Fear Universality and Doubt in Asset price movements
FFEAR, UNIVERSALITY AND DOUBT IN ASSET PRICEMOVEMENTS
IGOR RIVIN
Abstract.
We take a look the changes of different asset prices over variable pe-riods, using both traditional and spectral methods, and discover universality phe-nomena which hold (in some cases) across asset classes. Introduction
There has been a considerable amount of effort dedicated to understanding (and,ultimately, predicting) asset prices. We will give an idiosyncratic overview in Section2. In this note I describe a few experiments which both illustrate some of the short-comings of the current dogma and show surprising universality phenomena. I do notpretend to understand why this universality holds.Here is a brief outline of the rest of the paper.In Section 2 I give an overview of the current dogma.In Section 3 I look at the distribution of returns of a number of secturities overvarying lengths of time.In Section 4 I look at a somewhat more sophisticated way of computing statistics via the Hankel matrix ( trajectory matrix ) associated to the process.1.1.
Data.
In this paper we use the following data sets (all sets end on the ides ofMarch of 2018:(1) Alphabet, Inc - stock ticker
GOOG
We use data from the inception of thecurrent version of the company in March of 2014 to the present.(a) Alphabet Inc Stock price, log scale
Date : March 21, 2018.1991
Mathematics Subject Classification.
Key words and phrases. asset prices, universality, random matrices, Brownian motion, GeometricBrownian motion.The author would like to thank Andrew P. Mullhaupt for interesting comments. All the com-putations in this paper were conducting using the
Wolfram Mathematica system. The stock pricedata came from
Mathematica itself (via the
FinancialData[] mechanism, while the Bitcoin pricehistory came from Yahoo! Finance. All data terminates on March 16 2018. a r X i v : . [ q -f i n . M F ] M a r IGOR RIVIN
Figure 1.
Alphabet Inc Stock Price(2) Apple Computer, Inc - stock ticker
AAPL
We use data from 1991 to thepresent.(a) Apple Computer Stock price, log scale
Figure 2.
Apple Stock Price(3) General Electric, Inc – stock ticker GE We use data from 1962 to the present.(a) General Electric Stock price, log scale
EAR, UNIVERSALITY AND DOUBT IN ASSET PRICE MOVEMENTS 3
Figure 3.
GE Stock Price(4) Bitcoin - ticker
BTC
We use data from the time Bitcoin began trading in2010 to the present day.(a) Bitcoin exchange rate, log scale
Figure 4.
Bitcoin exchange rateThe three companies studied are all very large - this was intentional, since that meantthat we would not have any periods of low liquidity. It should, however, be notedthat before the decimalization in the early 2000s, the prices were discretized in muchlarger increments than later. 2.
A little history
The first model of stock returns is due to Louis Bachelier [Bac00, Bac12] has thefollowing components. The first is that the returns on different days are indepen-dent and identically distributed (i.i.d.) and the second is that the distribution ofreturns is Gaussian. The first assumption stems from an early form of the EfficientMarket Hypothesis (of which more anon), and the second from the philosophy thatthe price movement was due to a large number of small factors, and so some versionof the Central Limit Theorem would make the move Gaussian. It should be noted
IGOR RIVIN
Table 1.
Correlations of returns and magnitudes of returnsSeries Log return autocorrelation Absolute value of log return autocorrelationGOOG 0 . . − . . . . . . Now, there are a couple of major problems with all of the above.First, it is quite clear that whatever the distribution of returns is, it is not lognormal - if it were, market meltdowns like the ones in 1929, 1987, 2001, and 2008would occur far less frequently than they actually do (since these events are severalstandard deviations away from the mean). This is usually explained away by sayingthat the distribution of returns is sort of log-normal, but with fatter tails.Secondly, the day to day returns are clearly not independent. Table 1 shows auto-correlations of log returns of our four time series and also of the magnitudes of thelogs of autocorrelations.A look at Table 1 will tell us that while there might be some debate about thepredictive power of the return on day N of the return on day N + 1 (althoughautocorrelation of 6% as in the case of GOOG is certainly not negligible), there isabsolutely no doubt that the magnitude of the swing has some auto-regressive aspects.To deal with this problem, financial econometricians introduced another Brownian If the reader is keeping score, we are up to five Nobel prizes for Bachelier – one in physics andfour in Economics.
