carnets d’ordres pilotés par des processus de...
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Carnets d’ordres pilotés par des processus deHawkes
workshop sur les Mathématiques des marchés financiers en hautefréquence
Frédéric AbergelChaire de finance quantitative
fiquant.mas.ecp.fr/limit-order-books
14 Avril 2015
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 1/43
1 Stylized facts of limit order booksA word on dataArrival times of ordersCancellation of ordersIntraday seasonalityCompetitive liquidity
2 Mathematical modelling of limit order booksZero-intelligence models
Stability and long-time dynamicsLarge-scale limit of the price process
Modelling dependenciesHawkes processesApplication to order book modellingStability and long-time dynamics
3 Simulation with Hawkes processes
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 2/43
What is a limit order book ?
The limit order book is the list, at a given time, of all buy and selllimit orders, with their corresponding prices and volumes
The order book evolves over time according to the arrival of neworders
3 main types of orders:limit order: specify a price at which one is willing to buy (sell) a certainnumber of sharesmarket order: immediately buy (sell) a certain number of shares at thebest 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
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 3/43
What is a limit order book ?
The limit order book is the list, at a given time, of all buy and selllimit orders, with their corresponding prices and volumes
The order book evolves over time according to the arrival of neworders3 main types of orders:
limit order: specify a price at which one is willing to buy (sell) a certainnumber of sharesmarket order: immediately buy (sell) a certain number of shares at thebest 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
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 3/43
What is a limit order book ?
The limit order book is the list, at a given time, of all buy and selllimit orders, with their corresponding prices and volumes
The order book evolves over time according to the arrival of neworders3 main types of orders:
limit order: specify a price at which one is willing to buy (sell) a certainnumber of sharesmarket order: immediately buy (sell) a certain number of shares at thebest 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
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 3/43
Plan
1 Stylized facts of limit order booksA word on dataArrival times of ordersCancellation of ordersIntraday seasonalityCompetitive liquidity
2 Mathematical modelling of limit order booksZero-intelligence models
Stability and long-time dynamicsLarge-scale limit of the price process
Modelling dependenciesHawkes processesApplication to order book modellingStability and long-time dynamics
3 Simulation with Hawkes processes
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 4/43
Plan
1 Stylized facts of limit order booksA word on dataArrival times of ordersCancellation of ordersIntraday seasonalityCompetitive liquidity
2 Mathematical modelling of limit order booksZero-intelligence models
Stability and long-time dynamicsLarge-scale limit of the price process
Modelling dependenciesHawkes processesApplication to order book modellingStability and long-time dynamics
3 Simulation with Hawkes processes
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 5/43
Limit order book events
Timestamp Side Level Price Quantity33480.158 B 1 121.1 48033480.476 B 2 121.05 163633481.517 B 5 120.9 131833483.218 B 1 121.1 42033484.254 B 1 121.1 55633486.832 A 1 121.15 18733489.014 B 2 121.05 139733490.473 B 1 121.1 34233490.473 B 1 121.1 30433490.473 B 1 121.1 25633490.473 A 1 121.15 237
Table: Tick by tick data file sample
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 6/43
Trades
Timestamp Last Last quantity33483.097 121.1 6033490.380 121.1 21433490.380 121.1 3833490.380 121.1 48
Table: Trades data file sample.
These data are made available (for free) to the scientific communityhttp://fiquant.mas.ecp.fr/liquidity-watch
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 7/43
Trades and order book data processing
For each stock and each trading day:1 Parse the tick by tick data file to compute order book state variations;2 Parse the trades file and for each trade:
1 Compare the trade price and volume to likely market orders whosetimestamps are in [tTr −∆t , tTr + ∆t], where tTr is the trade timestampand ∆t is a predefined time window;
2 Match the trade to the first likely market order with the same price andvolume and label the corresponding event as a market order;
3 Remaining negative variations are labeled as cancellations.
Doing so, we have an average matching rate of around 85% for CAC 40stocks. As a byproduct, one gets the sign of each matched trade.
