## Thursday, November 23, 2017

### When you should jump out of a (Bitcoin) bubble

No reliable answer may be said, of course, but the math is still interesting

Let me assume that the reader agrees that there is some probability of a huge, fast enough, cataclysmic collapse of the Bitcoin price – when the long-term sentiment dramatically changes, most people agree that the future trend is "down" so they try to escape as quickly as possible, or some big ban in an important country is enacted etc. That's the bad news. Let's assume that the probability of the sudden death is described by the mathematics of the decay of a radioactive nucleus.

On the other hand, there are good news: Let's assume that before the sudden death, the Bitcoin price will grow exponentially. We neglect some 20% fluctuations away from the growing line which are the "normal error margins" in the Bitcoin world. OK, what does mathematics tell you about how quickly you should sell your Bitcoins and how much you should hold at each moment? Surely smooth mathematics recommends you some nice algorithm quantifying what percentage you should sell tomorrow, what percentage you should sell next month, and so on.

Well, it doesn't. It tells you something less smooth and simpler. ;-)

First, let's assume that the Bitcoin price in U.S. dollars follows an exponentially growing curve${\rm Price}(t) = {\rm Price}(0) \exp(t/t_0)$ where $$t_0$$, the time when the price increases $$2.718$$ times, is approximately equal to half a year in the recent year or two.

On the other hand, there is some probability that the Bitcoin hasn't died yet. The probability decreases exponentially with time:${\rm Prob}(t) = \exp(-t/t_1)$ where $$t_1$$ is another time scale. The probability of "survival" is taken to be $$1$$ at the moment $$t=0$$, now, because the Bitcoin hasn't collapsed yet; I checked it while I was writing the word "writing" and please accept my apologies if the sentences are already obsolete. ;-)

OK, so how quickly should you sell at time $$t$$? The answer is actually singular. You may simply compute the expectation value of the price of the Bitcoin at any moment $$t$$. It has an optimistic, growing factor corresponding to the exponential growth; but it also has a factor corresponding to the probability that you still own anything at all.$\langle {\rm Assets}(t) \rangle = {\rm Assets} (0) \cdot \exp(t/t_0-t/t_1).$ The exponent is simply $$t$$ multiplied by the difference $$1/t_0 - 1/t_1$$. If the difference is positive, the growth wins and you may neglect the probability of the sudden death. If you only care about the expectation value, you should hold the Bitcoin for as long as possible! On the other hand, if the difference is negative, the exponential factor from the decay wins, and you should sell all the Bitcoins now! It's this simple.

So with this simple model, mathematics doesn't give you any smooth "compromise". Instead, it tells you "everything or nothing". Depending on an inequality, the optimum strategy is either to sell immediately, or hold forever. This is just a manifestation of the fact that the Bitcoin "investment" is not a real investment but a maximally extreme gambling game.

Great. So is the difference positive or negative? It depends on the question whether the inequality$t_0 \lt t_1$ is true. If the lifetime associated with the "sudden death" radioactive decay is shorter than the timescale associated with the $$e$$-fold increase of the Bitcoin price, then you should sell immediately. Otherwise, you should hold for your dear life and only sell when you really need some money. Which of the scenarios is true?

I mentioned that the $$e$$-folding of the exponentially growing price could take place every half a year. So if the "rate of the sudden death" is smaller than "one death per half a year", then you may want to hold the Bitcoin for your dear life (HODL). In the real world, people don't care about the mean value only. They want to avoid significant risks, too. So many of them could prefer to sell earlier – or now – even if the inequality is slightly reversed. Also, they could lock some profits at some moment, keep the rest and so on. But no "canonical" prescription exists – the profile "when to sell gradually" is just a reflection of a curve encoding the person's decreasing tolerance to greater risks.

Even if we knew that the price gets multiplied by $$e$$ every six months, and we don't know that, we surely don't know the probability rate of the sudden death. It has never happened. The Bitcoin hasn't gotten a lethal blow yet, unlike some other currencies, so we just don't know. But what we do know is that the sudden death hasn't taken place for years. We could say that it hasn't taken place since the beginning, for 8 years i.e. 16 $$e$$-folding times, although because of an April 2013 crash by 61%, we could prefer to say that the safe period has only been some 9 $$e$$-foldings.

This longevity of the growing Bitcoin bubble may be viewed as circumstantial evidence that the bubble could continue to grow forever. And some uncritical promoters of the Bitcoin mania surely want to make the people believe it will be the case – this belief could perhaps act as a self-fulfilling prophesy.

Is it reasonable at all to believe that the Bubble will collapse at all, given the record of survival?

With the simple model above, two exponentials fighting each other, it's actually unreasonable. The experimental facts exclude the probability that the sudden death could beat the exponential price growth at roughly 5 sigma. Why? Because the Bitcoin price has already grown by a factor of one million since the Bitcoin was first traded. If the "sudden death" exponentially decreasing factor is more important, it means that the survival probability had to drop by a factor greater than one million, so it must be smaller than one millionth!

But if the probability is smaller than one millionth, it's very tiny and the assumption that the Bitcoin survived, despite this prediction of "at least one death per half a year in average", is excluded at 5 sigma, at least if we take the whole history since 2009 to be "a basically OK exponential growth".

