Modélisation mathématique des carnets d`ordres

Transcription

Modélisation mathématique des carnets d’ordres
Frédéric Abergel
Séminaire du laboratoire MICS
3 décembre 2015
Modélisation des carnets d’ordres
1/36
1
Elementary modelling of limit order books
Zero-intelligence models
Stability and long-time dynamics
Large-scale limit of the price process
Simulation with zero-intelligence models
2
Better order book modelling
Empirical evidence of the order flow dependency structure
Modelling with Hawkes processes
Simulation with Hawkes processes
2/36
What is a limit order book ?
The limit order book is the list, at a given time, of all buy and sell
limit orders, with their corresponding prices and volumes
The order book evolves over time according to the arrival of new
orders
3 main types of orders:
limit order: specify a price at which one is willing to buy (sell) a certain
number of shares
market order: immediately buy (sell) a certain number of shares at the
best available opposite quote(s)
cancellation order: cancel an existing limit order
Price dynamics
The price dynamics becomes a by-product of the order book dynamics
3/36
Plan
1
2
4/36
Mathematical modelling
A set of reasonable assumptions:
the limit order book is described as a point process;
several types of events can happen;
two events cannot occur simultaneously (simple point process).
The main questions of interest are
the stationarity and ergodicity of the order book;
the dynamics of the induced price...
... particularly, its behaviour at larger time scales.
⇒ Linking microstructure modelling and continuous-time finance
5/36
Some notations
The order book is represented by a finite-size vector of quantities
X(t ) := (a(t ); b(t )) := (a1 (t ), . . . , aK (t ); b1 (t ), . . . , bK (t ));
a(t ): ask side of the order book
b(t ): bid side of the order book
∆P: tick size
q: unit volume
P=
P A +P B
:
2
mid-price
A (p ), B (p ): cumulative number of sell (buy) orders up to price level
p
6/36
Notations
a7
PB
PA
S
a
a8
a6
a9
a5
a1 a2 a3 a4
b6
b4 b3 b2
b1
b5
b9
b7
b
b8
Figure: Order book notations
7/36
Plan
1
2
8/36
Zero-intelligence model
Inspired by Smith et al. (2003), Abergel and Jedidi (2013) analyzes an
elementary LOB model where the events affecting the order book are
described by independent Poisson processes:
±
λM
;
q
±
λL ∆ P
Li± : arrival of a limit order at level i, with intensity i
;
q
M ± : arrival of new market order, with intensity
Ci± : cancellation of a limit order at level i, with intensity λC
i
+
ai
and
q
|bi |
q
⇒ Cancellation rate is proportional to the outstanding quantity at
each level.
λCi
−
Under these assumptions, (X(t ))t ≥0 is a Markov process with state
space S ⊂ Z2K .
9/36
Stability of the order book
Stationary order book distribution
±
} > 0, then
Abergel and Jedidi (2013) If λC = min1≤i ≤N {λC
i
(X(t ))t ≥0 = (a(t ); b(t ))t ≥0 is an ergodic Markov process. In particular
(X(t )) has a unique stationary distribution Π. Moreover, the rate of
convergence of the order book to its stationary state is exponential.
The proof relies on the use of a Lyapunov function V (a, b) = ai + |bi |
The proportional cancellation rate helps a lot... and so do the boundary
conditions!
P
P
10/36
Stability of the order book
Stationary order book distribution
±
} > 0, then
Abergel and Jedidi (2013) If λC = min1≤i ≤N {λC
i
(X(t ))t ≥0 = (a(t ); b(t ))t ≥0 is an ergodic Markov process. In particular
