“Those of you in the old school who believe this is a bubble simply have not understood the new mathematics of the blockchain…. Bubbles are mathematically impossible in this new paradigm. ” — John McAfee, 46th(Maybe) President of the United States of America Well, when John stated above, most of those who involved in cryptocurrency investing, at least as far as I saw, laughed it off and believed such claim made no sense. Empirically, this hasn't aged well either, with bitcoin now hanging around 3500 USD per unit. However, if we look at this matter purely mathematically and statistically, he was actually more right than those who thought otherwise. The definition of Financial Bubble is already flawed, bubble usually means the price of underlying hugely inflated from its fundamental value under a very short time and became unsustainable, followed by eventual reverting back to fundamental price,a.k.a burst. But what if none knows the fundamental value, how much higher can be said as overly inflated, and under what time horizon? If one can not answer with some degree of certainty to the questions above, how can one be sure that what you saw is a bubble? I don't think anyone knows the true answer to the questions above. Only in hindsight, people find it obvious that the “bubble” had burst. Even though bubbles are hard to detect, otherwise there should not be any bubble. It is still statistically possible for a financial bubble to form and burst IF(and a very big IF), the statistical distribution of underlying’s return meet three conditions. 1st, the Mean is definite. 2nd, the Variance is definite. 3rd, the shape of the distribution is definite and symmetric. Above criterion implies that, sequentially, (1) some sort of fundamental value exists, (2) there can be overpricing (though hard to tell), (3) Given enough data, for financial series, this means enough time, it returns to the fundamental value. (well, this last point is easily misunderstood and not rigorous per say but take it as for now) So far, the best statistical analysis that was done on the bitcoin I have seen is none other but the great . He pointed out that bitcoin’s return fits into a symmetric, finite mean infinite variance stable distribution. Taleb himself If the true shape of bitcoin’s return is indeed as Taleb pointed out, the criterion for a detectable bubble were all met except one — the variance is infinite, meaning in the “new mathematics of blockchain” there’s nothing such as being overpriced! Also, all current attempts at pricing bitcoin will not be efficient enough because noise to signal ratio can be infinitely large. But which also indicates that under such heuristic, “Bubbles are mathematically impossible in this new paradigm”.( I guess those who questioned John simply has not understood the depth of his thought ; ). While I do not have a complete picture of Taleb’s (or John’s)work, I managed to replicate some of the best fits, see below. (Data sourced from ) coinmetrics.io Here’s a closers look of each individual fit. Bitcoins log-return from 2013–5–5 till 2018–11–17 was divided into 200 tiles and fitted against 200 tiles created by the corresponding pdf, hand-picked top 5 besting fitting distributions out of near a hundred different shapes, via means of minimizing SSE. Out of 5 best fits, both T distribution and Cauchy have an undefinable variance.Empirically, the average daily bitcoin log-return is around 0.002. Though debatable about the true distribution of bitcoins log-return, I am fairly confident that the log-return of bitcoin is extremely fat-tailed. So fat that it’s unseen and unprecedented by any asset that ever existed. A 90% decrease followed by extreme rebound isn’t that unlikely, it is even bound to happen given long enough time. Such volatility means investing in bitcoin isn’t for the faint-hearted and will ultimately hinder bitcoin’s quests to become the world’s sovereignless currency reserve, of such volatility in time, it is an acceptable store of value, As for not only Cult level faith on the long term value of bitcoin is required, timing of your investment is also extremely important. As I previously mentioned, If one brought bitcoin at 20,000 USD, given current level and an expected log-return of 0.002/day, bitcoin is not to touch that level in the next 2 or 3 years. Also if measured by classic risk-return measurements such as Sharpe ratio, Bitcoin is only an OK-ish investment, mostly due to extreme volatility. GIVEN persistency NOT expected But there are schools of bitcoin evangelist who believe the volatility of bitcoin will eventually die down. For example, studies presented by suggested that bitcoins volatility will die down as soon as this year. See chart below, directly copied over from Willy’s Study. willy woo Source: where Willy extrapolated Bitcoins historical highest volatility. https://woobull.com/bitcoin-volatility-will-match-major-fiat-currencies-by-2019/ Here’s a more recent chart on bitcoin’s volatility. 