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Random Walks, Liquidity Molasses and Critical Response in Financial Markets

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Stock prices are observed to be random walks in time despite a strong, long-term memory in the signs of trades (buys or sells). Lillo and Farmer have recently suggested that these correlations are compensated by opposite long-ranged fluctuations in liquidity, with an otherwise permanent market impact, challenging the scenario proposed in Quantitative Finance, 2004, 4, 176, where the impact is instead transient, with a power-law decay in time. The exponent of this decay is precisely tuned to a critical value, ensuring simultaneously that prices are diffusive on long time scales and that the impact function is nearly lag independent. We provide new analysis of empirical data that confirm and make more precise our previous claims. We show that the power-law decay of the bare impact function comes both from an excess flow of limit order opposite to the market order flow, and to a systematic anti-correlation of the bid-ask motion between trades, two effects that create a 'liquidity molasses' which dampens market volatility.
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arXiv:cond-mat/0406224v2 [cond-mat.other] 10 Jun 2004
Random walks, liquidity molasses and
critical response in financial markets
Jean-Philippe Bouchaud, Julien Kockelkoren, Marc Potters
November 24, 2008
Science & Finance, Capital Fund Management, 6-8 Bvd Haussmann
75009 Paris, France
Abstract
Stock prices are observed to be random walks in time despite a strong,
long term memory in the signs of trades (buys or sells). Lillo and Farmer
have recently suggested that these correlations are compensated by op-
posite long ranged fluctuations in liquidity, with an otherwise permanent
market impact, challenging the scenario proposed in Quantitative Finance
4, 176 (2004), where the impact is transient, with a power-law decay in
time. The exponent of this decay is precisely tuned to a critical value,
ensuring simultaneously that prices are diffusive on long time scales and
that the response function is nearly constant. We provide new analysis of
empirical data that confirm and make more precise our previous claims.
We show that the power-law decay of the bare impact function comes both
from an excess flow of limit order opposite to the market order flow, and
to a systematic anti-correlation of the bid-ask motion between trades, two
effects that create a ‘liquidity molasses’ which dampens market volatility.
1
1 Introduction
The volatility of financial assets is well known to be too much large compared to
the prediction of Efficient Market Theory [1] and to exhibit intriguing statistical
anomalies, such as intermittency and long range memory (for recent reviews,
see [2, 3, 4, 5]). The availability of all trades and quotes on electronic markets
makes it possible to analyze in details the intimate mechanisms leading to these
anomalies. In a previous paper [6], we have proposed, based on empirical data,
that the random walk nature of prices (i.e. the absence of return autocorrelations)
is in fact highly non trivial and results from a fine-tuned competition between
liquidity providers and liquidity takers. In order not to reveal their strategy,
liquidity takers must decompose their orders in small trades that are diluted in
time over a several hours to several days. This creates long range persistence in
the ‘sign’ of the market orders (i.e. buy, ε= +1 or sell ε=1) [7, 8, 6, 9].
This persistence should naively lead to a positive correlations of the returns and
a super-diffusive behaviour of the price [6, 9]. However, liquidity providers act
such as to create long range anti-persistence in price changes: liquidity providers
make their profit on the bid-ask spread but lose money when the price makes large
excursions, in which case they sell low and have to buy high (or vice versa) for
inventory reasons. Both effects rather precisely compensate and lead to an overall
diffusive behaviour (at least to a first approximation), such that (statistical)
arbitrage opportunities are absent, as expected. We have shown in [6] that this
picture allows one to understand the temporal structure of the market impact
function (which measures how a given trade affects on average future prices),
which was found to first increase, reach a maximum and finally decrease at large
time, reflecting the mean-reversion action of liquidity providers.
The above picture was recently challenged by Lillo and Farmer [9]. Although
they also find long memory (i.e., non summable power-law correlations) in the sign
of market orders, they claim that the compensating mechanism that leads to un-
correlated returns is not the slow, mean-reverting influence of liquidity providers
suggested in [6]. Rather, they argue that long range liquidity fluctuations, cor-
related with the order flow, act to suppress the otherwise permanent impact of
market orders and make the price diffusive.