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 8/43
Plan
1 Stylized facts of limit order booksA word on dataArrival times of ordersCancellation of ordersIntraday seasonalityCompetitive liquidity
2 Mathematical modelling of limit order booksZero-intelligence models
Stability and long-time dynamicsLarge-scale limit of the price process
Modelling dependenciesHawkes processesApplication to order book modellingStability and long-time dynamics
3 Simulation with Hawkes processes
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 9/43
Arrival time of orders
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 2 4 6 8 10 12 14
Em
pir
ical
den
sity
Interarrival time
BNPP.PALognormal
ExponentialWeibull
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 10/43
Plan
1 Stylized facts of limit order booksA word on dataArrival times of ordersCancellation of ordersIntraday seasonalityCompetitive liquidity
2 Mathematical modelling of limit order booksZero-intelligence models
Stability and long-time dynamicsLarge-scale limit of the price process
Modelling dependenciesHawkes processesApplication to order book modellingStability and long-time dynamics
3 Simulation with Hawkes processes
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 11/43
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
1 10 100 1000 10000
Pro
babi
lity
func
tions
Lifetime for cancelled limit orders
Power law ∝ x-1.4
BNPP.PAFTE.PA
RENA.PASOGN.PA
Figure: Distribution of estimated lifetime of cancelled limit orders.
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
1 10 100 1000 10000
Pro
babi
lity
func
tions
Lifetime for executed limit orders
Power law ∝ x-1.5
BNPP.PAFTE.PA
RENA.PASOGN.PA
Figure: Distribution of estimated lifetime of executed limit orders.
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 12/43
Plan
1 Stylized facts of limit order booksA word on dataArrival times of ordersCancellation of ordersIntraday seasonalityCompetitive liquidity
2 Mathematical modelling of limit order booksZero-intelligence models
Stability and long-time dynamicsLarge-scale limit of the price process
Modelling dependenciesHawkes processesApplication to order book modellingStability and long-time dynamics
3 Simulation with Hawkes processes
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 13/43
0
0.5
1
1.5
2
2.5
3
30000 35000 40000 45000 50000 55000 60000 65000
Num
ber
of m
arke
t ord
ers
subm
itted
in ∆
t=5
min
utes
Time of day (seconds)
BNPP.PABNPP.PA quadratic fit
FTE.PAFTE.PA quadratic fit
Figure: Normalized average number of market orders in a 5-minute interval.
0
0.5
1
1.5
2
2.5
3
3.5
30000 35000 40000 45000 50000 55000 60000 65000
Num
ber
of li
mit
orde
rs s
ubm
itted
in ∆
t=5
min
utes
Time of day (seconds)
BNPP.PABNPP.PA quadratic fit
FTE.PAFTE.PA quadratic fit
Figure: Normalized average number of limit orders in a 5-minute interval.
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 14/43
Plan
1 Stylized facts of limit order booksA word on dataArrival times of ordersCancellation of ordersIntraday seasonalityCompetitive liquidity
2 Mathematical modelling of limit order booksZero-intelligence models
Stability and long-time dynamicsLarge-scale limit of the price process
Modelling dependenciesHawkes processesApplication to order book modellingStability and long-time dynamics
3 Simulation with Hawkes processes
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 15/43
Competitive liquidity
Muni Toke (2011)
Figure: Evidence of liquidity providing
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Competitive liquidity
Muni Toke (2011)Anane et al. (2015)
Figure: Evidence of liquidity taking
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 17/43
Plan
1 Stylized facts of limit order booksA word on dataArrival times of ordersCancellation of ordersIntraday seasonalityCompetitive liquidity
2 Mathematical modelling of limit order booksZero-intelligence models
Stability and long-time dynamicsLarge-scale limit of the price process
Modelling dependenciesHawkes processesApplication to order book modellingStability and long-time dynamics
3 Simulation with Hawkes processes
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 18/43
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
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 19/43
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
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 19/43
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 = PA+PB
2 : mid-price
A(p),B(p): cumulative number of sell (buy) orders up to price levelp
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 20/43
Notations
a
9a
1a 3a 4a2a5a
6a
7a
8a
b
9b
8b
7b
6b
5b
1b2b3b4b
APBP S
Figure: Order book notations
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 21/43
Plan
1 Stylized facts of limit order booksA word on dataArrival times of ordersCancellation of ordersIntraday seasonalityCompetitive liquidity
2 Mathematical modelling of limit order booksZero-intelligence models
Stability and long-time dynamicsLarge-scale limit of the price process
Modelling dependenciesHawkes processesApplication to order book modellingStability and long-time dynamics
3 Simulation with Hawkes processes
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 22/43
Zero-intelligence model
Inspired by Smith et al. (2003), Abergel and Jedidi (2013) analyzes anelementary LOB model where the events affecting the order book aredescribed by independent Poisson processes:
M±: arrival of new market order, with intensityλM±
q;
L±i : arrival of a limit order at level i, with intensityλL±
i ∆P
q;
C±i : cancellation of a limit order at level i, with intensity λC+
iai
qand
λC−i|bi |
q⇒ Cancellation rate is proportional to the outstanding quantity ateach level.