EAR, UNIVERSALITY AND DOUBT IN ASSET PRICE MOVEMENTS 5 motion to control the volatility, and this work (more specifically, the definition of aGARCH stochastic process) has produced yet another Nobel Memorial Prize (this oneto Robert Engle and Clive Granger). Interestingly, to the author’s knowledge, theresulting technology is not used by finance practitioners, due to its poor predictivepower.2.1.
State of the art, as she is spoke.
In the end, we have the following takeaways:a) The logarithms of the returns are modeled as Gaussian, except with fat tails.b) The returns on different days are kind of sort of independent random variables,butc) The return time series is not stationary (”heteroscedatic”, because that soundsreally cool).Not a very satisfying state of affairs.3.
The distributions
For our first attempt at enlightenment, we will do the simplest thing possible andlook at the distributions of logs of daily returns of our chosen instruments. Foreach of them we will look at the histogram of returns and also at a kernel-smoothedversion of the distribution (since it is conceivable that the discretization artifacts ofthe histogram will blind us to the truth.A) GE log return distribution
Figure 5.
GE log returnsB) AAPL log return distribution
Figure 6.
AAPL log returns
IGOR RIVIN
C) GOOG log return distribution
Figure 7.
GOOG log returnsD) BTC log return distribution
Figure 8.
Bitcoin log returnsA quick look at the graphs is enough to convince us that the only thing the dis-tributions have in common with the Gaussian distribution is the unimodality and(rough) symmetry - in fact, the GOOG,AAPL, and BTC distributions are visibly not symmetric about the mean. The one for GE is a bit more so.All of them have a more”triangular” shape around the mean than the Gaussian (which is relatively flat inthe neighborhood of the origin). To check that the distributions are, indeed, verydifferent from the Gaussian, we will look at the QQ(quantile-quantile) plots. For twocontinuous distributions d , d the points in such a plot are the points ( c ( q ) , c ( q )) , where c ( q ) is the inverse of the cumulative distribution function of d applied to q and c is the inverse of the cumulative distribution function of d likewise applied to q. If d = d , the qq plot will simply be the line x = y. If d and d differ by scaleand or location (that is, d ( x ) = d ( ax + b )) the qq plot will be a straight line. If theright tail of d is fatter than that of d , the graph will veer higher than the x = y line, and similarly for the left tail.As a warm-up we will look at the qq plot of Bitcoin log-returns vs the standardGaussian N (0 ,
1) - see Figure 9. It is quite clear that our original impression wascorrect, and the Bitcoin returns are nothing like log-normal. However, Bitcoin is anew instrument, so who can tell what is going on with it, so let’s look at our equitystalwarts - see Figure 10. We see again that these returns are nothing like log-normal.
EAR, UNIVERSALITY AND DOUBT IN ASSET PRICE MOVEMENTS 7
Figure 9.
Q-Q plot of Bitcoin log returns vs the standard Gaussian.
Figure 10.
Q-Q plot of AAPL, GOOG, and GE log returns vs thestandard Gaussian.
Figure 11.
Q-Q plot of AAPL, GOOG, and GE log returns vs Bitcoinlog returns.Let us see if the Bitcoin returns really are qualitatively different from the largecap stock returns. We see the Q-Q plots in Figure 11. We see that our conjectureabout the idiosyncrasies of Bitcoin returns was justified. Finally, let us look at howdifferently distributed the returns of our large cap triad are. The results are in Figure12:Now we are in for a bit of a surprise: the distributions are essentially the same, upto scaling and location. This is our first universality:
Conjecture 3.1 (First Universality Conjecture) . The distributions of (at least largecap) equities are essentially the same, and differ only in scale and location.
IGOR RIVIN
Figure 12.
Q-Q plots of AAPL, GOOG, and GE log returns vs eachother.The study of daily returns distribution is an inherently static activity, and so wemust look for more sophisticated tools to look at the dynamics of the situation.4.
The trajectory matrix and its spectrum
Suppose we want to predict stock prices (who doesn’t?) and we want to applysome out of the box machine learning algorithm. If all we have are past stock prices,then the usual way to approach it is to structure our data as follows: Let us assumethat our series of log returns is X = X , X , . . . , X n . . . . If our look back horizon is k time periods, then our inputs are laid out as the matrix: H k,n = à X X . . . X k X X . . . X k +1 . .. . .. . . . . .. X n X n +1 . . . X k + n − í , while our outputs are just the vector ( X k +1 , . . . , X n + k . The matrix H k,n is known inthe time series community as the trajectory matrix . It is a Hankel matrix (meaningthat the ij -th element of the matrix depends only on i + j ), and it is reasonably clearthat its singular values will have some relationship with the properties of the timeseries X . Indeed, the study of this relationship is a whole area of time series analysis -this subarea is known as SSA - Singular Spectrum Analysis. See, for example [GZ13].Singular spectrum analysis generally deals with k (cid:28) n, and is used for divining local properties of the time series, but we will take a bird’s eye view, and make k verylarge. So large that the matrix will be square, with both n and k being half as large EAR, UNIVERSALITY AND DOUBT IN ASSET PRICE MOVEMENTS 9 as our entire history. We will call this matrix H . We will study the spectrum of H and see if we find anything interesting.4.1. Eigenvalues of Random Matrices.