The aim of this paper is to explain in more details the differences and simi-
larities between these conflicting pictures, and to present new data that support
our original assertions [6]. While our previous paper mainly discussed on the
case of France-Telecom, we also present a more systematic account of our main
observables for a substantial set of stocks from the Paris Bourse. We also give a
much more precise qualitative and quantitative description of the way liquidity
providers manage, on average, to mean-revert the price by monitoring the flow of
limit orders. We therefore argue that liquidity providers create a kind of ‘liquidity
molasses’ that stabilises the volatility of financial markets, which is indeed the
traditional role given to market makers.
2
2 The impact of trades on prices
2.1 Formulation of the problem
In the following, we will consider follow the dynamics of prices in trade time n(i.e.
each distinct trade increases nby one unit) and define prices pnas the midpoint
just before the nth trade: pn= (an+bn)/2, where anand bnare, respectively, the
ask price and the bid price corresponding to the last quote before the trade. We
assume that the price can be written in general as:
pn=X
n<n
G(n, n|εn, Vn,Sn) (1)
where Gdescribes the impact at time nof a trade at time n, of sign and volume
εn, Vn, knowing that the order book at time nis in a certain state Sn(specified
by the list of all prices and volumes of the limit orders). The assumption we
made in [6] is that the impact function Gcan be decomposed into an average,
systematic part in the direction of the trade, plus fluctuations:
G(n, n|εn, Vn,Sn)εnG(n, n|Vn) + ξ(n, n),(2)
where the function Gwas furthermore assumed to by time translation invariant1
and factorisable as: G(n, n|Vn) = f(Vn)G0(nn). The last assumption is
motivated by theoretical and empirical results [10, 16, 6], where f(V) is found
to be a power-law with a small exponent f(V)Vβ[14, 15] or a logarithm
f(V)ln V[16, 6]. The noise term ξ(n, n) is uncorrelated with the εnand has
a variance (nn)D. The final form of the model proposed in [6] therefore reads:
pn=X
n<n
G0(nn)εnln Vn+ξ(n, n).(3)
The main finding of [6] is that the bare impact function G0() must decay with
the time lag in order to compensate for the long range correlation in the ε, in
other words that the impact of a single trade is transient rather than permanent.
In their recent work, Lillo and Farmer [9] argue that it is rather the fluctuations
in liquidity (encoded in the instantaneous shape of the order book Sn), that are
crucial. Their model amounts to write pnas:
pn=X
n<n
εnVβ
n
λ(Sn)+ξ(n, n),(4)
with β= 0.3 and where λis the instantaneous liquidity of the market. The
difference between V.3and ln Vis actually not relevant; rather, the crucial dif-
ference between Eq. (3) and Eq. (4) is that the impact is transient in the former
1This is probably only an approximation since time of the day, for example, should matter.
3
case and permanent (but fluctuating) in the latter case, a point on which we will
comment later.
The argument of ref. [9] in favor of the second model, Eq. (4) goes in two
steps: first, they propose, as a proxy of the instantaneous liquidity λn, the volume
vnat the best price (i.e. ask for buys and bid for sells): see [9] section VI B. They
then study the time series of rn=εnVβ
n/vnand find that linear correlations have
nearly completely disappeared, at variance with the unrescaled series εnVβ
nthat
exhibit the problematic long range correlations. Their conclusion is therefore
that “the inclusion of the time varying liquidity term apparently removes long-
memory”. Here, we want to refute this interpretation based on three independent
sets of arguments: a) we show that Eq. (4) has less explicative power than Eq.
(3); b) Eq. (4) leads to an average response function (see [6] and below) that
significantly increases with time lag, at variance with data and c) the absence
of linear correlations observed in rnis an artefact coming from the very large
fluctuations of the volume at the best price. Note that our data concerns stocks
from Paris Bourse rather than the LSE stocks studied in [9]. However, we do not
expect major qualitative differences between the two markets.
