# The Bitcoin Crash and How Nature Works

2 years of bitcoin 2103-2015

Moreover, several Bitcoin exchanges have shut down due to hacks and/or criminal activity, and key BitCoin players are on trial. This begs the question:

Why is BitCoin Crashing ?

Markets crash all the time.  Stock markets.  Currency Markets.   Even the Dutch Tulip market of 1637 crashed–although even this is still hotly debated.

Fraud?  Speculation?  Mania?  Lack of Regulation?  What gives?

### Market Crashes: the EconPhysics Perspective

In 1996, researchers at the University of Chicago[1] and elsewhere [2] independently proposed that market crashes resemble physical crashes such as earthquakes, ruptures, and avalanches.  Such phenomena arise from long range, collective fluctuations — i.e. herding behavior — that overtakes a system as it approaches a critical point…and/or in the aftershocks.

The theory relies upon a fairly generous application of the theory of critical phenomena and the Renormalization Group (RG) theory.  It provides both a qualitative description of the phenomena and quantitative predictions such as when a crash may occur or how long a bear market will continue.

Most notably, the strongest proponent, Didier Sornette, accurately estimated the long bear run of the Japanese markets–what he terms a Financial Anti-Bubble:

He has written numerous papers, a book, and even a TED Talk.

We also introduced these ideas in a previous post:  Noisy Time Series II: Earth Quakes, Black Holes, and Machine Learning

### Power Laws and Symmetry Breaking

It is well known that financial markets display non-Gaussian, long tail, power law-like behavior.  After a crash model, the price series itself $p(t)$may follow a power law

$p(t)=A+B(t-t_{c})^{\alpha}$

where $t_{c}$ is the time of the crash, the time $t>t_{c}$, and $\alpha$ is a real number.

We can fit the recent Bitcoin EOD (End of Day) prices to a power law, starting at it’s peak on Nov 11, 2013:

We obtain a modestly good least squares fit, with $\alpha=0.32$ and $R^{2} = 0.77$.  Clearly there is a lot of noise–or is there?

RG theory suggests that near a crash, the power law may become complex (i.e a form of  ‘symmetry breaking’).  We will now sketch how this comes about, using the nobel prize winning Renormalization Group theory.

#### Complex Power Laws and the Renormalization Group

When examining a price series $p(t)$, or some other phenomena, say $F(x)$, a critical point occurs when we observe a rapid, sharp spike–i.e. Bitcoin’s fast rise up to Nov 29, 2013.

Here, we will find that as $x\rightarrow 0$

$F(x)=A+B(x)^{\alpha}$,

where $\alpha$ is a complex critical exponent

We say $F(x)$ becomes singular near a crash when there exists a k-th derivative that becomes infinite as $x\rightarrow 0$:

$\lim_{x\rightarrow0}\dfrac{d^{k}F(x)}{dx^{k}}=0$

We also know that near a critical point, physical systems exhibit scale invariant (i.e. fractal) behavior. The noise, or fluctuations, we observe on small time scales $x=t-t_{c}$ look just like fluctuations on larger time scales $\phi(x)$.

We call $\phi(x)$ the RG flow map.  The flow map describes these change of time scales, and the RG equation describes the invariances.

The fundamental RG equation (eqn) is:

$x\rightarrow\phi(x)$

$F(x)\rightarrow g(x)+\dfrac{1}{\mu}F(\phi(x))$

We assume $F(x)$ is continuous–which is non trivial for price series –and that $\phi(x)$ is differentiable. $g(x)$ is the non-singular (regular) part of $F(x)$.

The magic of the RG theory is that it allows us to describe the behavior near a critical point knowing only the flow map $\phi(x)$ and the regular function $g(x)$.  The formal solution is an infinite order power series

$F(x)=\underset{n\rightarrow\infty}{\lim}f^{n}(x)$,

$f^{n}(x)=\sum_{i=1}^{n}\dfrac{1}{\mu^{i}}g(\phi^{[i]}(x))$

We will sketch a very simple, leading order approximation to $F(x)$ that leads to a complex power law.

#### RG Theory and Discrete Scale Invariance

We start by first assuming power law behavior (at lowest order):

$F_{0}(x)\sim x^{\alpha}$

We seek the simplest RG solution.  Let us assume the RG flow map is linear in $x$:

$\phi(x)=\lambda x+\cdots$,

where $\lambda > 1$ (for technical reasons to ensure the critical point is an unstable fixed point of the RG flow).

