Square root law for price impact:
Empirical evidence and theory
Jonathan Donier
(CFM - LPMA, University Paris 6)
joint work with J. Bonart, I. Mastromatteo and J.-P. Bouchaud
Imperial College London,
March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Summary
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
I. Introduction to price impact and LLOB
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
By definition, trading affects the shape of supply and demand
BUT HOW ?
To determine the properties of supply and demand, we need to probe it...
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Numerical results [animation]
yt
ρB
ρA
Evolution in presence of a metaorder mt , in the LLOB framework.
yt =
1
L
Z
0
Square root law for price impact: Empirical evidence and theory
t
dsms
p
2
e
(yt −ys )
− 4D(t−s)
(1)
4πD(t − s)
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Relevance of price impact
Why is this issue relevant?
I Theory (I): Relevant, because price impact is a way to probe the supply and
demand curves, so as to determine their properties;
I Theory (II): Because price impact is the mechanism through which prices absorb
information encoded in trades; because it is the core ingredient of many
agent-based models that aim to study price formation;
I Practice (I): Price impact is a cost for traders, which they need to accurately
control in order to optimize their execution;
I Practice (II): For regulators, price impact controls stability.
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Some historical results on financial markets
Empirical results on MI
Evidence that dates back to
1997 (!) shows that impact
Evidence dating back to
has a concave shape
1997 (!) of:
shows that impact
(roughly) independent
period,
orders. T
market
Small ticks
pants) a
Large ticks
δ=1/2
metaord
δ=1
bit of co
identify
-2
there are
10
the conc
tioned.
trades b
becomes
e.g., [3],
-3
10
needed t
-4
-3
-2
-5
10
10
10
10
disappea
Q/V
Q/V
any case
A squ
FIG. 1. The impact[from
of Tóth
metaorders
for Capital
Fund
et
al
(2011),
impact
of
≈
500
000
trades]
[from
Tóth et alproprietary
(2011), impact
of ≈
000 markets,
sequences]in the period
highly n
Management
trades
on 500
futures
classical
from June 2007 to December 2010. Impact is measured here as
impact !
the average execution shortfall of a metaorder of size Q. The
of as a s
data base contains nearly 500 000 trades. We show !=" vs Q=V
on a log-log scale, where " and V are the daily volatility and
that the e
daily volume measured the day the metaorder is executed. The
high im
blue curve is for large tick sizes, and the red curve is for small
that trad
tick sizes. For large ticks, the curve can
be well
fit with
# ¼ March
0:6, 4, 2015
Imperial
College
London,
tenth of
2
has a concave shape
(roughly) independent of:
Impact
10
I Maturity
- Venue
- Maturity
I Historical period
- Historical period
I Geographical- area
Geographical area
[see Torre (1997), Almgren et al.
(2003), Moro et al. (2009), Tóth et al.
[see Torre (1997), Almgren (2011),
et al.(2003),
Gomes,Moro
Waelbroeck (2014),
et al. (2009), Tóth et al.(2011),
Gomes,
Bershova,
Rakhlin (2013), IM et al.
(2014),
X. Brokmann
Waelbroeck (2014),Bershova,
Rakhlin
(2013), et al. (2014),
Zarinelli et al. (2015)]
Mastromatteo et al.(2014), X. Brokmann et al.
(2014),Zarinelli et al. (2015)]
Square root law for price impact: Empirical evidence and theory
∆/σ
I(Q)/
I Venue
B. TÓTH et al.
-1
Jun. 2007 - Dec. 2010
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Some historical results on financial markets
Evidence that dates back to
1997 (!) shows that impact
has a concave shape
(roughly) independent of:
I Venue
I(Q) = Y σ
I Maturity
Q
V
1/2
(2)
I Historical period
I Geographical area
I(Q)
price change
V
daily traded volume
executed volume
Y
Y-ratio
daily volatility
(adimensional ∼ 1)
[see Torre (1997), Almgren et al.(2003), Moro
et al. (2009), Tóth et al.(2011), Gomes,
Q
Waelbroeck (2014),Bershova, Rakhlin (2013),
Mastromatteo et al.(2014), X. Brokmann et al.
(2014),Zarinelli et al. (2015)]
Square root law for price impact: Empirical evidence and theory
σ
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
II. General insights on Bitcoin, data and metaorders
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
General aspects of the Bitcoin/USD market
I Crypto-currency, exchanged against usual currencies on limit order books,
I Power law distribution of volumes traded and traders wealth,
I Power law distribution of returns,
I Unpredictable price changes.
