Prediction of Bitcoin bubble-Metcalfe’s Law and LPPLS Model
Bitcoin, Metcalfe’s Law, LPPLS model, Market-to-Metcalfe Ratio
Background
It’s a big problem across time that discovering the economic bubble and predicting when to burst. Comparing with other traditional economical methods, Log-Periodic Power Law (LPPL) model has shown its reliability in many fields. It has successfully predicted the 2008 oil bubble, the U.S. real estate bubble, and the 2009 Chinese stock market bubble, and so on. Professor Sornette, the founder of the LPPL model, has applied a modified Log-Periodic Power Law model (LPPLS) to predict the bubble in the bitcoin market. Except for LPPLS model, a ratio called Market-to-Metcalfe, based on the generalized Metcalfe’s Law is used by him as well.
Bitcoin, as the first digital decentralized cryptocurrency, was introduced by pseudonymous Satoshi Nakamoto in 2008 [1]. Its market capitalization is over 50 billion currently, and even exceeded over 200 billion at its peak. Besides, there are more than 2000 cryptocurrencies. Therefore, much academic research on cryptocurrency valuations and growth mechanisms has emerged. This article makes references to Wheatley and Sornette’s paper [2] and is based on Bitcoin’s data from 2013–4–28 to 2018–12–10. We will do the valuation based on Metcalfe’s Law model and introduce a prediction tool-LPPLS model. Finally, a warning mechanism under Market-to-Metcalfe Ratio will be built.
How to value bitcoin: Metcalfe’s Law
It’s widely discussed that the number of network users could be regarded as a measure for valuation. Metcalfe’s Law, which was mentioned in the 1980s and formulated in the 1993s, states that the effect/value of a network is proportional to the square of the number of connected users in the network. This law has held true in many network systems, even Facebook values itself based on Metcalfe’s Law. We will try to assess bitcoin market cap under this law as well. Before valuation, some evidence that Metcalfe’s Law holds in the bitcoin market is needed.
The logarithm formula is
Use some measures in linear regression — R²(adjusted) measures to which degree the regression model could explain the true data, which varies from 0 (irrelevant) to 1 (perfect fit); while p-value measures to which degree the fitting model could be accepted.
R² is calculated to be 0.62, which is acceptable since our data sample is over 2000 and the negligible P-value indicates that the linear fitting model is reasonable. Although the pattern looks like a hook, it is evenly distributed on both sides. Therefore, Metcalfe’s Law holds here and the formula is ln(p) = 3.525 + 1.547ln(u).
Before building the model under Metcalfe’s Law, we introduce an ecological-type non-linear regression which is used in Wheatley and Sornette’s paper [2]for an active address(u) over time(t):
Figure 2 shows the regression result. Both high correlation or negligible residual sum-of-squares (SSR) would suggest that this non-linear model fits perfectly, this could be seen in Figure 2.
Actual Model:
In Figure 3, we attempt to assess the market capitalization using Metcalfe’s Law curve. The non-linear regression & Metcalfe’s Law curve is used as a reference line. These two curves are combined with the original market-cap curve and two bound lines (normalize the time).
More precisely, the black curve (Metcalfe’s Law valuation curve) is plotted using the Metcalfe’s Law formula ln(p) = 3.525 + 1.547ln(u), and data with time(t) against active address(u). The blue line (Metcalfe’s Law’s reference line) is obtained by using the original Metcalfe’s Law formula and calculating u with non-linear regression.
Additionally, parameters of the blue line is changed to construct the bound curves; upper bound: ln(p) = 3+1.66ln(u)(subjective estimate), and lower bound (2015.1.1 to 2017.3.31): ln(p) = 3.855 + 1.449ln(u).
It appears that the actual market cap curve fluctuates around the Metcalfe’s Law valuation curve. If the Metcalfe’s Law holds here, two peaks which are overvalued may be caused by some bubble; and the valley, which is undervalued may be due to the panic after a bubble burst. Furthermore, Metcalfe’s Law valuation curve (black curve) does not deviate too far from the reference line, which points to a healthy user growth model (volatility & exponential increase).
