Why Do Markets Crash? Bitcoin Data Offers Unprecedented Insights

Abstract

Crashes have fascinated and baffled many canny observers of financial
markets. In the strict orthodoxy of the efficient market theory, crashes must
be due to sudden changes of the fundamental valuation of assets. However,
detailed empirical studies suggest that large price jumps cannot be explained
by news and are the result of endogenous feedback loops. Although plausible, a
clear-cut empirical evidence for such a scenario is still lacking. Here we show
how crashes are conditioned by the market liquidity, for which we propose a new
measure inspired by recent theories of market impact and based on readily
available, public information. Our results open the possibility of a dynamical
evaluation of liquidity risk and early warning signs of market instabilities,
and could lead to a quantitative description of the mechanisms leading to
market crashes.

Full text

Why Do Markets Crash ? Bitcoin Data Offers
Unprecedented Insights
Jonathan Donier1,2,3and Jean-Philippe Bouchaud1,4
1: Capital Fund Management, 23-25 Rue de l’Universit´e, 75007 Paris, France
2: Laboratoire de Probabilit´es et Mod`eles Al´eatoires, Universit´e Pierre et Marie Curie (Paris 6), 4
Place Jussieu, 75005 Paris
3: Ecole des Mines ParisTech, 60 Boulevard Saint-Michel, 75006 Paris
4: CFM-Imperial Institute of Quantitative Finance, Department of Mathematics, Imperial College,
180 Queen’s Gate, London SW7 2RH
Abstract
Crashes have fascinated and baffled many canny observers of financial markets. In
the strict orthodoxy of the efficient market theory, crashes must be due to sudden
changes of the fundamental valuation of assets. However, detailed empirical studies
suggest that large price jumps cannot be explained by news and are the result of
endogenous feedback loops. Although plausible, a clear-cut empirical evidence for
such a scenario is still lacking. Here we show how crashes are conditioned by the
market liquidity, for which we propose a new measure inspired by recent theories of
market impact and based on readily available, public information. Our results open
the possibility of a dynamical evaluation of liquidity risk and early warning signs of
market instabilities, and could lead to a quantitative description of the mechanisms
leading to market crashes.
Contents
1 Introduction 2
2 Anatomy of April 10, 2013 crash 2
3 Three definitions of “liquidity” 6
4 Comparing the liquidity measures 8
5 Discussion – Anticipating crashes? 10
1

1 Introduction
Why do market prices move? This simple question has fuelled fifty years of academic debate,
reaching a climax with the 2013 Nobel prize in economics, split between Fama and Shiller who
promote radically different views on the question[32]. Whereas Fama argues that markets
are efficient and prices faithfully reflect fundamental values, Shiller has shown that prices
fluctuate much more than what efficient market theory would suggest, and has insisted
on the role of behavioural biases as a source of excess volatility and price anomalies. Of
particular importance is the origin of the largest changes in prices, aka market crashes, that
may have dire consequences not only for market participants but also for the society as a
whole [35]. It is fair to say that after centuries of market folly [25, 22, 34, 30], there is
no consensus on this issue. Many studies [15, 21, 11] have confirmed the insight of Cutler,
Poterba & Summers [12] who concluded that [t]he evidence that large market moves occur on
days without identifiable major news casts doubts on the view that price movements are fully
explicable by news.... The fact that markets appear to crash in the absence of any remarkable
event suggests that destabilising feedback loops of behavioural origin may be at play[33, 24,
20, 7]. Although plausible, a clear-cut empirical evidence for such an endogenous scenario
is still lacking. After all, crashes are not that frequent and a convincing statistical analysis
is difficult, in particular because of the lack of relevant data about the dynamics of supply
and demand during these episodes.
In this respect, the Bitcoin[29, 1, 6] market is quite unique on many counts. In particular,
the absence of any compelling way to assess the fundamental price of Bitcoins makes the
behavioral hypothesis highly plausible. For our purpose, the availability of the full order
book1at all times provides precious insights, in particular before and during extreme events.
Indeed, at variance with most financial markets where participants hide their intentions, the
orders are placed long in advance by Bitcoin traders over large price ranges. Using two highly
informative data-sets – the trade-by-trade MtGox data between December 2011 and January
2014, and the full order book data over the same period – we analyse in depth the liquidity of
the Bitcoin market. We find that what caused the crash was not the selling pressure per se,
but rather the dearth of buyers that stoked the panic. Following up on this observation, we
show that three different liquidity measures that aim at quantifying the presence of buyers
(or sellers) are highly correlated and correctly predict the amplitude of potential crashes.
Whereas two of them are direct probes of the prevailing liquidity but difficult to access on
financial markets, the third one – which is also firmly anchored theoretically[14] – only uses
readily available, public information on traded volumes and volatility, and is therefore a
promising candidate for monitoring the propensity of a market to crash.
2 Anatomy of April 10, 2013 crash
Amongst all crashes that happened on the Bitcoin and for which we found some data, the
April 10, 2013 crash is probably the most interesting one since on that day the price dropped
by more than 50% of its value in a few hours. At that time, MtGox was by far the leading
1The order book is the record of all intentions to buy or sell at a given point in time, each volume coming
with an offering price.
2