Almost like a usual market, except...
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
General aspects of the Bitcoin/USD market
I Crypto-currency, exchanged against usual currencies on limit order books,
I Power law distribution of volumes traded and traders wealth,
I Power law distribution of returns,
I Unpredictable price changes.
Almost like a usual market, except...
I One exchange (MtGox) with market share > 80% (at that time) and few
correlated product/derivatives,
I Very large spread and fees,
I Few professionals (no significant HFT, market making, almost no brokerage...),
I Trading intentions are displayed much longer in advance !
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Data
I Snapshots of the whole order book of MtGox every 10 min since 2011
I MtGox full trading report (7M trades) with anonymized IDs.
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Metaorders
Position of a given trader vs time and zoom on a metaorder
I Times series decomposition rather irrelevant due to the irregular nature of the
time series.
I Method used:
I
I
I
I
for each trader, spot periods of inactivity (>1h)
define the start of a metaorder as the first trade after this period
continue until either a new inactivity period or a position reversal
this eliminates some sequences, but also mean-reversion biases
We consider only aggressive orders to limit adverse selection biases (since we don’t
know the target volume to execute).
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Results on metaorders
Metaorders size distribution
Metaorders duration distribution
1
Cumulative distribution
Cumulative distribution
1
0.1
0.01
0.1
0.01
0.001
0.001
1
10
100
1
1000
10
Volume in Bitcoins
1000
10000
Average execution paths
1
1
% of volume executed
Cumulative distribution
100
Duration in s
Metaorders participation rates distribution
0.1
0.01
0.001
0.0001
1e-07 1e-06 1e-05 0.0001 0.001 0.01
small t, small Q
small t, large Q
large t, small Q
large t, large Q
y = x (TWAP)
0.8
0.6
0.4
0.2
0
0.1
1
Participation rate
0
0.2
0.4
0.6
0.8
1
Time (rescaled)
(Bottom right) Execution speed is on average constant on [0, T ] => this limits
selection biases and is a sign of poorly strategic behaviour regarding execution
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Results on metaorders
Metaorders during a metaorder (#)
4
Metaorders during a metaorder (volume)
1
Same sign
Opposite sign
Net
Volume (BTC/s)
5
#
3
2
1
0
-60
-40 -20
0
20 40 60 80
Time (% of meta-order duration)
100
0.8
Same sign
Opposite sign
Net
0.6
0.4
0.2
0
-60 -40 -20 0 20 40 60 80 100
Time (% of meta-order duration)
Metaorders are positively correlated with the aggressive imbalance of other traders,
but the effect is not dynamical (the correlation remains constant on [0, T ]).
Thus the impact picture will not come from some dynamical synchronization between
agents (who for instance would all try to exploit the same signal at the same time, resulting in a sharper increase
in price when the signal is released).
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
III. Market impact on the Bitcoin
Square root law for price impact: Empirical evidence and theory
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
[Donier and Bonart, 2014]
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Typical impact shape
Abrupt rise of the price to I(Q) then decay
Price path
Price
I(Q)
Execution
0
After execution
T(Q)
Square root law for price impact: Empirical evidence and theory
Time
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Market impact on the Bitcoin
What would we expect on the Bitcoin market?
Reminder: fees are high, market is immature, agents are amateur, EMH is not the rule
on such scales...
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Market impact on the Bitcoin
Over 4 decades, impact I(Q) = h(pT − p0 ) | Qi is square root
Impact of metaorders on the price
Idem, including in-trajectory points
100
Buy meta-orders
Sell meta-orders
δ = 0.5
10
1
0.1
0.01
0.1
1
10
100 1000 10000
Volume in Bitcoins
Impact in %
Impact in %
100
Buy meta-orders
Sell meta-orders
δ = 0.5
10
1
0.1
0.01
0.1
1
10
100 1000 10000
Volume in Bitcoins
Remark: For the right plot, metaorders are regularly sub-sampled in quantiles of volumes (every
2.5%) so that every trajectory has equal weight.
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Market impact on the Bitcoin
Question: is it possible that this comes from a conditioning between the
executed volume and the price signal?