We will ignore the regular solution $g(x)$ and write the simplified RG eqn:

$F(\lambda x)=\mu F(x)$

For our simple power law, we have

$(\lambda x)^{\alpha}=\mu x^{\alpha}$

This is easy to satisfy if

$\dfrac{\lambda^{\alpha}}{\mu}=1$

or

$\alpha=\dfrac{log(\mu)}{log(\lambda)}$

We seek the most general solution, applicable to discrete physical systems, such as earthquakes, ruptures in materials–and financial market crashes.   That is, we expect $\lambda$ to take on discrete values:

$\lambda\in[\lambda_{1},\lambda_{2},\cdots]$

We call this Discrete Scale Invariance (DSI)

The most general solution of DSI is

$F(x)=x^{\alpha}P(\dfrac{\log x}{\log \lambda})$

where $P()$ is a general periodic function.  As above, we express $P()$ as the limit of an infinite power series

$P_{n}(x)=\sum_{i=1}^{n}c_{n}\exp\left[2n\pi i\left(\dfrac{\log x}{\log\lambda}\right)\right]$

This is a classic Weierstrass function–a pathological function that, in the infinite limit, is continuous everywhere and yet differentiable nowhere.  It is used to model fractals, natural systems that are scale invariant, and highly discontinous but structured time series.

In a crash scenario, we expect that the true power series for $P(x)$ will diverge; i.e. the k-th derivative explodes.  So here, we only take the leading term of our linearized model

$exp\left[2n\pi i\left(\dfrac{\log x}{\log\lambda}\right)\right]$

Note that it takes the form

$F(x)=F_{0}(x)p(\log F_{0}(x))$

so $F_{0}(x)$, simple power law behavior, is like the regular part of the solution.

We now see that we have a complex critical exponent $\alpha$:

$\dfrac{\lambda^{\alpha}}{\mu}=e^{2\pi n}$

or

$\alpha=\dfrac{log(\mu)}{log(\lambda)}+i\dfrac{2\pi n}{log(\lambda)}$

We consider the real part of the complex power law

$Re[x^{\alpha+i\beta}]=x^{\alpha}cos(\beta\log x)$

which gives the following log periodic formula for the price series

$p(t)=A+B(t-t_{c})^{\alpha}(1+C\cos(\omega\log(t-t_{c})+\phi))$

Instead of fluctuating randomly around the underlying power law drift, the price series exhibits log-periodic oscillations–the DSI signature of a crash.

This model appears to work for 1-2 oscillations before (or after) the crash.  To model the longer time anti-bubble behavior, Sornette has developed an extended formula, based on the third-order Landau expansion (as opposed to, say, additional terms in the power series defined above).  We leave these details and further analysis for later.

#### DSI Fit of the Bitcoin Crash

We now fit the latest Bitcoin behavior, assuming the crash / Bear market started on Nov 29, 2013 :

We readily find a DSI fit, with $R^2=0.89$.

It is actually quite difficult to get a very tight fit, and usually advanced methods are needed. This is a simple, crude fit done to illustrate the basic ideas.

We see that Bitcoin, like other financial markets, displays log periodic behavior, characteristic of a self-organized crash.

These kinds of crashes are not caused by external events or bad players–they are endemic to all markets and result from the cooperative actions of all participants.

We can try to detect them, perhaps even profit off them, but it is unclear how to avoid these Self-organized Critical points that wreck havoc on our finances.

### Conclusion: How Nature Works

Bitcoin is an attempt to create a new kind of currency–a CryptoCurrency–that is free from both institutional control and individual corruption.  It is hoped that we can avoid the kind of devastating inflation, devaluation, and crashes that occur all too frequently in our current worldwide banking systems.  The promise is that Bitcoin, backed by the Blockchain, can remove our dependence on a single, faulty institution and replace it with a decentralized, distributed form of trust.  But

“Real life operates at a critical point between order and chaos”

Our analysis here shows that even the Bitcoin economy still appears to behave like a traditional market–prone to kinds of crashes that frequently arise in all natural self organized systems.

This Self-Organized Criticality is perhaps just how nature works [6,7].

And while Bitcoin is certainly interesting and exciting, perhaps it too is subject to the natural laws of physics.

#### References

[1] 1995  James A. Feigenbaum and Peter G.O. Freund,  Discrete Scaling in Stock Markets Before Crashes

[4] D. Sornette, Critical market crashes

[6] 1999 A. Johansen and  D. Sornette, Financial “Anti-Bubbles”: Log-Periodicity in Gold and Nikkei collapses

[7] Anders Johansen and Didier Sornette, Evaluation of the quantitative prediction of a trend reversal on the Japanese stock market in 1999 , 2000

[8] 2003 Wei-Xing Zhou and  Didier Sornette  Renormalization group analysis of the 2000–2002 anti-bubble in the US S&P500 index: explanation of the hierarchy of five crashes and prediction

[9] 20122 Didier Sornette, Ryan Woodard, Wanfeng Yan, and Wei-Xing Zhou  Clariﬁcations to Questions and Criticisms on the Johansen-Ledoit-Sornette Bubble Model

[11] Per Bak,  “How Nature Works: The Science of Self-Organized Criticality”,  1996

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