Now, it should be noted that randommatrix theory is a huge subject, and there are generally two kinds of invariants tolook at: bulk invariants (the spectral measure of our favorite class of asymptoticallyvery large matrices), and fine invariants (these include the spacings between adjacenteigenvalues, or the behavior of eigenvalues on the ”edge” of the spectrum. It is oftentrue that fine invariants are more universal than bulk invariants (for example, thereare strong connections between spacings between eigenvalues of very large Hermitianmatrices and spacings between zeros of the Riemann zeta function - see, for examplethe classic book by N. Katz and P. Sarnak - [KS99]. In any case, nothing muchseems to be known for spacings of large symmetric Hankel matrices, but there areresults on the limiting spectral measure by Bryc, Dembo, and Jiang - [BDJ06] - thespectral measure of a large square Hankel matrix coming from a series of i.i.d. randomvariables (with finite variance) X does approach a limit, which is a funny lookingbimodal distribution, though aside from its general shape, it seems that not so manyproperties of this distribution are known. Of course, we can easily sample from it, bygenerating a large sample from the standard Gaussian distribution, constructing thetrajectory matrix, and computing its eigenvalues. Here are the results:A) i.i.d. Gaussian trajectory matrix (3000 × Figure 13.
Random Hankel Matrix spectral densityB) i.i.d. Gaussian trajectory matrix (3000 × Figure 14.
Random Hankel Matrix spacings distribution eigenvalues of Hankel matrices can actually be computed in time O ( n log n ) . ) The careful reader will see that the spacings distribution is showing the characteristic“level repulsion” phenomenon - the peak of the distribution is away from zero, just asit is for the popular GOE, GUE, and GSE ensembles of matrices – [Meh04]. However,the distribution is none of those three. Below are some diagrams which drive the pointhome:A) Spacing distributions.
Figure 15.
Spacing distributions for standard ensembles andrandom Hankel MatricesB) i.i.d. Gaussian trajectory matrix (3000 × Figure 16. pairwise quantile-quantile plots of spacing distributionsThe sharp-eyed observer will note that the Hankel spacing distribution has fattertails than the other ones.4.2.
Back to the stocks.
Our journey is now nearing the end. For our next act, wegenerate maximal trajectory matrices for our data sets, and see what happens.First, the bulk eigenvalue distributions (Figure 17)How close are these to the “mother” random Hankel distribution? A look at Figure21 tells us that they are clearly different from their parent (though less different inthe case of GE, which is a sign that the differences might be a function of not-quitelarge enough sample size.)
EAR, UNIVERSALITY AND DOUBT IN ASSET PRICE MOVEMENTS 11
Figure 17.
Bulk eigenvalue distribution of trajectory matrices.
Figure 18.
Bulk eigenvalue distribution of trajectory matrices.The spacings are reasonably those of the parent distribution, as shown below. Notethat BTC is no longer an outlier.A) Spacing distributions of trajectory matrices
Figure 19.
Spacing distributionsB) Quantile plots of spacing distributions of trajectory matrices vs random Han-kel
Figure 20.
Spacing distributions vs random HankelFinally, we see how the spectra of the various trajectory matrices compare. First,the bulk distributions (Figure 21)And then the spacings (Figure 22): The bulk distributions look very close, thespacings distributions a little less so, but we have again eliminated the apparentlylarge differences between instruments (even going across asset classes). This leads usto another conjecture:
Conjecture 4.1 (spectral universality) . The spectral distribution and the spectralspacing distribution of trajectory matrices exhibits universality, across asset classes.
Figure 21.
Bulk eigenvalue distribution of trajectory matrices.
Figure 22.
Eigenvalue spacing distribution of trajectory matrices.
EAR, UNIVERSALITY AND DOUBT IN ASSET PRICE MOVEMENTS 13
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Mathematics Department, Temple University and The Cryptos Fund
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