exchange (its market share was over 80% on the BTC/USD spot market) so our data-set
captures a large fraction of the investors’ behaviour. Intuitively, the main driver of market
crashes is the mismatch between the aggregate market order flow imbalance (O, defined
below) that becomes strongly negative and the prevailing liquidity on the buy side, i.e. the
density of potential buyers below the current price. Whereas the former quantity can be
easily reconstructed from the series of trades, the notion of “prevailing liquidity” is only at
best ambiguous. It is only when the price starts heading down, that one expects most of the
interested buyers to declare themselves and post orders in the order book. Therefore, the
liquidity cannot in principle be directly inferred from the information the publicly available
order book. The dynamic nature of liquidity has been clearly evidenced[38, 8], and has
led to the notion of “latent” liquidity that underpins recent theories of impact in financial
markets [37, 26, 27, 14].
0
10
20
30
40
50
60
70
80
30 35 40 45 50 55
Volume (kBTC)
Price
0
10
20
30
40
50
60
70
80
30 35 40 45 50 55
Volume (kBTC)
Price
φ
LOB (φ)
Cumulated demand
Cumulated supply
Figure 1: Instantaneous cumulated order book. Snapshot of the cumulated supply and
demand displayed on the order book, with a graphical representation of the order book
liquidity LOB(φ) defined in Def. 1.
However, Bitcoin is quite an exceptional market in this respect, since a large fraction of
the liquidity is not latent, but actually posted in the order book – possibly resulting from
less strategic participants on a still exotic market – and thus directly observable (see Fig. 1).
This allows us to test in detail the respective roles of aggregate imbalance and liquidity in
the triggering of market crashes. We first study the “aggressive” order flow defined as the
aggregated imbalance of market orders for every 4 hours window between January 2013 and
August 2013. In fact, two definitions are possible. One is defined as the average of the signed
number of Bitcoin contracts sent as market orders2,OB=Piiqi,where each iis a different
market order of sign iand number of contracts qi, and the sum runs over consecutive trades
in a 4 hour window. The second is the volume imbalance expressed in USD: O$=Piiqipi,
2A market order is an order to trade immediately at the best available price. Because of this need for
immediacy, one often refers to them as aggressive orders.
3

where piis the i-th transaction price. These two quantities are shown in Fig. 2 and reveal
that (a) large sell episodes are more intense than large buy episodes; (b) when expressed in
Bitcoin, the sell-off that occurred on April, 10 (of order of 30,000 BTC on a 4h window) is
not more spectacular than several other sell-offs that happened before or after that day; (c)
however, when expressed in USD, the April 10 sell-off indeed appears as an outlier.
-5
-4
-3
-2
-1
0
1
2
3
4
01/2013 02/2013 03/2013 04/2013 05/2013 06/2013 07/2013 08/2013
Imbalance (mUSD)
Time
-50
-40
-30
-20
-10
0
10
20
30
40
01/2013 02/2013 03/2013 04/2013 05/2013 06/2013 07/2013 08/2013
Imbalance (kBTC)
Time
4h imbalance (mUSD)
crash value
4h imbalance (kBTC)
crash value
Figure 2: Order flow imbalances in USD and BTC. Top: Aggressive imbalance in order
flow Piivi(where i=±1 is the sign of the transaction, and viits volume in Bitcoins),
aggregated over periods of 4 hours between January 2013 and August 2013, expressed in Bit-
coins. April 10, 2013 (for which the realised imbalance is represented as a dashed horizontal
line) does not appear as an outlier. Bottom: aggressive imbalance in order flow Piivipi,
aggregated over periods of 4 hours between January 2013 and August 2013 and expressed in
USD. April 10, 2013 now clearly appears as an outlier.
The difference between OBand O$originates from the fact that a large fraction of this
selling activity occurred at the peak of the “bubble” that preceded the crash, see Fig. 3,
top. The BTC price rose from $13 in early January to $260 just before the crash. In
Fig. 3, we represent a “support” level p40k
Ssuch that the total quantity of buy orders
between p40k
Sand the current price ptis 40,000 BTC, see Fig. 1. One notices that the price
dramatically departed from the support price during the pre-crash period, which is a clear
4

