-
(π+2γδ)
2γ
-1/γ
ω
1
γ
γ
0
(π+2γδ)
γ
1
γ
γ
0
1
γ
γ
(t-t0)
1
γ
γ
(t-t0)
2γ
K
p0
2.3 Generalized charged capacitor growth
2.3 Generalized charged capacitor growth
In[]:=
ν=.γ=.FullSimplifyDSolvep'[t](k+αSin[ω+δ])1-,p[t0]==p0,p[t],t
γ
(t-t0)
ν
p[t]
K
Out[]=
p[t]InverseFunction-Hypergeometric2F11,,1+,#1&k(-t+t0)--p0Hypergeometric2F11,,1+,
1
ν
1
ν
ν
#1
K
(t-t0)αExpIntegralE,-ω-ExpIntegralE,ω
-δ
2δ
-1+γ
γ
γ
(t-t0)
-1+γ
γ
γ
(t-t0)
2γ
1
ν
1
ν
ν
p0
K
2.4 Combined generalized logistic and charged capacitor growth
2.4 Combined generalized logistic and charged capacitor growth
In[]:=
ν=.γ=.λ=.FullSimplifyDSolvep'[t](k+αSin[ω+δ])1-,p[t0]==p0,p[t],t
γ
(t-t0)
λ
(p[t])
ν
p[t]
K
Out[]=
p[t]InverseFunction&k(-t+t0)-+Hypergeometric2F11,,,
Hypergeometric2F11,,1+,
1-λ
ν
1-λ
ν
ν
#1
K
1-λ
#1
-1+λ
(t-t0)αExpIntegralE,-ω-ExpIntegralE,ω
-δ
2δ
-1+γ
γ
γ
(t-t0)
-1+γ
γ
γ
(t-t0)
2γ
1-λ
p0
1-λ
ν
1-λ+ν
ν
ν
p0
K
-1+λ
2.4.1 Combined logistic and charged capacitor growth
2.4.1 Combined logistic and charged capacitor growth
In[]:=
γ=.λ=.FullSimplifyDSolvep'[t](k+αSin[ω+δ])1-,p[t0]==p0,p[t],t
γ
(t-t0)
λ
(p[t])
p[t]
K
Out[]=
p[t]InverseFunction&Beta,1-λ,0+(t-t0)2kγ+αExpIntegralE,-ω-ExpIntegralE,ω
Hypergeometric2F11,1-λ,2-λ,
#1
K
1-λ
#1
K(1-λ)
-λ
p0
λ
p0
K
p0
K
-δ
δ
2δ
-1+γ
γ
γ
(t-t0)
-1+γ
γ
γ
(t-t0)
2Kγ
2.4.2 Combined Gompertz and charged capacitor growth
2.4.2 Combined Gompertz and charged capacitor growth
In[]:=
γ=.λ=.FullSimplifyDSolvep'[t](k+αSin[ω+δ])Log,p[t0]==p0,p[t],t
γ
(t-t0)
λ
(p[t])
K
p[t]
Out[]=
p[t]InverseFunction-ExpIntegralEi(-1+λ)Log&k(t-t0)+-KExpIntegralEi(-1+λ)Log
-(-1+λ)Log+Log[#1]
K
#1
K
#1
(t-t0)αExpIntegralE,-ω-ExpIntegralE,ω
-δ
2δ
-1+γ
γ
γ
(t-t0)
-1+γ
γ
γ
(t-t0)
2γ
-λ
K
p0
-λ
p0
K
p0
3 Results
3 Results
3.1 Bitcoin carrying capacity K = $476, 190 and no oscillation amplitude α = 0
3.1 Bitcoin carrying capacity K = $476, 190 and no oscillation amplitude α = 0
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 (https://8marketcap.com/metals/, retrieved February 20, 2022). 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 gives the Bitcoin carrying capacity K=$476,190.
