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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 n′is 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′)≡εn′G(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(n−n′). 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 εn′and has

a variance (n−n′)D. The ﬁnal form of the model proposed in [6] therefore reads:

pn=X

n′<n

G0(n−n′)εn′ln Vn′+ξ(n, n′).(3)

The main ﬁnding of [6] is that the bare impact function G0(ℓ) 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

εn′Vβ

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+ℓ[6]:

R(ℓ) = h(pn+ℓ−pn)·εni,(5)

The quantity R(ℓ) 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

ℓlater. 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(ℓ) [6]. This quantity is

plotted in Fig. 1 for Carrefour in 2001, 2002. As emphasized in [16, 6], R(ℓ) is

found to weakly increase up to a maximum beyond which it decays back and can

even change sign for large ℓ(see Figs. 2, 3). For other stocks, or other periods,

the maximum of R(ℓ) is not observed, and R(ℓ) is seen to increase (although

always rather mildly, at most by a factor 3) with ℓ: 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(ℓ) = h(pn+ℓ−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(ℓ) (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(ℓ) (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(ℓ) increases for all ℓ(see e.g. CGE),

whereas for other stocks R(ℓ) reaches a maximum before becoming negative (see

e.g. ACA). The dotted line correspond to R(ℓ) = 0.

5

1 10 100 1000

l

0

0.02

0.04

0.06

R(l)

Figure 3: Response function R(ℓ) (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(ℓ) increases for all ℓ(see e.g. GLE),

whereas for other stocks R(ℓ) reaches a maximum before becoming negative (see

e.g. CA, for ℓ > 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(ℓ), RV(ℓ), RLF (ℓ) and R∗

LF (ℓ),

(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 ℓ).

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 (ℓ) = h(pn+ℓ−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 (ℓ) is in fact a factor 3 smaller than

RV(ℓ) (see also the quantity Zin Table 2, last column).

The second interesting quantity is:

R∗

LF (ℓ) = * n+ℓ−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 (ℓ),

at variance with the true R(ℓ), sharply grows with ℓ, 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(ℓ) 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(ℓ) 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)∝vµ−1exp(−v/v0) with µ > 1,

so that the most probable values correspond to v∼1, 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(ℓ) 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(ℓ) is chosen adequately. In [6], it was shown that if the correla-

tion of the ε’s decays as ℓ−γ, then G0(ℓ) should also decay, at large times, as a

power-law ℓ−βwith β≈(1 −γ)/2. For β > (1 −γ)/2, the price is subdiﬀusive

(anti-persistent) and the response function R(ℓ) has a maximum before becoming

negative at large ℓ. 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(ℓ) can in fact be extracted from empirical

data by using the following exact relationship:

R(ℓ) = hln ViG0(ℓ) + X

0<n<ℓ

G0(ℓ−n)C(n) + X

n>0

[G0(ℓ+n)−G0(n)] C(n).(9)

where:

C(ℓ) = hεn+ℓεnln 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(ℓ) is ﬁrst ﬂat or rises very slightly with ℓbefore indeed decaying, for ℓ≫1,

like a power law, with βgiven in Table 2. The ﬁt used to extract the value of βis

Gf

0(ℓ) = Γ0/(ℓ2

0+ℓ2)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(ℓ). 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(ℓ), extracted from

Rand Cusing Eq.(9), and the ﬁt Gf

0(ℓ) = Γ0/(ℓ2

0+ℓ2)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(ℓ) = h(pn+ℓ−pn)2i ≈ Dℓ;∀ℓ, (11)

as demonstrated in Figs. 9 and 10. The fact that D(ℓ) is strictly linear in ℓis 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) Γ0ℓ0β 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(ℓ) is ﬁtted as: G0(ℓ) = Γ0/(ℓ2

0+ℓ2)β/2, and

C(ℓ) = C0/ℓγ, both in the range ℓ= 2 →2000. pD(1) and Γ0are in cents of

Euros. The * means that the ﬁt of G0for small ℓis 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)/ℓ (in Euros) vs. ℓfor several stocks. Apart from BN

and FP (for which the tick size is large), this quantity is roughly constant with ℓ,

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)/ℓ (in Euros) vs. ℓfor 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+ℓ−pn=

n+ℓ−1

X

k=n

[∆Mpk+ ∆Lpk]≡(pn+ℓ−pn)M+ (pn+ℓ−pn)L.(12)

One can then measure the response function restricted to price changes due

to market orders:

RM(ℓ) = h(pn+ℓ−pn)M·εni,(13)

and compare it (see Fig. 11) to R(ℓ). We observe for all stocks that RM(ℓ)

and R(ℓ) have the same overall shape. For FTE, for example, RM(ℓ) also

bends down and becomes negative for large ℓ. But since by deﬁnition

∆Mpk=εkGkwith Gk≥0 (a buy market order can only move the price

up or leave it unchanged), the fact that RM(ℓ) 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(ℓ)>R(ℓ) for ℓnot 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+ℓ, 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 h∆Mpn·

∆Mpn+ℓi(full line), and h∆Mpn·∆Lpn+ℓi(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+ℓ(and between ∆Lpnand ∆Lpn+ℓ), 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(ℓ) (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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