sign that Bitcoin price was engaged in a bubble. Although the liquidity expressed in USD
was actually increasing during that period (see Fig. 3, middle), the BTC price increased
even faster, resulting in a thinner and thinner liquidity on the buy side of the order book
expressed in BTC, see Fig. 3, bottom. This scenario is precisely realised in some Agent
Based Models of markets[18].
0
50
100
150
200
250
300
16/03/2013 23/03/2013 30/03/2013 06/04/2013 13/04/2013 20/04/2013
Price
Time
0
2
4
6
8
10
16/03/2013 23/03/2013 30/03/2013 06/04/2013
Liquidity LOB(φ) (mUSD)
Time
0
50
100
150
200
16/03/2013 23/03/2013 30/03/2013 06/04/2013
Liquidity LOB(φ) (kBTC)
Time
Actual price
Support price
φ= 10%
φ= 20%
φ= 50%
φ= 10%
φ= 20%
φ= 50%
Figure 3: Liquidity and support price. Top: Actual price pt(blue) vs. support price p40k
S
(red) defined as the price that would be reached if a typical sell-off of 40,000 BTC was to occur
instantaneously. Note that p40k
Sis ≈50% below the price ptjust before the crash, explaining
the order of magnitude of the move that happened that day. Middle (resp. Bottom): Buy
volume LOB(φ) in USD (resp. BTC) in the order book, during the months preceding the
crash of April 10, 2013, measured as the volume between the current price ptand pt(1 −φ)
where φ= 10%, 20% and 50%. One can see that for any quantile the liquidity in USD
tended to increase by an overall factor '2 during the period, while the liquidity in BTC
was decreased by a factor '2−3 as an immediate consequence of the bubble.
5

The conclusion of the above analysis, that may appear trivial, is that the crash occurred
because the price was too high, and buyers too scarce to resist the pressure of a sell-off. More
interesting is the fact that the knowledge of the volume present in the order book allows one
to estimate an expected price drop of ≈50% in the event of a large – albeit not extreme –
sell-off. Of course, the possibility to observe the full demand curve (or a good approximation
thereof) is special to the Bitcoin market, and not available in more mainstream markets
where the publicly displayed liquidity is only of order 1% of the total daily traded volume.
Still, as we show now, one can built accurate proxies of the latent liquidity using observable
quantities only, opening the path to early warning signs of an impeding crash.
3 Three definitions of “liquidity”
More formally, the market liquidity measure discussed above is defined as:
Definition 1 The order-book liquidity LOB (φ)(on the buy side) is such that (cf. Fig. 1
above):
Zpt
pt(1−φ)
dpρ(p, t) := LOB(φ),(1)
(and similarly for the sell-side). In the above equation, ptis the price at time tand ρ(p, t)
is the density of demand that is materialised on the order book at price pand at time t.
Conversely, the price drop −φ∗ptexpected if a large instantaneous sell-off of size Q∗
occurs is such that:
φ∗=L−1
OB(Q∗),(2)
where L−1
OB is a measure of illiquidity.
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70
Liquidity-adjusted imbalance (%)
L O
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70
Actual drop (%)
Imbalance (kBTC)
O
Figure 4: Forecast of crashes amplitudes using order book volumes. For the 14 most extreme
negative returns that have occurred between Jan 1, 2013 and Apr 10, 2013, we compare the
realised return with: (Left) the net imbalance OBduring the period (usually a few hours)
and (Right) the liquidity-adjusted imbalance L−1
OB(OB). This illustrates the relevance of the
LOB liquidity measure to predict the amplitudes of crashes – even in the most extreme cases.
6