Data=;
Figure 1. Assuming no oscillation amplitude α=0, the empirical price p_E,logistic growth p_L,Gompertz growth p_G,charged capacitator growth p_C,combined logistic and charged capacitor growth p_LC and p_LC, and combined Gompertz and charged capacitor growth p_GC and p_GC, for July 23, 2010–June 21, 2021 (panels a and c) and until January 1, 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.
λ
λ
Figure 1 a, 1 b
Figure 1 a, 1 b
In[]:=
Out[]=
x
Out[]=
476190
1+9.61806×
6
10
-k(-2010.56+x)
Out[]=
{k1.28132}
Out[]=
Function{x},
476190
1+9.61806×
6
10
-1.28132(-2010.56+x)
Out[]=
476190
-k(-2010.56+x)
1.03971×
-7
10
Out[]=
{k0.163572}
Out[]=
Function{x},476190
-0.163572(-2010.56+x)
1.03971×
-7
10
Out[]=
476190-476190.
k(2010.56-x)
476190
Out[]=
{k1177.83}
Out[]=
Function{x},476190-476190.
0.00247344(2010.56-x)
3.2 Bitcoin carrying capacity K = $23, 809, 524 and no oscillation amplitude α = 0
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 gives the Bitcoin carrying capacity K=$23,809,524.
Figure 2. Assuming no oscillation amplitude α=0, the empirical price p_E, logistic growth p_L, Gompertz growth p_G, charged capacitator growth p_C, combined logistic and charged capacitor growth p_LC and p_LCλ, and combined Gompertz and charged capacitor growth p_GC and p_GCλ, for July 23, 2010–June 21, 2021 (panels a and c) and until January 1, 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.
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
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.
3.5 Bitcoin carrying capacity K = $476, 190 and oscillation amplitude α >= 0
3.5 Bitcoin carrying capacity K = $476, 190 and oscillation amplitude α >= 0
3.6 Bitcoin carrying capacity K = $23, 809, 524 and oscillation amplitude α >= 0
3.6 Bitcoin carrying capacity K = $23, 809, 524 and oscillation amplitude α >= 0
3.7 Estimating bull/bear market local maxima 4, 5, 6, 7, 8 when K = $476, 190 and K = $23, 809, 524
3.7 Estimating bull/bear market local maxima 4, 5, 6, 7, 8 when K = $476, 190 and K = $23, 809, 524
Table 1 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 p_L,Gompertz growth p_G, charged capacitor growth p_C, combined logistic and charged capacitor growth p_LC and p_LCλ, and combined Gompertz and charged capacitor growth p_GC and p_GCλ.
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.
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.
Notes: Each cell has four rows. The dates are on the format month/day/year for the least squares method and the weighted least squares method in rows 1 and 2 when K=$476,190. Rows 3 and 4 are for the least squares method and the weighted least squares method when K=$23,809,524. After each date the price follows in $. Asymptotic means that no local maximum exists, but that the Bitcoin price p approaches K asymptotically. At the bottom each cell has 13 rows, i.e. seven rows with dates and six rows with local maxima expressed as expected value ± standard deviation in days and $. Row 1 gives the date for both methods and both K=$476,190 and K=$23,809,524. Rows 2 and 8 apply for both methods when K=$476,190. Rows 3 and 9 apply for the least squares method when K=$476,190. Rows 4 and 10 apply for the weighted least squares method when K=$476,190. Rows 5 and 11 apply for both methods when K=$23,809,524. Rows 6 and 12 apply for the least squares method when K=$23,809,524. Rows 7 and 13 apply for the weighted least squares method when K=$23,809,524.
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.
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.
Notes: The structure is the same as for Table 1, but applies for local minima instead of local maxima.
4 Conclusion
4 Conclusion
The motivation for this article is the apparently unpredictable Bitcoin price evolution since January 3, 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 .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.
The parameters in the differential equation growth models are estimated with the least squares method against the July 23, 2010–June 21, 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.
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, 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.
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