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Market impact on the Bitcoin
I The whole price trajectories
during impact pt , t = 0...T (for
given Q, T ) are square root of
time (not only the end points);
I This suggests the existence of a
microscopic mechanism to
produce this shape.
Square root law for price impact: Empirical evidence and theory
150BT C
600BT C
10000BT C
δ = 0.5
1
Impact in %
the scatter plot (I(Q), Q) does
not come from a conditioning of
Q on the price signal: it is a
trajectory effect;
Impact in %
I Thus, the square root form for
Trading rate: 10 BTC/s
10
0.1
0.01
1
10
100
1000
Volume in Bitcoins
10000
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Bid, ask, traded price: What relevance?
800
600
400
200
0
-200
-400
-600
-1500 -1000
500
400
300
200
100
0
-100
-200
-1500 -1000
250
200
150
100
50
0
-50
-100
-150
-1500 -1000
Impact on the price
2000
0.008
0.004
0
-0.004
-0.008
-0.012
-0.016
-1500 -1000
-500
0
500
1000
1500
2000
2000
0.006
0.004
0.002
0
-0.002
-0.004
-0.006
-0.008
-0.01
-1500 -1000
-500
0
500
1000
1500
2000
-500
0
500
1000
1500
2000
-500
0
500
Time (s)
1000
1500
2000
Impact in %
Imbalance (BTC)
Order flow imbalance
1600
1200
800
400
0
-400
-800
-1200
-1600
-1500 -1000
Meta-order
Others
All
-500
-500
0
0
500
500
1000
1000
1500
1500
-500
0
500
1000
1500
2000
-500
0
500
Time (s)
1000
1500
2000
0.003
0.002
0.001
0
-0.001
-0.002
-0.003
-0.004
-0.005
-0.006
-1500 -1000
0.003
0.002
0.001
0
-0.001
-0.002
-0.003
-0.004
-0.005
-0.006
-1500 -1000
Avg Exec
Ask
Bid
Some impact trajectories for given Q,T. The "price" means nothing here: Only the
best opposite is relevant for our purposes and give a square root, which supports
mechanical theories of impact as LLOB.
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Opposite side dynamics after the trade
I After the execution is completed, the opposite side reverts;
I For isolated trades, it almost reverts to the initial price
Square root law for price impact: Empirical evidence and theory
[see also Brokmann et al., 2014]
.
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Execution speed
Cost increases with execution speed for isolated metaorders
Remark: For non-isolated metaorders, changing the execution horizon T changes the
amount of correlation with the markets: we would observe the wrong effect!
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Impact pre-factor
0.002
f
√ Y
σD / VD
f
Y
-ratio
0.0015
0.001
0.0005
0
2011-09-01
2012-03-01
2012-09-01
Time
4
2013-09-01
√
f
Y := Y
(σD / VD )−1
rolling mean
3.5
3
Y -ratio
2013-03-01
2.5
2
1.5
1
0.5
0
2011-09-01
2012-03-01
2012-09-01
Time
2013-03-01
2013-09-01
√
√
(Top) Impact pre-factor I(Q)/ Q vs usual normalization σD / VD (Bottom)
Residual Y-ratio
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Lessons from the Bitcoin study
What did we learn here?
I A constant pressure on the price lifts it as a square root of time;
I The square root holds at all scales, in particular far below the spread and the
fees;
I The impact of isolated orders reverts to zero (or close); the part of the impact
that appears permanent is only due to correlation with the market overall
direction.
I Because of the microstructure of the Bitcoin market, EMH cannot be the
determinant of impact. More mechanical mechanisms must be at stake.
I This study strongly supports the LLOB theory
Square root law for price impact: Empirical evidence and theory
[Donier, Bonart, Mastromatteo and Bouchaud, 2014]
.
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
IV. Zooming-out: Bubbles and crashes
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
We have a full description of what happens at the microscopic scale
CAN WE GO FURTHER ?
Let us investigate some macroscopic features: Bubbles, crashes...
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Volume
Liquidity on the order book fluctuates... [animation]
1
Price
...which should reflect at all scales!
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Macroscopic liquidity: Definition
Let us introduce a macroscopic definition of liquidity LOB :
Z
pt
dpρ(p, t) := LOB (φ) ,
(3)
Snapshot of cumulated volume (BTC)
pt (1−φ)
80000
Cumulated demand
Cumulated supply
70000
60000
50000
40000
LOB (φ)
30000
20000
10000
φ
0
30
35
40
45
50
55
Price
This definition is meaningful on the Bitcoin where liquidity is displayed long in
advance (as opposed to financial markets).