10 days standard deviations on bitcoins daily log-diff, rolling Combining the two charts above, one can observe that indeed we are experiencing lower and lower maximum volatility over the years. But to conclude that the volatility of bitcoin will die down based on these historical data is, rigorously speaking, flawed and mathematically unverifiable. There are ways, however, to test whether the data follows the same distributional shape over time. ( ) for the statistically informed or interested, I am referring to the test of stationarity&ergodicity, I skipped stationarity test, the log-diff process is stationary but not ergodic, you may verify it on your own Log-return of bitcoin from 2013 till 2018, in two graphs Here are a few lines of code to test whether the distributional shape of bitcoins log-diff is the same over years. import statsmodels.sandbox.stats.runs as runsimport numpy as npimport pandas as pd df=pd.read_csv('btc.csv', sep=',' ,index_col=0,header=0,encoding="utf-8-sig")P=df["price(USD)"]dP=np.diff(np.log(P))d1=dP[:int(len(dP)/2)]d2=dP[int(len(dP)/2)+1:]runs.runstest_2samp(d1,d2)ties detected#null hypothesis that d1 and d2 are from the same distribution is rejected #download coinmetrics bitcoin data Out[61]: (-5.116167239251087, 3.1180662016738176e-07) Here I used Wald-Wolfowitz run test to measure whether the distributional shape of bitcoin is consistent over a different period of time. Statistically, the best conclusion I can draw from it is that the shape and volatility of bitcoin’s log-diff over time does not stay the same. I am not able to say whether the volatility has decreased over time or has it increased. But as long as the shape changes over time. it gives hope to discover the true measurement on bitcoins’ price process and modeling. While at this stage I am not able to rigorously testify what can be the true model behind bitcoin’s price process, There is one scenario and modeling choice that converged well on a theoretically decreasing volatility. Namely, Metcalfe’s law and the growth of underlying follows a logistic population growth trajectory. See blew. V is defined as Bitcoin’s network value, N as total users of the Bitcoin protocol modeled as a logistic population growth function. we made a simplistic assumption that price of bitcoin is monotonical linearly equal to Network value. the variance from population growth rate as the single source of error. No matter how you model it, the model can never rival the complexity of real word but like all else, errors add up and at least Metcalfe's law can be somewhat used to explain part of the Bitcoin price process, we are confident the errors from that part will be gradually diminished. IF Above also lies in the rigid setting that bitcoin’s success rate is 100%, which, obviously, is a subject of much debate. Interestingly enough, the empirical variance of bitcoin log-diff is times of the empirical variance of the log-diff of the usage dynamics. As seen from above, I deducted that the variance of bitcoin is times to that of active addresses. hmmm, I am at least directionally right….. 5.8 4 Reference: _An economic bubble or asset bubble (sometimes also referred to as a speculative bubble, a market bubble, a price…_en.wikipedia.org Economic bubble - Wikipedia _Beginning in 2018, there has been a large sell-off of cryptocurrencies. From January to February 2018, the price of…_en.wikipedia.org Cryptocurrency bubble - Wikipedia _Population dynamics is the branch of life sciences that studies the size and age composition of populations as…_en.wikipedia.org Population dynamics - Wikipedia _Bitcoin was created in 2009 by a mysterious character who claimed it to be a payments network. But unlike most other…_woobull.com Bitcoin volatility will match major fiat currencies by 2019 https://coinmetrics.io/ Grazzini, Jakob (2012) ‘Analysis of the Emergent Properties: Stationarity and Ergodicity’ Journal of Artificial Societies and Social Simulation 15 (2) 7 . doi: 10.18564/jasss.1929 Wald, A. B. R. A. H. A. M., & Wolfowitz, J. (1943). An exact test for randomness in the non-parametric case based on serial correlation. , (4), 378–388. The Annals of Mathematical Statistics 14

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