2.2 Response functions
We first start by recalling the definition of the average response function, as
the correlation between the sign of a trade at time nand the subsequent price
difference between nand n+[6]:
R() = h(pn+pn)·εni,(5)
The quantity R() measures how much, on average, the price moves up condi-
tioned to a buy order at time 0 (or how a sell order moves the price down) a time
later. Note that because of the temporal correlations between the ε’s, this quan-
tity is not the above market response to a single trade G0() [6]. This quantity is
plotted in Fig. 1 for Carrefour in 2001, 2002. As emphasized in [16, 6], R() is
found to weakly increase up to a maximum beyond which it decays back and can
even change sign for large (see Figs. 2, 3). For other stocks, or other periods,
the maximum of R() is not observed, and R() is seen to increase (although
always rather mildly, at most by a factor 3) with : see Figs. 2,3. As will be clear
below, this difference of behaviour can actually be understood within our model.
In Fig. 4, we also plot three other, similar quantities. The first is the (nor-
malized) correlation between the price change and εnln Vn:
RV() = h(pn+pn)·εnln Vni
hln2Vni1/2(6)
which has a similar shape but is distincly larger than Ritself, showing that,
as expected, the variable εnln Vnhas a larger explicative power than εnitself.
4
1 10 100 1000 10000
l
0
0.01
0.02
R(l)
Carrefour 2002
Carrefour 2001
Figure 1: Response function R() (in Euros) for Carrefour in the periods 2001
and 2002.
1 10 100 1000
l
−0.01
0.01
0.03
R(l)
Figure 2: Response function R() (in Euros) for stocks from Paris Bourse in
2002. From top to bottom: EN, EX, FTE, ACA, CGE. (See Table 1 for the
stocks code). Note that for some stocks R() increases for all (see e.g. CGE),
whereas for other stocks R() reaches a maximum before becoming negative (see
e.g. ACA). The dotted line correspond to R() = 0.
5
1 10 100 1000
l
0
0.02
0.04
0.06
R(l)
Figure 3: Response function R() (in Euros) for other stocks from Paris Bourse
in 2002. From top to bottom: FP, BN, GLE, MC, CA, VIE. (See Table 1 for the
stocks code). Note that for some stocks R() increases for all (see e.g. GLE),
whereas for other stocks R() reaches a maximum before becoming negative (see
e.g. CA, for > 5000).
Code Stock name Av. price Av. tick Av. spread # trades
ACA Cr´edit Agricole 19.63 0.01 0.0408 379,000
BN Danone 132.50 0.1 0.154 351,000
CA Carrefour 48.54 0.0268 0.0578 555,000
CGE Alcatel 9.85 0.01 0.015 1,020,000
EN Bouygues 29.69 0.01 0.0413 240,000
EX Vivendi 27.47 0.0126 0.0287 979,000
FP Total 152.27 0.1 0.136 759,000
FTE France-Telecom 21.04 0.01 0.024 1,051,000
GLE Soci´et´e G´en´erale 61.80 0.043 0.0735 499,000
MC LVMH 47.71 0.0209 0.0566 437,000
VIE Vivendi Env. 29.75 0.01 0.0452 226,000
Table 1: Selection of stocks studied in this paper, with the average price, tick size and
bid-ask spread in Euros in 2002. We also give the total number of trades in 2002. The
results reported here qualitatively hold for most other stocks from Paris Bourse, but
also other exchanges (see [6, 9]).
6
100101102103
l
10−2
10−1
100
R(l)
R(l)
RV(l)
RLF(l)
RLF*(l)
Figure 4: Four different ‘response functions’ R(), RV(), RLF () and R
LF (),
(see text) in Euros for BN in 2002. This plot shows (a) that the Lillo-Farmer
variable rnhas a weak explicative power (see RLF – dashed line) and (b) that
their permanent impact model leads to a considerable over-estimation of the true
response function (see R
LF – dashed-dotted lines, showing a 30 times increase
with ).
In order to test the Lillo-Farmer model, we have also computed two further
quantities. One is the normalized correlation between the Lillo-Farmer variable
rn=εnVβ
n/vnand the empirical price change:
RLF () = h(pn+pn)·rni
hr2
ni1/2.(7)
This quantity measures the explicative power of rn, and can be directly compared
to Rand RV. As can be seen in Fig. 4, RLF () is in fact a factor 3 smaller than
RV() (see also the quantity Zin Table 2, last column).