(X(t )) has a unique stationary distribution Π. Moreover, the rate of
convergence of the order book to its stationary state is exponential.
The proof relies on the use of a Lyapunov function V (a, b) = ai + |bi |
The proportional cancellation rate helps a lot... and so do the boundary
conditions!
An extension of this result holds for general cancellation rates tending to
infinity up to a change of Lyapunov function, see Abergel et al. (2015)
P
P
10/36
A general approach to study price asymptotics
Combining ergodic theory and martingale convergence
The ergodicity of the order book allows for a direct study of the
asymptotic behaviour of the price, based on Foster-Lyapunov-type
criteria Meyn and Tweedie (1993)Glynn and Meyn (1996) and the
convergence of martingales Ethier and Kurtz (2005):
the evolution of the price is dPt =
PK 0
i =1
the rescaled, centered price is P̃tn ≡
Fi (Xt )dNti ;
R nt PK 0
i =1 Fi (Xs )λi ds
0
√
Pnt −
n
its predictable quadratic variation is < P̃ n , P̃ n >t =
R nt PK 0
0
2
i =1 (Fi (Xs )) λi ds
n
R nt PK 0
ergodicity ensures the convergence of
0
2
nt
as n → ∞
11/36
A general approach to study price asymptotics
Combining ergodic theory and martingale convergence
The ergodicity of the order book allows for a direct study of the
asymptotic behaviour of the price, based on Foster-Lyapunov-type
criteria Meyn and Tweedie (1993)Glynn and Meyn (1996) and the
convergence of martingales Ethier and Kurtz (2005):
the evolution of the price is dPt =
PK 0
i =1
the rescaled, centered price is P̃tn ≡
Fi (Xt )dNti ;
R nt PK 0
i =1 Fi (Xs )λi ds
0
√
Pnt −
n
its predictable quadratic variation is < P̃ n , P̃ n >t =
R nt PK 0
0
2
n
R nt PK 0
ergodicity ensures the convergence of
0
2
nt
as n → ∞
This approach is easily extended to state-dependent intensities
A similar approach - in a somewhat different modelling context - is used
in recent papers by Horst and co-authors Horst and Paulsen (2015).
11/36
A general approach made precise I
Write as above
Pt = P0 +
t
XZ
Fi (Xs )dNsi
0
i
and its compensator
Qt =
XZ
h=
λi Fi (Xs )ds .
0
i
Define
t
X
λi Fi (X )
i
and let
a .s .
α = lim
t →+∞
Z
Z
1X t i
λ (Fi (Xs ))ds =
h (X )Π(dX ).
t i 0
Finally, introduce the solution g to the Poisson equation
Lg = h − α
12/36
A general approach made precise II
and the associated martingale
t
Z
Zt = g (Xt ) − g (X0 ) −
Lg (Xs )ds ≡ g (Xt ) − g (X0 ) − Qt + αt .
0
Then, the deterministically centered, rescaled price
P̄tn ≡
Pnt − αnt
√
n
converges in distribution to a Wiener process σ̄W . The asymptotic
volatility σ̄ satisfies the identity
σ̄2 = lim
t →+∞
Z
XZ
1X t i
λ ((Fi −∆i (g ))(Xs ))2 ds ≡
λi ((Fi −∆i (g ))(X ))2 Π(dx ).
t i 0
i
where ∆i (g )(X ) denotes the jump of the process g (X ) when the process
N i jumps and the limit order book is in the state X .
13/36
Plan
1
2
14/36
Shape of the order book
Figure 2 represents the average shape of the order book. The agreement
between the simulated shape and the empirical one is fairly good.
2000
Model
Data
1500
Quantity (shares)
1000
500
0
−500
−1000
−1500
−2000
15/36
Some quantities of interest
(a)
FTE
LVMH
SCHN
AXAF
CAGR
UNBP
Log hAi (5) (data)
4
PRTP VIV
ISPA
GSZ
BNPP
SASY TOTF
ALUA
SEVI
EDF
VIE
DANO
CARR
EAD
ESSI
PERP
AIRP
OREP
BOUY
SGEF
SGOB
MICP
TECF
VLLP
SOGN
ALSO
PUBP
ACCP
SAF
RENA
PEUP
CAPP
LAFP
3
2
1
STM
0.5
1
1.5
2
2.5
3
Log hAi (5) (model)
3.5
4
4.5
(b)
Log hSi (data)
2
STM
RENA
LAFP
CAPP
PEUP
SOGN
ACCP
PUBP
ALSO
SAF
TECF
VLLP