So if we want to rationally believe that the "sudden death" is actually more important for the mean value than the exponential growth, we need to use a model that suppresses the sudden death of a "young" Bitcoin, but increases it for an "older" Bitcoin. The simplest modification is to replace the exponent in the survival probability by a quadratic function:${\rm Prob}(t) = \exp(-t/t_1-t^2 / t_2^2)$ Again, it's been normalized to be equal to one at $$t=0$$. So it's really the conditional probability that it will survive up to time $$t$$ assuming that it has survived up to $$t=0$$, now. With the quadratic Ansatz for the exponent, the expectation value of your net worth will have the exponent which is also a quadratic function,$t\zav{\frac{1}{t_0} - \frac{1}{t_1} } - \frac{t^2}{t_2^2}.$ Note that we chose a negative sign of the quadratic term in order for the sudden death to become more likely as the Bitcoin is getting older. We need $$t_1$$ to be longer or much longer than half a year, in order to avoid the 5-sigma contradiction with the fact that the Bitcoin hasn't died yet. OK, the graph of the quadratic function above is an "upside down" parabola with a maximum at some value of $$t$$ given by the vanishing derivative of the function$\zav{\frac{1}{t_0} - \frac{1}{t_1} } = \frac{2t}{t_2^2}.$ So the maximum is at$t = \zav{\frac{1}{t_0} - \frac{1}{t_1} } \frac{t_2^2}{2}$ Now, the recommendation for the timing of your sales is still easy in principle. Sell at time $$t$$! This is where the expectation value of your assets is maximized. So if $$t$$ is positive, you should wait and sell at that time. If $$t$$ calculated above is negative, you were lucky that the Bitcoin hasn't collapsed yet, but you should sell immediately because the expectation value of your assets are already decreasing because of the growing risk of the sudden death! ;-)

OK, again, which of these answers is correct? Obviously, again, I won't be able to give you any unambiguous answer but the answer may be translated to various inequalities. For example, for the sake of simplicity, let's neglect the term $$1/t_1$$ in the expression above so that the probability of the "sudden death" is dominated by the quadratic term, and let's imagine we would redo the whole analysis with the variable $$T$$ that obeys $$T=0$$ at the moment when Nakamoto released the currency in 2009.

Well, if that were so, the condition $$T\lt {\rm now}$$ (8 years after Nakamoto) i.e. you should sell immediately is basically equivalent to$\frac{t_2^2}{2\cdot \text{8 years}} \gt t_0$ where $$t_0$$ is half a year, as I mentioned. So the question is whether $$t_2$$ – sort of a time scale where the quadratic term becomes important in the sudden death – is greater than $$\sqrt{8}$$ years. Is it? Again, I don't know. Nobody knows. But with this model, both options are totally plausible and there is no good enough way to disfavor either possibility. Because these are just two semi-infinite lines in a parameter space, you could estimate that the probability that the inequality $$t_2\lt \sqrt{8}\,{\rm years}$$ is comparable to 50%.

With this approach, you may say that the probability is some 50% that you should HODL and 50% that you should sell immediately.

At the end, the mathematics is useless in practice because you don't know the right model and even if you knew it, you don't know the values of the parameters. But one general lesson can be learned from the simple calculations, anyway: the answers are bound to be extreme. Either the best idea is to sell immediately or it is to hold for your dear life. Nobody knows for sure.

If you don't really want to sacrifice what you already have, you should sell immediately. If you can afford to lose it but you prefer big profits, you might want to hold because there's some chance that your holding will be exponentially more valuable than it is now. At any rate, if the Bitcoin remains unbacked, the character of the game won't change with time – analyses like one above were relevant years ago and they may be relevant in a few years as well if the Bitcoin manages to survive. The gambling will remain qualitatively the same. If the Bitcoin price increases further, however, it will become a gambling game involving a greater amount of money.

I actually do think that most people in the game will understand that they may very realistically lose everything. This will reduce the inflow of the money into this game at some point. Before the collapse, there may be a preparation, perhaps some plateau. People will notice that the doubling hasn't taken place for several months if not a year, graphs like "peak Bitcoin" may become a good approximation of the time series, and before the perceived global maximum, speculators start to sell as quickly as possible, thus quickly reversing the long-term expectations of everybody, and the sudden death comes suddenly, indeed.

Your humble correspondent is in no way sure that the sudden death will already arrive on December 11th or so when the CME futures are first traded. It's possible that the best probability I could assign to this proposition is still smaller than 50%. But I am confident that this probability of a December 11th sudden death, or soon afterwards, is much higher than almost all people are led to believe.

The general competition between the two rates may be viewed as a message for smart traders. Imagine that sometime in the future, in March 2018, there will be a risk of a sudden death, e.g. a complete ban of cryptocurrencies in the U.S. If the probability of the sudden death were greater than 10% in the next week, it would make complete sense to sell all cryptocurrency holdings because the expectation value of the holdings would be predicted to drop over that week – the decrease from the sudden death risk would beat the positive predicted exponential growth.

There are no pressures that would push the Bitcoin above any positive threshold. Some percentage of the Bitcoin traders know what they're doing and they're roughly calculating the expectation values above. So once the probability of the death exceeded the expected gains from the exponential growth, I find it plausible that the sales by the rational traders could drive the Bitcoin down to any value, perhaps below \$100 or further. Some people could always expect a recovery but only some percentage of the capital would be ready to re-enter the market after such a heart attack. If there were another heart attack, even the very long-term, "after many heart attacks" expected trend could become negative and almost everyone could try to sell – most of the people would do it too late.