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Macroscopic liquidity: Fact I
LOB correctly predicts the amplitude of crashes:
OB
Actual drop (%)
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
10
L−1
OB (OB )
R2 = 0.75
20
30
40
50
Imbalance (kBTC)
Square root law for price impact: Empirical evidence and theory
60
70
0
0
R2 = 0.994
10
20
30
40
50
60
Liquidity-adjusted imbalance (%)
70
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Macroscopic liquidity: Fact II
A support price can be defined as φ∗ such that φ∗ = L−1
(Q ∗ ), where Q ∗ := 40kBTC
OB
(typical large sell-off):
300
250
Actual price
Support price
Price
200
150
100
50
0
16/03/2013
23/03/2013
Square root law for price impact: Empirical evidence and theory
30/03/2013
Time
06/04/2013
13/04/2013
20/04/2013
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Macroscopic liquidity: Fact III
LOB is well tracked by the theoretical and empirical impact pre-factors:
√
I LI := I(Q)/ Q
√
I LTH := σD / VD
20
√ Y-ratio (LI )
σ/ ADV (LT H )
Liquidity (LOB )
0
% of drop if a typical sell-off happens
-20
-40
-60
-80
-100
-120
-140
-160
11/2012
01/2013
Square root law for price impact: Empirical evidence and theory
03/2013
05/2013
Time
07/2013
09/2013
11/2013
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Macroscopic liquidity: Fact IV
√
The theoretical impact pre-factor LTH := σD / VD is a good predictor of tomorrow’s
liquidity (much better that Amihud’s ILLIQ measure):
LI
120
R2 = 0.88
120
100
100
80
80
60
60
40
40
20
0
R2 = 0.86
40
60
20
0
20
40
ILLIQ
60
80
100
120
0
0
20
R2 = 0.74
120
120
100
100
80
80
60
60
40
40
20
0
LT H
80
LT H (lagged)
R2 = 0.83
40
80
100
120
100
120
20
0
20
40
60
80
Square root law for price impact: Empirical evidence and theory
100
120
0
0
20
60
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Conclusions: Bubbles and crashes
Based on a microscopic understanding
of liquidity and price impact, we proposed a
√
measure of liquidity σD / VD (σD : daily volatility, VD : daily volume) that
I Is publicly available,
I Detects bubbles,
I Correctly predicts the amplitude of potential crashes,
I Largely outperforms Amihud’s popular ILLIQ measure σD /VD .
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
Conclusions: Overall
I
Continuous trading has a universal effect on th eorder book shape.
I
It makes it grow linearly next to the price.
I
A microscopic understanding of liquidity allows for a prediction of
extreme macroscopic events (bubbles, crashes...).
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
One last picture...
Order book...
Average volume
In the LLOB
theory
Avg shape bid
Avg shape ask
Linear interpolation
Relative price
Average volume
On the Bitcoin
Avg shape bid
Avg shape ask
Linear interpolation
Relative price
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
References
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015
Introduction: Price impact and LLOB
Data, metaorders
Market impact on the Bitcoin
Bubbles and crashes
Conclusions
References
I J Donier, J Bonart, A Million Metaorder Analysis of Market Impact on the
Bitcoin, arXiv 1412.4503 (2014)
I X Brokmann, E Serie, J Kockelkoren, J-P Bouchaud, Slow decay of impact in
equity markets, arXiv 1407.3390 (2014)
I B.Tòth, Y Lempérière, C Deremble, J de Lataillade, J Kockelkoren, J-P
Bouchaud, Anomalous price impact and th critical nature of liquidity in financial
markets, Phys. Rev. X (2011)
I I Mastromatteo, B.Tòth, J-P Bouchaud, Agent-based models for latent liquidity
and concave price impact, Phys. Rev. E 89 (2014)
I I Mastromatteo, B.Tòth, J-P Bouchaud, Anomalous impact in reaction-diffusion
models, Phys. Rev. Lett. (2014)
I J Donier, J Bonart, I Mastromatteo, J-P Bouchaud, A fully consistent, minimal
model for non-linear market impact, arXiv 1412.0141 (2014)
Square root law for price impact: Empirical evidence and theory
Imperial College London, March 4, 2015