The second interesting quantity is:
R
LF () = * n+1
X
n=n
rn!·εn+.(8)
The quantity measures a fictitious average response function, which would
follow if the price dynamics was given by Eq. (4). We see in Fig. 4 that R
LF (),
at variance with the true R(), sharply grows with , as a consequence of the
correlation of the ε’s which are not compensated by a fluctuating liquidity. As
we have mentioned in [6], the response function R() is a very sensitive measure
7
100101102103
l
10−3
10−2
10−1
100
C(l)
Fit l−0.61
rn correlations
1/vn correlations
C(l)
Figure 5: Sign correlations C() for BN, showing a long range, power-law decay,
and comparison between the smaller and faster decaying correlation of the rnand
the 1/vn, showing that the former is dominated by the weak correlations between
small order volumes, and not by a compensation between market order flows and
limit order flows.
of the dynamics of prices that allows one to reveal subtle effects, beyond the
simple autocorrelation of price changes (see also below).
Finally, we show in Fig. 5 the rapid fall of the autocorrelation of the variables
rn, that was argued by Lillo and Farmer to be a strong support to their model [9].
Unfortunately, this effect is not relevant and is due to the fact that the volume
at the best price has large fluctuations. For example, in the case of FTE, the
distribution of vis found to be well-fit by P(v)vµ1exp(v/v0) with µ > 1,
so that the most probable values correspond to v1, whereas the mean value
is 3000 [11]. Since vnappears in the denominator of rn, it is clear that the rn
correlations are dominated by times where the volume at bid/ask is particularly
small; these small values show little autocorrelations (see Fig. 5).2
2.3 The bare impact function and price diffusion
We conclude from Fig. 4 that although the variables rnare indeed close to being
uncorrelated, they do not provide an adequate basis to interpret the dynamics
of real price. Our transient impact model, on the other hand, allows one to
reconcile the absence of autocorrelations in price changes with the observed non
2After discussions, Lillo and Farmer have agreed that their results on LSE stocks are in fact
compatible with the above interpretation.
8
100101102103
l
10−2
10−1
100
C(l)
ACA
CA
EX
FP
Figure 6: Plot of the sign correlations C() for a selection of four stocks, showing
the long-ranged nature of these correlations. See also Table 3.
monotonous shape of the average response function, provided the bare impact
function G0() is chosen adequately. In [6], it was shown that if the correla-
tion of the ε’s decays as γ, then G0() should also decay, at large times, as a
power-law βwith β(1 γ)/2. For β > (1 γ)/2, the price is subdiffusive
(anti-persistent) and the response function R() has a maximum before becoming
negative at large . For β < (1 γ)/2, on the other hand, the price is superdiffu-
sive (persistent) and the response function monotonously increases (see Fig. 10
of [6]). The short time behaviour of G0() can in fact be extracted from empirical
data by using the following exact relationship:
R() = hln ViG0() + X
0<n<ℓ
G0(n)C(n) + X
n>0
[G0(+n)G0(n)] C(n).(9)
where:
C() = hεn+εnln Vni,(10)
a correlation function that can also be measured directly (see Figs. 5,6).
Eq. (9) gives a set of linear equations relating R,G0and Cthat can easily be
solved for G0. The result is plotted in Fig. 7 for different stocks. One sees that
G0() is first flat or rises very slightly with before indeed decaying, for 1,
like a power law, with βgiven in Table 2. The fit used to extract the value of βis
Gf
0() = Γ0/(2
0+2)β/2which is similar, but not identical to, the one proposed in
[6]. The advantage of the present fit is that it matches quite well the rather flat
initial behaviour of G0(). We also give in Table 2 the value of other quantities
such as the exponent γgoverning the decay of the εcorrelations. A very similar
9
1 10 100 1000
l
10−3
10−2
G0(l)
CA
EX
FP
ACA
Figure 7: Comparison betwen the empirically determined G0(), extracted from
Rand Cusing Eq.(9), and the fit Gf
0() = Γ0/(2
0+2)β/2, used to extract the
parameters given in Table 2, for a selection of four stocks: ACA, CA, EX, FP.
shape for G0can be observed for all stocks; fluctuations around the critical line
β= (1 γ)/2 (see Fig. 8) are enough to explain the fact that Rsometimes has
a maximum, sometimes not.