MICP
ESSI
SGOB
OREP
AIRP
BOUY
PERP
EAD
SGEF
CARR
DANO
SEVI
PRTP
EDFALUA
VIE
BNPP
SASY
TOTF
ISPA
GSZ
VIV
UNBP
AXAF
CAGR
SCHN
LVMH
FTE
1.5
1
0.5
0.2
0.4
0.6
0.8
1
1.2 1.4
Log hSi (model)
1.6
1.8
2
(c)
SOGN
σ∞ (data)
1.2
RENA
STM
1
LAFP
PEUPSGOB
ALSO
CAPP
SGEF
MICP
VLLPAIRP
ACCP
ALUA
CARR
OREP
PUBP TECF
BNPP
DANO
SASY
TOTF
BOUY
ESSI SAF
EDF
ISPA
GSZ
PERP
PRTP
EAD
VIE
AXAF
CAGR
VIV
SEVI
UNBP
LVMH
SCHN
FTE
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
σ∞ (model)
1
1.2
16/36
Signature plot
Figure 4 shows the signature plots computed on our simulations
compared to the empirical ones. Signature plots are computed for both
the trade prices and the mid-prices, and in both event and calendar time.
Trades (Sim.)
Mid−price (Sim.)
Trades (Data)
Mid−price (Data)
1.2
1
0.25
0.8
σh2
0.2
0.15
0.6
0.1
0.05
0.4
0
20
40
60
120
140
160
80
100
0.2
0
20
40
60
80
100
time) des carnets d’ordres
Séminaire du laboratoireTime
MICS lag h (trade
Modélisation
180
200
17/36
Summarizing the zero intelligence models
Easy to model
Nice mathematical properties
Easy to calibrate and simulate
A good descriptor of the shape of the book and the retun distibution
Fails to reproduce the very high frequency structure of volatility
Fails to reproduce the low frequency volatility
18/36
Plan
1
2
19/36
Plan
1
2
20/36
A finer description of the order flow
Notation
M, L , C, O
Definition
market order, limit order, cancellation, any order.
Mbuy , Msell
buy/sell market order.
0
0
, Msell
Mbuy
buy/sell market order that does not change the mid price:
i. e. order quantity < best ask/bid available quantity.
1
1
Mbuy
, Msell
buy/sell market order that changes the mid price:
i. e. order quantity ≥ best ask/bid available quantity.
Lbuy , Lsell
buy/sell limit order.
0
0
Lbuy
, Lsell
buy/sell limit order that does not change the mid price:
i. e. order price ≤ / ≥ best bid/ask price.
1
1
Lbuy
, Lsell
buy/sell limit order that changes the mid price:
i. e. order price > / < best bid/ask price.
21/36
Measuring dependencies in the order flow: passive
orders
0
|Lbuy
0
|Lsell
0
|Cbuy
0
|Csell
0
|Mbuy
0
|Msell
1
|Lbuy
1
|Lsell
1
|Cbuy
1
|Csell
1
|Mbuy
1
|Msell
0
Lbuy
41.37
9.61
17.91
25.18
22.17
5.60
32.39
7.65
25.02
21.48
28.27
11.83
0
Lsell
9.64
41.79
25.88
17.98
5.33
21.14
8.06
29.94
19.09
23.28
9.60
23.05
0
Cbuy
16.00
21.95
40.67
6.04
4.75
10.61
0.21
26.04
35.70
5.42
7.38
33.36
0
Csell
22.40
16.12
5.98
41.30
9.94
5.01
25.27
0.22
4.96
34.70
28.12
7.24
0
Mbuy
2.90
1.61
1.39
1.79
34.64
0.72
4.84
5.63
0.96
0.76
3.11
1.04
0
Msell
1.58
2.96
1.74
1.42
0.70
34.32
5.58
5.62
0.67
1.16
1.02
3.13
|O
22.82
22.93
19.80
20.03
2.99
3.00
Table: Conditional probabilities (in %) of occurrences per event type
22/36
Measuring dependencies in the order flow: aggressive
orders
0
|Lbuy
0
|Lsell
0
|Cbuy
0
|Csell
0
|Mbuy
0
|Msell
1
|Lbuy
1
|Lsell
1
|Cbuy
1
|Csell
1
|Mbuy
1
|Msell
|O
1
Lbuy
2.35
1.02
1.20
2.08
7.68
0.53
1.42
1.39
8.34
3.20
11.52
6.79
1
Lsell
1.12
2.29
2.34
1.27
0.65
7.19
1.57
1.36
3.59
7.88
7.98
9.34
1
Cbuy
0.02
1.05
1.49
0.37
0.55
1.48
5.80
1.42
0.72
0.63
0.90
1.05
1
Csell
1.08
0.02
0.37
1.49
1.31
1.10
1.77
5.39
0.35
0.75
0.87
1.81
1
Mbuy
1.39
0.15
0.56
0.51
11.86
0.42
2.44
12.37
0.48
0.18
0.67
0.66
1
Msell
0.16
1.44
0.47
0.60
0.42
11.88
10.65
2.96
0.12
0.57
0.55
0.70
2.07
2.12
0.85
0.88
1.27
1.26
Table: Conditional probabilities (in %) of occurrences per event type
23/36
Interpreting the results
0