Since the reference line (blue curve) could be regarded as a rough model for valuation, it is evident that when the mining process finishes, the final price would approach 3,000USD/BTC. Here, the two bounds of the exponential function reflect the price 1,000 and 7,000. The final price and the range between two bounds can become higher and wider respectively due to the behavior of investment and speculation.
Try to predict bubble bursting: LPPLS model
LPPL model describes the behavior of a speculative bubble and predicts the critical time by which the bubble must burst. The model was brought up in 1996 by Didier Sornette — a statistical physicist and geophysicist, also a professor in finance at Swiss Federal Institute of Technology in Zurich currently. He and his team later modified it for the Bitcoin market [2].
The modified model’s (LPPLS) formula is
where t_c is the critical time by which the bubble must burst [2]. we run the model from period 2013–4–28 to 2013–12–15 (2058 days in total) and find that the output t_c depends on the initial t_c set before. For instance, output t_c is 2097 (2019–1–23) when initial t_c is 2088, and output t_c is 2124 (2019–2–19) when initial t_c is 2118. In our perspective, since the output depends on the initial value, the model does not seem to be reliable. This is despite Wheatley and Sornette implying time series to the residual, which improves the goodness of fitting.
Warning mechanism: Market-to-Metcalfe Ratio
In the traditional economic market, experts prefer to do analysis or research based on some index like Standard＆Poors Stock Price Indexes and Dow Jones Stock Price Indexes, so we introduce a ratio as an index here and apply LPPLS model to it.
this ratio is created by Wheatley and Sornetteto and called Market to-Metcalfe Value (MMV) Ratio and its formula is[2]
The plot for this ratio is in figure 4.
Besides, In consideration that the critical time in LPPLS model is only an estimator, a warning mechanism seems to be more reasonable than the prediction.
thermal imaging, a late-model technology, has been widely used in many fields, like the Calendar-Heatmap in Meteorology. In view of its intuitive and comparability, We will create a warning mechanism in the form of Calendar-Heatmap.
In view of the emergence and bubbles bursting, We focused on the periods where the price increases. The latest period (2017–03–31 to 2017–12–18) have been chosen for analysis, and some results are displayed in Figure 5.
The patterns (log result) shown above, represents the time interval from when the last date of the data was run by the model and when the bubble is predicted to burst. Red represents a shorter interval and green represents a longer interval.
The actual forecast starts from 2017–05–01 since the data in April is regarded as historical data for modeling. The inference is stated as follows: the accuracy of the prediction increases when we look at monthly (or weekly) compared to daily results (output). For instance, there are many red or orange alerts in the first half of May, but in actual fact, there was only a short drop from 5–25 to 5–28. On the other hand, red or orange alerts cover the whole of November and the first half of December, resultingly, the actual bubble burst happened on 12–18. It seems that this new measure is suitable to create a warning mechanism during a price increase period.
Limitation to MMV
Compared to the LPPLS model, Market-to-Metcalfe (MMV) Ratio is more reliable and practical. However, there are some limitations: it does not work for a short period since it needs historical data endogenously (simulation for 7 parameters). Furthermore, there are two preconditions (1) Metcalfe’s Law must hold and (2) it’s within a period where price increases.
Discussion
To improve the model further, we have two proposals (1) include time series for residual to improve the goodness of fitting and introduce the confidence interval to guarantee the accuracy of MMV-LPPLS model (Wheatley and Sornette have mentioned in their article)[2], and (2) modification for MMV ratio or Metcalfe’s Law.
Besides, our models do not consider the factors other than trading, like policy and market sentiment. it’s emerging that semantic emotional analysis or some other machine learning methods are used to construct models.
Is there a warning now?
We have run our warning mechanism for the increasing period of recent 20 days (2018–12–16 to 2019–1–6) and the result suggests that nowadays is a short increase and it may drop in the near future.
References
[1] Satoshi Nakamoto. Bitcoin: A peer-to-peer electronic cash system. 2008.
[2] Spencer Wheatley, Didier Sornette, Tobias Huber, Max Reppen, and Robert N Gantner. Are bitcoin bubbles predictable? combining a generalized metcalfe’s law and the lppls model. 2018