Correspondingly, the vicinity of the critical line ensures that the price has a
diffusive behaviour, as is indeed confirmed by measuring the variance of price
changes:
D() = h(pn+pn)2i ≈ Dℓ;ℓ, (11)
as demonstrated in Figs. 9 and 10. The fact that D() is strictly linear in is of
course tantamount to saying that price increments are uncorrelated.
2.4 Economic interpretation of the shape of the bare im-
pact
The economic interpretation of the non monotonic behaviour of G0is as follows.
Suppose that you are a liquidity provider, making profits on the bid-ask spread
and losses on large price excursions, and that you see a flow of buy orders coming.
In the absence of news and for typical buy volumes,3the natural strategy is, on
short times, to biais the ask price up to be able to sell higher while there are
3The following discussion is intended to describe typical situations. Obviously, if the buy
volume is anomalously large, liquidity providers would anticipate some insider information and
react differently.
10
0 0.2 0.4 0.6 0.8
γ
0.1
0.2
0.3
0.4
0.5
β
Fit parameters
β=(1−γ)/2
Figure 8: Scatter plot of the exponents β, γ extracted from the fit of G0and C.
These exponents are seen to lie in the vicinity of the critical line β= (1 γ)/2
(dotted line), as expected from the nearly diffusive behaviour of prices (see Fig.
9), and [6].
Stock qD(1) Γ00β C0γ Z
ACA 1.69 0.63 16.3 0.44 0.58 0.125 0.35
BN 7.9 1.75 3.1 0.26 0.81 0.61 0.37
CA 3.13 0.71 7.4 0.22 0.83 0.57 0.27
CGE 0.84 0.20 8.9 0.275 0.49 0.35 0.18
EN 2.75 0.66 9.2 0.27 0.83 0.57 0.27
EX 1.79 0.47 15.3 0.26 0.45 0.40 0.20
FP 7.0 1.46 2.2 0.15 0.79 0.69 0.28
FTE 3.9 0.47 20.3 0.30 0.52 0.41 0.23
GLE 4.37 0.73 0.7* 0.13 0.86 0.58 0.28
MC 3.47 0.67 3.1 0.19 0.95 0.58 0.26
VIE 2.8 0.38 0.25* 0.12 0.75 0.63 0.26
Table 2: Summary of the different quantities and fit parameters for 11 stocks of the
Paris Bourse during the year 2002. G0() is fitted as: G0() = Γ0/(2
0+2)β/2, and
C() = C0/ℓγ, both in the range = 2 2000. pD(1) and Γ0are in cents of
Euros. The * means that the fit of G0for small is not very good. The last column
measures the relative explicative power of the Lillo-Farmer variable, compared to our
own: Z=RLF (1)/R(1).
11
1 10 100 1000 10000
l
0.03
0.05
0.07
(D(l)/l)1/2
Figure 9: Plot of qD()/ℓ (in Euros) vs. for several stocks. Apart from BN
and FP (for which the tick size is large), this quantity is roughly constant with ,
showing that prices are to a very good approximation diffusive, even on shortest
times scales. From top to bottom: BN, FP, GLE, FTE, MC.
1 10 100 1000 10000
l
0
0.01
0.02
0.03
(D(l)/l)1/2
Figure 10: Plot of qD()/ℓ (in Euros) vs. for all other (smaller tick) stocks.
From top to bottom: CA, VIE, EN, EX, ACA, CGE.
