# Last edited on 2015-04-22 03:10:50 by stolfilocal [b]A descriptive sum-of-bubbles model for the price of bitcoin[/b] Figuer 1 below shows that the price history of bitcoin can be described fairly accurately by a model consisting of the sum of several idealized [i]bubbles[/i], each consisting of an exponential [i]rise[/i], an optional flat [i]plateau[/i], and an exponential [i]tail[/i]. [url=][img][/img][/url] [size=9pt][ Figure 1. A bubble model for the price of bitcoin, showing the actual price (grey), the modeled price (green), and the individual bubbles. The brown line near the bottom is the ratio of the actual price to the model price. Click on the image for a full-size version ][/size] Figures 2 amd 3 below show two of those bubbles in linear scale. Note that, on a log scale plot, such as Figure 1, the rise and tail of a single bubble are plotted as straight lines, but they are distorted into curves when added to other bubbles. [url=][img][/img][/url] [size=9pt][ Figure 2. The bubble "2011-06" that peaked around June 08, 2011. Click on the image for a full-size version ][/size] [url=][img][/img][/url] [size=9pt][ Figure 3. The bubble "2013-04" that has a plateau from about April 5th to May 31, 2013. Click on the image for a full-size version ][/size] The above model has 10 such bubbles. The first one (red) accounts for the price between 2010-07-15 to 2010-09-30, and is essentially flat at about 0.06 $/BTC in that period. All other bubbles have a relatively fast ascending rise, with the price increasing between 2% and 10% per day. Two of them have a flat plateau. All have a decaying tail, except the "2012-08" bubble that apparently continued to grow slowly (at about 0.1%/day ) after the end of the rise phase. In the mathematical model each bubble extends over the entire range of dates considered, but is plotted only while it makes a significant contribution to the model price, specifically at least 5% of it. Under this criterion, each bubble effectively starts to be relevant and noticeable a few months before the peak, and (except for the "2012-08" one) stop being relevant a few months after the peak. [b]Descriptive, not predictive[/b] The model is meant to be purely [i]descriptive[/i], not [i]predictive[/i]. That is, it aims only to provide a succint description of the historical prices, without attempting to predict future prices. If anything, it is "anti-predictive", because it implies that the past bubbles have exponentially decreasing relevance for the future prices, and any future bubbles cannot be detected until they are well underway. Until mid-2014, it was widely claimed that the price would continue to grow, as it has done in the past, by a sequence of bubbles spaced roughly 9 months apart, with exponentially increasing peak amplitudes. However, this model suggests that such "exponential bubble train model" may be just an illusion. The good fit of our model to the price, particularly between successive bubbles, suggests that bubbles aer added rather than multiplied. It follows that each new bubble is only noticeable when its amplitude is substantially greater than the sum of all tails of the previous bubbles. If a bubble like the "2011-06" one occurred today, for example, lifting the price by only 10 dollars, it would not be discernible at all. Therefore, if bubbles actually occurred at random intervals and with random amplitudes, this masking effect would give the impression that bubble amplitudes are increasing. The bubble "2012-08" is exceptional, in that the best fit was obtained by assuming that it continued growing at a slow but positive rate (0.1%/day) after its fast rise ended. It is not possible to tell whether that rise continued after january 2013, when the next bubble began to dominate. If it did, it may be contributing 30 $/BTC to the price today, and thus may be a slowly increasing "floor" that wil dominate again after the other bubbles have decayed. [b]Interpretation of bubbles as market openings[/b] However, I believe that this model is also [i]explanatory[/i], to the extent tht each bubble can be interpreted as a surge in demand due to the opening of some new [i]market[/i]. Each market may be another community of users, isolated from the others by national, language, or legal barriers; or a new use of bitcoins. The exponential rise part of a bubble would then be due to the spread of demand in that market by "contagion", possibly amplified by media coverage and speculative demand. The end of the rise would be due to saturation of that market. (In some bubbles one can see oscillations extending for a month or two after the end of the rise, presumably caused by panic and recovery among the speculators. These oscllations have not been included in the model yet.) The plateau part of each bubble would be due to a period of relatively constant demand after the peak, while a decreasing tail could be due to gradual decrease of the demand, e.g. for disappointment, government repression, etc. In particular, the bubble labeled "2013-11" ("Beijing 1") was almost certainly due to the opening of the Mainland Chinese market after the major exchanges Hubi and OKCoin started operating in Beijing, and a report about bitcoin was featured in mainstream Chinese media. Local reports attribute the huge demand to a large contingent of amateur speculators that used to day-trade in commodities like tea or garlic, and found bitcoin more attractive for that purpose because of its higher volatility. That is one of the fastest-rising bubbles in the model (about 10%/day). It peaked at 2013-11-29, when the Central Bank of China (PBoC) intervened and banned the use of bitcoin in e-commerce and forbade financial institutions from dealing with it. That bubble then started decaying at about 0.55%/day. The earlier bubble "2013-04" ("Sanghai 1"), that has a plateau from about April 5th to May 31, 2013, may have been created by demand in China, too; but by BTC-China in Shanghai, possibly catering to a different community. BTC-China started operating well before that bubble, but in early 2013 it recruited Bobby Lee as CEO, a Stanford alumnus who formerly worked for Walmart. BTC-China's trading volume started growing exponentially in the first 3 months of 2013, and it was leading the price increase during that period. In the model, it is assumed that the contribution of that demand to the price remained constant for 2 months and then started to decay. The price recovered between July and September; this recovery was modeled as a separate bubble "2013-08" that peaked around August 31, 2013; although it may have been a recovery of the demand in the same market. Note that the relation between price increase and demand depends on the liquidity of the market, which is not known, and must be non-linear. Therefore, the magnitude of each bubble should not be interpreted as a measure of the demand, but only of its effect on the price. This in interpretation would confirm the claim that the model is not predictive, because the opening of markets in the past would hardly influence the opening of new markets in the future. In particular, to have another bubble with magnitude above 1000 $/BTC, bitcoin woudl have to conquer some market with demand comparable to the Chinese one. While there are conjectural candidates for such a market, the past history of the price cannot have much influence on that future event. [b]How the model was constructed[/b] A typical bubble has 4 parameters: the rate of change per day during the rise part (always greater than 1), the date of the peak, the rate of change per day during the tail (less than 1, except for the "2010-07" and "2012-08" bubbles), and the magnitude at the peak. Some bubbles had one additional parameter, the duration of the plateau phase. The number of bubbles in the model (10) and all bubble parameters except the peak magnitude were determined initially by hand, inspecting the price chart. They were then tweaked by variosu means to improve the fitting. Once the other parameters were chosen, the bubble magnitudes were then determined by weighted least squares fitting of their linear combination to the observed prices P(i). Each price datum P(i) was assigned a weight W(i) = 1/P(i), in order to simulate the effect of fitting the model in log scale, while actually using linear scale. The input prices and the component bubbles were smoothed with a Hann window spanning 2 weeks.