An a posteriori comparison between realised returns and the liquidity-adjusted imbalance
for the 14 most extreme negative returns that have occurred between Jan 1, 2013 and Apr
10, 2013 is shown in Fig. 4. It shows that the quantity L−1
OB(OB) nearly perfectly matches
crashes amplitudes, vindicating the hypothesis that most of the liquidity is indeed present
in the visible order book for the Bitcoin.
However, as recalled above, the visible order book on standard financial markets usually
contains a minute fraction of the real intentions of the agents. Therefore the use of LOB(φ)
deduced from the observable order book would lead to a tremendous underestimation of
the liquidity in these markets [31, 38]. Liquidity is in fact a dynamic notion, that reveals
itself progressively as a reaction (possibly with some lag) to the incoming order flow [38, 8].
Another definition of liquidity, that accounts for the progressive appearance of the latent
liquidity as orders are executed, is based on a measure of market impact. With enough
statistics, the average (relative) price move I(Q) = h∆p/piinduced by the execution of
a meta-order3can be measured as a function of their total volume Q. Since these meta-
orders are executed on rather long time scales (compared to the transaction frequency), it is
reasonable to think that their impact reveals the “true” latent liquidity of markets [37, 26,
27, 14]. This leads us to a second definition of liquidity, based on market impact:
Definition 2 The impact liquidity LI(φ)is defined as the volume of a meta-order that
moves, on average, the price ptby ±φpt, or, more precisely, LI(φ)is fixed by the condition:
I(LI(φ)) = φ, (3)
since the impact I(Q)is usually measured in relative terms. As above, the price drop expected
if a large sell-off of volume imbalance Q∗occurs is simply given by L−1
I(Q∗) = I(Q∗).
The problem with this second definition is that it requires proprietary data with suffi-
cient statistics, available only to brokerage firms or to active asset managers/hedge funds. It
turns out to be also available for Bitcoin [13] – see below. However, a very large number of
empirical studies in the last 15 years have established that the impact of meta-orders follows
an extremely robust “square-root law”[36, 2, 28, 13, 5, 37, 23, 4, 19, 26, 9]. Namely, inde-
pendently of the asset class, time period, style of trading and micro-structure peculiarities,
one has:
ITH(Q)≈Y σdrQ
Vd
,(4)
where Yis an a-dimensional constant of order unity, Vdis the daily traded volume and σd
is the daily volatility. This square-root law has now been justified theoretically by several
authors, building upon the notion of latent liquidity[37, 26, 27, 14] (see Ref. [16] for an
alternative story). Assuming that the above functional shape of market impact is correct
leads to a third definition of liquidity:
Definition 3 The theoretical liquidity LTH(φ)is the theoretical volume of a meta-order
required to move the price ptby ±φptaccording to formula Eq. 4 above, i.e.:
ITH(LTH (φ)) = φ. (5)
3A meta-order is a sequence of individual trades generated by the same trading decision but spread out
in time, so as to get a better price and/or not to be detected[37].
7

Together with Eq. (4), this amounts to consider σd/√Vdas a measure of market illiquidity.
Clearly, since both σdand Vdcan be estimated from public market data, this last definition of
liquidity is quite congenial. It was proposed in Ref. [10] as a proxy to obtain impact-adjusted
marked-to-market valuation of large portfolios, and tested in Ref. [23] on five stock market
crashes, with very promising results. However, there is quite a leap of faith in assuming
that our above three definitions are – at least approximately – equivalent. This is why the
Bitcoin data is quite unique since it allows one to measure all three liquidities LOB,LI,LTH
and test quantitatively that they do indeed reveal the very same information.
4 Comparing the liquidity measures
We measured the order book liquidity LOB by averaging the volume present at all prices
in the buy side of the order book on disjoint 15 minutes periods. The empirical impact
is obtained following Ref. [13] by measuring the full I(Q), obtained as an average over all
meta-orders of a given volume Qon a given day. Finally, the theoretical impact Eq. 4 is
obtained by measuring both the traded volume of the day Vdand the volatility 4σd.
-140
-120
-100
-80
-60
-40
-20
0
20
11/2012 01/2013 03/2013 05/2013 07/2013 09/2013 11/2013
% of drop if a typical sell-off happens
Time
I(Q)/√Q(L−1
I)
σd/√Vd(L−1
T H )
Liquidity (L−1
OB )
Figure 5: Comparison between the three (il-)iquidity measures. Parallel evolution of the
three price drops φ∗deduced from our three estimates of illiquidity L−1
OB,L−1
I,L−1
TH defined
above. The estimates based on L−1
I,L−1
TH have been rescaled by a factor 6.104to match the
average order book data prediction.
4defined as σ2
d=1
TPT
t=1 0.5ln (Ht/Lt)2−(2ln(2) −1) ln (Ct/Ot)2where Ot/Ht/Lt/Ctare the
open/high/low/close prices of the sub-periods [17].
8