12
clients eager to buy. However, you now have a net short position on the stock
that you want to eventually shift back to zero. So you would like to buy back, in
the near future, at the cheapest possible price. In order to prevent the price from
going up, you can/should do two things: a) create a barrier to further price rises
by placing a large number of sell orders at the ask, off which the price will bounce
back down b) place bid orders as low as possible. Both effects act to create a
liquidity molasses that mean revert the price towards its initial value. Both these
effects can actually be observed directly on the data.
a) One observes a strong correlation between a buy (resp. sell) market
order moving the price up and the subsequent appearance of limit orders
at the ask (resp. bid) [17, 9]. If a ‘wall’ of limit orders appears at the
ask while the bid remains poorly populated, the probability that the price
moves down upon the arrival of further market orders becomes larger than
the probability to move up. One can visualize this effect more clearly by
separating the total price change into two components: price variations due
to market orders, ∆Mpn, corresponding to the change of mid-point between
the quote immediately prior and the quote immediately posterior to the
n-th trade, and price variations due to limit orders, ∆Lpncorresponding to
changes of mid-points in-between trade nand trade n+ 1. By definition,
pn+pn=
n+1
X
k=n
[∆Mpk+ ∆Lpk](pn+pn)M+ (pn+pn)L.(12)
One can then measure the response function restricted to price changes due
to market orders:
RM() = h(pn+pn)M·εni,(13)
and compare it (see Fig. 11) to R(). We observe for all stocks that RM()
and R() have the same overall shape. For FTE, for example, RM() also
bends down and becomes negative for large . But since by definition
Mpk=εkGkwith Gk0 (a buy market order can only move the price
up or leave it unchanged), the fact that RM() decreases implies that Gk
is anticorrelated with εnεk. In other words, sell orders following buy orders
impact the price more than buy orders following buy orders, as expected if
the order book fills in more on the ask side than on the bid side after a buy
market order (and, of course, similarly for the sell side).
b) there is an anticorrelation between buy orders and the subsequent motion
of the bid-ask in-between trades. This is seen both from the fact that
RM()>R() for not too large (see Fig. 11), implying that the response
function restriced to limit orders is negative. Furthermore, one can study
the correlation between a market order induced price change ∆Mpnand a
later limit order price change ∆Lpn+, which is found to be negative (as
13
0 100 200 300
−0.2
−0.1
0
0.1
0.2
0.3
1 10 100 1000
l
−0.02
−0.01
0.00
0.01
R(l)
R(l)
RM(l)
Figure 11: Main figure: Comparison between the full response R(circles) and
the response restricted to market order induced price changes RM(squares), for
FTE in 2002. Inset: Integrated correlation functions, corresponding to hMpn·
Mpn+i(full line), and hMpn·Lpn+i(dotted line). The former is clearly
positive, and is compensated by the negative correlation between market orders
induced shifts and subsequent changes in the mid-quotes.
also reported in [6, 9]). This compensates the positive correlations between
Mpnand ∆Mpn+(and between ∆Lpnand ∆Lpn+), that would otherwise
lead to a superdiffusion in the price.
In order to make our point even more clearly, it is useful to emphasize the
antagonist forces present in financial markets:
The ideal world for liquidity providers is a stable, fixed average price that
allows them to earn the bid-ask spread at every round-turn. Volatility is the
enemy4, liquidity molasses is the solution: a vanishing long term impact (i.e.
G0() = 0) is a way to limit the volatility of the market and to increase
the liquidity provider gains. Reducing the volatility of financial markets
is in fact the traditional role given to market makers in non electronic
markets. Note that we do not assume any kind of collusion between liquidity
providers: they all, individually, follow a perfectly reasonable strategy.
4Insider information is also the liquidity provider enemy, but this situation is rather rare on
the scale of the thousands of trades happening every day on each single liquid stock. However,
creating a liquidity wall is indeed risky for the liquidity provider in the case where some true
information motivates the market orders. In that case, the insider can use his information
without impacting the price.
14
Conversely, permanent impact is what the liquidity taker should hope for:
if the price rises because of his very trade but stays high until he sells back,
his impact is not really a cost. On the other hand, if the price deflates
back after having bought it, it means that he paid to much for it.5The
correlations created by splitting his bid in small quantities also help keeping
the price up.