: reinforces the consensus that the stock is not moving down. This
Lbuy
0
.
increases the probability of Lbuy
0
: decreases the available liquidity on the buy side. Other participants
Cbuy
0
may feel less comfortable posting buy orders, and the probability of Cbuy
1
and Cbuy
increases.
0
0
Mbuy : increases the probability of Mbuy
. This can be explained by order
splitting - large orders are split into smaller pieces that are more easily
executed - and the momentum effect - other participants following the
1
1
move. The increase of the probability of Mbuy
and Lbuy
is also explained
by the momentum effect.
1
Lbuy
: improves the offered price to buy the stock. The first major effect
1
observed is a big increase in the probability of Msell
- this is the market
taking effect. The second effect is a large increase in the probability of
1
Cbuy
- the new liquidity is rapidly cancelled. This might reflect a market
manipulation, where agents are posting fake orders.
24/36
Interpreting the results
1
Cbuy
: a total cancellation of the best buy limit increases the probability of
1
Lbuy ; other participants re-offer the liquidity at the previous best buy price.
1
It also increases the probability of Lsell
, when a new consensus is
concluded by the market participants at a lower price.
1
Mbuy
: consumes all the offered liquidity at the best ask. This increases
1
the probability of Lsell
when some participants re-offer the liquidity at the
1
same previous best ask price. It also increases the probability of Lbuy
,
when a new consensus is concluded by the market participants at a
higher price. This is the market making effect.
25/36
Plan
1
2
26/36
The failures of zero-intelligence models
Zero-intelligence models fail to capture the dependencies between
various types of orders:
clustering of market orders;
interplay between liquidity taking and providing;
leverage effect,
as has been previously shown.
27/36
Hawkes processes
Hawkes processes provide an ad hoc tool to describe the mutual
excitations of the arrivals of different types of orders.
j
j
In D dimensions, the process Ns has a stochastic intensity λt such that
λjt
=
λj0
+
D Z
X
p =1
t
φjp (t − s )dNsp .
(2.1)
0
A simplifying choice is the exponential kernel
φjp (s ) = αjp exp(−βjp s )
(2.2)
leading to markovian processes. A classical result states that the process
is stationary iff the spectral radius of the matrix
[
αjp
]
βjp
(2.3)
is < 1, see Massoulié (1998).
28/36
Large-time behaviour
Results similar to those obtained in the zero-intelligence case hold
Abergel and Jedidi (2015)
Large-time behaviour for Hawkes-process driven LOB
Under the usual stationarity conditions for the intensities, there
P
P
P
exists a Lyapunov function V = |ai | + |bi | + Uk λk and the LOB
converges exponentially to its stationary distribution Π
The rescaled, (deterministically) centered price converges to a
Wiener process
The proofs rely on the same decomposition as in the Poisson arrival
case, thanks to the fact that the proportional cancellation rate remains
bounded away from zero
Some extra care is required to prove that the solution to the Poisson
equation is in L2 (Π(dx ))
29/36
Plan
1
2
30/36
Introducing dependencies in the order flow
As an example, Muni Toke (2011) study variations of the model below
A better order book model
λM (t )
λL (t )
= λM
0 +
Z
t
αMM e −βMM (t −s ) dN M (s ),
0
Z t
= λL0 +
αLL e −βLL (t −s ) dN L (s )
0
Z t
+
αML e −βML (t −s ) dN M (s ).