These three estimates allow us to compare, as a function of time (between November
2012 and November 2013) the expected price drop for a large sell meta-order of size – say –
Q∗= 40,000 BTC, see Fig. 5. We have rescaled by a constant factor the predictions based
on LIand LTH, so as to match the average levels. The agreement is quite striking, and shown
in a different way in Fig. 6 as a scatter plot of L−1
OB vs L−1
Ior L−1
TH, either on the same day,
or with a one day lag. As coinciding times, the R2of the regressions are ≈0.86 and only
fall to ≈0.83 with a day lag, meaning that one can use past data to predict the liquidity of
tomorrow. As a comparison, when using instead Amihud’s[3] measure of illiquidity σd/Vd,
one obtains R2of resp. 0.74 and 0.71.
0
20
40
60
80
100
120
0 20 40 60 80 100 120
L−1
I/L−1
OB R2= 0.88
0
20
40
60
80
100
120
0 20 40 60 80 100 120
L−1
T H /L−1
OB R2= 0.86
0
20
40
60
80
100
120
0 20 40 60 80 100 120
ILLIQ/L−1
OB R2= 0.74
0
20
40
60
80
100
120
0 20 40 60 80 100 120
L−1
T H (lagged)/L−1
OB R2= 0.83
Figure 6: Regression of the actual (il-)liquidity against the different (il-)liquidity measures.
Regression of the actual illiquidity L−1
OB on three same-day illiquidity measures (after rescaling
so that the samples means coincide): The direct measure of orders market impact L−1
I, the
publicly available measure L−1
TH that corresponds to the theoretical and empirical impact, and
the well-known Amihud ILLIQ measure [3]. Both L−1
Iand L−1
TH outperform ILLIQ (R2≈0.86
vs. 0.74). Note that a high predictability remains when lagging L−1
TH by one day (R2≈0.83
vs. 0.71). The regression slopes for the four graphs are, respectively: 0.9,0.95,0.87 and 0.93.
That the estimates based on LIand LTH match is no surprise since the square-root law
was already tested with a high degree of precision on the Bitcoin[13]. But that the theoretical
measure of liquidity LTH based on easily accessible market data is able to track so closely the
information present in the whole order book is truly remarkable, and suggests that one can
indeed faithfully use LTH on markets where reliable information on the latent order book is
absent (as is the case for most markets).
9

5 Discussion – Anticipating crashes?
Thanks to the unique features of the Bitcoin market, we have been able to investigate
some of the factors that determine the propensity of a market to crash. Two main features
emerge from our study. First, the price level should lie within a range where the underlying
demand (resp. supply) is able to support large – but expected – fluctuations in supply (resp.
demand). When the price is clearly out of bounds (for example the pre-April 2013 period for
Bitcoin) the market is unambiguously in a precarious state that can be called a bubble. Our
main result allows one to make the above idea meaningful in practice. We show that three
natural liquidity measures (based, respectively, on the knowledge of the full order book, on
the average impact of meta-orders, and on the ratio of the volatility to the square-root of
the traded volume, σd/√Vd) are highly correlated and do predict the amplitude of a putative
crash induced by a given (large) sell order imbalance.
Since the latter measure is entirely based on readily available public information, our
result is quite remarkable. It opens the path to a better understanding of crash mechanisms
and possibly to early warning signs of market instabilities. However, while we claim that the
amplitude of a potential crash can be anticipated, we are of course not able to predict when
this crash will happen – if it happens at all. Still, our analysis motivates better dynamical
risk evaluations (like value-at-risk), impact adjusted marked-to-market accounting [10] or
liquidity-sensitive option valuation models. As a next step, a comprehensive study of the
correlation between the realised crash probability and σd/√Vdon a wider universe of stocks –
expanding the work of Ref.[23] – would be a highly valuable validation of the ideas discussed
here.
Acknowledgements: We thank A. Tilloy for his insights on the Bitcoin and for reading
the manuscript; P.Baqu´e for reading the manuscript; and J. Bonart for useful discussions.
Bitcoin trades data are available at http://api.bitcoincharts.com/v1/csv/. Bitcoin order
book data have been collected by the authors and are available on request.
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