These are the basic ingredients ruling the competition between liquidity providers
and liquidity takers. The subtle balance between the positive correlation in the
trades (measured by γ) and the liquidity molasses induced by liquidity providers
(measured by β) is a self-organized dynamical equilibrium. Its stability comes
from two counter-balancing effects: if the liquidity providers are too slow to re-
vert the price (β < (1 γ)/2), then the price is superdiffusive and liquidity
providers lose money on average [21]; therefore they increase β. If the mean
reversion is too strong (β > (1 γ)/2), the resulting long term anticorrelations
is an incentive for buyers to wait for prices to come back down to continue buy-
ing. Liquidity takers thereby spread their trading over longer time scales, which
corresponds to smaller values of γ.
A dynamical equilibrium where β(1 γ)/2 therefore establishes itself
spontaneously, with clear economic forces driving the system back towards this
equilibrium. Interestingly, fluctuations around this critical line leads to fluctu-
ations of the local volatility, since persistent patches correspond to high local
volatility and antipersistent patches to low local volatility (see also [22] for a
similar mechanism). Extreme crash situations are well-known to be liquidity cri-
sis, where the liquidity molasses effect disappears temporarily, destabilising the
market (on that point, see the detailed recent study of [12, 18]).
Finally, the mean-reverting nature of the response function is of crucial im-
portance to understand the influence of volume and execution time on the actual
impact of trading on prices (on this point, see [19, 20]).
3 Summary and Conclusion
The aim of this paper was to challenge Lillo and Farmer’s suggestion that the
strong memory in the signs of trades is compensated by liquidity fluctuations,
with an otherwise permanent market impact, and confirm the more subtle sce-
nario proposed in our previous paper [6], in which the impact is transient, with
a power-law decay in time. The exponent is precisely tuned to a critical value,
ensuring simultaneously that prices are diffusive on long time scales and that the
response function is nearly constant. Therefore, the seemingly trivial random
walk behaviour of price changes in fact results from a fined-tuned competition
5The salesman knows nothing about what he is selling, save that he is charging a great deal
too much for it. (Oscar Wilde)
15
between two opposite effects, one leading to super-diffusion – the autocorrelation
of market order flow; the other leading to sub-diffusion – the decay of the bare
impact function, reflecting the mean-reverting nature of the limit order flow. We
have shown that mean reversion comes both from an excess flow of limit order
opposite to the market order flow, and to a systematic anti-correlation of the
bid-ask motion between trades. Note that in the above picture, the random walk
nature of prices and their volatility are induced by the trading mechanisms alone,
with no reference to real news. These of course should also play a role, but proba-
bly not as important as pure speculation and trading that lead to excess volatility
(see the discussion and references in [6]).
The above fine tuning is however, obviously, not always perfect, and is ex-
pected to be only approximately true on average. Breakdown of the balance
between the two effects can lead either to large volatility periods and crashes
when the liquidity molasses disappears, or to low volatility periods when mean-
reverting effects are strong. The small imbalance between the two effects there-
fore leads to different shapes of R() (monotone increasing or turning round and
changing sign). As emphasized in [6], our finding that the absence of arbitrage
opportunities results from a critical balance between antagonist effects might jus-
tify several claims made in the (econo-)physics literature that the anomalies in
price statistics (fat tails in returns described by power laws [23, 24], long range
self similar volatility correlations [3, 5], and the long ranged correlations in signs
[6, 9]) are due to the presence of a critical point in the vicinity of which the market
operates (see e.g. [25], and in the context of financial markets [26, 27, 28]). From
a more practical point of view, we hope that the present detailed picture of mar-
ket microstructure could help understanding the mechanisms leading to excess
volatility, and suggest ways to control more efficiently the stability of financial
markets.
Acknowledgments
We want to thank Matthieu Wyart and Yuval Gefen for many inspiring discus-
sions and ideas about this work. We also thank Doyne Farmer and Fabrizio Lillo
for many comments and e-mail exchanges that allowed to clarify a lot the present
paper.
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19
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... In Refs. [4,5,7], Bouchaud et al. use a self-response function that only depends on the time lag τ . This function measures how much, on average, the price moves up (down) at time τ conditioned to a buy (sell) order at time zero. ...
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