0
MM and LL effect for clustering of orders
ML effect as observed on data
No global LM effect observed on data
31/36
Numerical results on the order book: fitting
Fitting and simulation
Model
µ0
αMM βMM
HP
0.22
LM
0.22
MM
0.09
1.7
6.0
MM LL
0.09
1.7
6.0
MM LM
0.12
1.7
6.0
MM LL LM 0.12
1.7
5.8
Common parameters: m1P
λ0
αLM
βLM
αLL
1.69
0.79 5.8
1.8
1.69
0.60
1.7
0.82 5.8
1.8
0.02 5.8
1.8 1.7
= 2.7, ν1P = 2.0, s1P = 0.9
V
= 275, m2V = 380
1
C
λ = 1.35, δ = 0.015
βLL
6.0
6.0
Table: Estimated values of parameters used for simulations.
32/36
Impact on arrival times
Figure: Empirical density function of the distribution of the interval times between a market
order and the following limit order for three simulations, namely HP, MM+LL, MM+LL+LM,
compared to empirical measures. In inset, same data using a semi-log scale
33/36
Impact on the bid-ask spread
0.3
Empirical BNPP.PA
Homogeneous Poisson
Hawkes MM
Hawkes MM+LM
0.25
100
-1
10
0.2
10-2
10-3
0.15
-4
10
10-5
0.1
10-6
0
0.05
0.1
0.15
0.2
0.05
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Figure: Empirical density function of the distribution of the bid-ask spread for three
simulations, namely HP, MM, MM+LM, compared to empirical measures. In inset,
same data using a semi-log scale. X-axis is scaled in euro (1 tick is 0.01 euro).
34/36
References I
Abergel, F., Anane, M., Chakraborti, A., Jedidi, A., and Muni Toke, I.
(2015). Limit order books. Cambridge University Press. to appear.
Abergel, F. and Jedidi, A. (2013). A mathematical approach to order book
modelling. International Journal of Theoretical and Applied Finance,
16:1350025.
Abergel, F. and Jedidi, A. (2015). Long-time behavior of a Hawkes
process-based limit order book. SIAM Journal on Financial
Mathematics, 6:1026–1043.
Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes:
Characterization and Convergence. Wiley, Hoboken.
Glynn, P. W. and Meyn, S. P. (1996). A Liapounov bound for solutions of
the Poisson equation. Annals of Probability, 24:916–931.
Horst, U. and Paulsen, M. (2015). A law of large numbers for limit order
books. arXiv:1501.00843.
Massoulié, L. (1998). Stability results for a general class of interacting
point processes dynamics, and applications. Stochastic Processes
and their Applications, 75:1–30.
35/36
References II
Meyn, S. and Tweedie, R. (1993). Stability of Markovian processes III:
Foster-Lyapunov criteria for continuous-time processes. Advances in
Applied Probability, 25:518–548.
Muni Toke, I. (2011). Market making in an order book model and its
impact on the spread. In Abergel, F., Chakrabarti, B. K., Chakraborti,
A., and Mitra, M., editors, Econophysics of Order-driven Markets,
pages 49–64. Springer, Milan.
Smith, E., Farmer, J. D., Gillemot, L., and Krishnamurthy, S. (2003).
Statistical theory of the continuous double auction. Quantitative
Finance, 3:481–514.
36/36

Modélisation mathématique des carnets d`ordres

Transcription

Documents pareils

Postdoctoral Research Fellow « Organic synthesis of new additives

Language and Gender

Plaquette DigiCosme EN ( PDF - 385.2 kb)

DESCRIPTIFS DES COURS 2016-2017 Seminar 4 : Introduction to

[Descartes e il Seicento 5] Jean-Robert Armogathe, Vincent Carraud

4. L` animal a... Le poisson a . Le cochon d`Inde a . L

4228 Holland Dr Des Moines, IA 50310 | Home for Sale | Real

Congrè

Imagine you are working for the Scottish Tourist Board and you are

DESCRIPTIFS DES COURS 2016-2017