Under these assumptions, (X(t))t≥0 is a Markov process with statespace S ⊂ Z2K .
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 23/43
Order book dynamics
dai(t) = −
q − i−1∑k=1
ak
+
dM+(t) + qdL+i (t) − qdC+
i (t)
+ (JM−(a) − a)idM−(t) +K∑
i=1
(JL−i (a) − a)idL−i (t)
+K∑
i=1
(JC−i (a) − a)idC−i (t),
dbi(t) = similar expression,
where JM± , JL±i , and JC±i are shift operators corresponding to the effect oforder arrivals on the reference frame
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 24/43
The shift operator
For instance, the shift operator corresponding to the arrival of a sellmarket order is
JM−(a) =
0, 0, . . . , 0︸ ︷︷ ︸k times
, a1, a2, . . . , aK−k
,with
k := inf{p :
p∑j=1
|bj | > q} − inf{p : |bp | > 0}.
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 25/43
Infinitesimal generator
Lf (a; b) = λM+
(f([ai − (q − A(i − 1))+]+; JM+
(b))− f)
+K∑
i=1
λL+
i (f(ai + q; JL+
i (b))− f)
+K∑
i=1
λC+
i ai(f(ai − q; JC+
i (b))− f)
+ λM−(f(JM−(a); [bi + (q − B(i − 1))+]−
)− f
)+
K∑i=1
λL−i (f
(JL−i (a); bi − q
)− f)
+K∑
i=1
λC−i |bi |(f
(JC−i (a); bi + q
)− f)
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 26/43
Stability of the order book
Stationary order book distribution
Abergel and Jedidi (2013) If λC = min1≤i≤N{λC±i } > 0, then
(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 ofconvergence of the order book to its stationary state is exponential.
The proof relies on the use of a Lyapunov functionThe proportional cancellation rate helps a lot... and so do the boundaryconditions!
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 27/43
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 theasymptotic behaviour of the price, based on Foster-Lyapunov-typecriteria Meyn and Tweedie (1993)Glynn and Meyn (1996) and theconvergence of martingales Ethier and Kurtz (2005):
the evolution of the price is dPt =∑K ′
i=1 Fi(Xt )dNit ;
the rescaled, centered price is P̃nt ≡
Pnt−∫ nt
0
∑K ′i=1 Fi(Xs)λids√
n
its predictable quadratic variation is < P̃n, P̃n >t =∫ nt
0
∑K ′i=1(Fi(Xs))
2λidsn
ergodicity ensures the convergence of∫ nt
0
∑K ′i=1(Fi(Xs))
2λidsnt as n → ∞
This approach is easily extended to state-dependent intensitiesA similar approach - in a somewhat different modelling context - is usedin recent papers by Horst and co-authors Horst and Paulsen (2015).
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 28/43
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 theasymptotic behaviour of the price, based on Foster-Lyapunov-typecriteria Meyn and Tweedie (1993)Glynn and Meyn (1996) and theconvergence of martingales Ethier and Kurtz (2005):
the evolution of the price is dPt =∑K ′
i=1 Fi(Xt )dNit ;
the rescaled, centered price is P̃nt ≡
Pnt−∫ nt
0
∑K ′i=1 Fi(Xs)λids√
n
its predictable quadratic variation is < P̃n, P̃n >t =∫ nt
0
∑K ′i=1(Fi(Xs))
2λidsn
ergodicity ensures the convergence of∫ nt
0
∑K ′i=1(Fi(Xs))
2λidsnt as n → ∞
This approach is easily extended to state-dependent intensitiesA similar approach - in a somewhat different modelling context - is usedin recent papers by Horst and co-authors Horst and Paulsen (2015).
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 28/43
A general approach made precise I
Write as above
Pt = P0 +∑
i
∫ t
0Fi(Xs)dNi
s
and its compensator
Qt =∑
i
∫ t
0λiFi(Xs)ds.
Defineh =
∑i
λiFi(X)
and let
α =a.s.lim
t→+∞
1t
∑i
∫ t
0λi(Fi(Xs))ds =
∫h(X)Π(dX).
Finally, introduce the solution g to the Poisson equation
Lg = h − α
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 29/43
A general approach made precise II
and the associated martingale
Zt = g(Xt ) − g(X0) −
∫ t
0Lg(Xs)ds ≡ g(Xt ) − g(X0) − Qt + αt .
Then, the deterministically centered, rescaled price
P̄nt ≡
Pnt − αnt√
n
converges in distribution to a Wiener process σ̄W . The asymptoticvolatility σ̄ satisfies the identity
σ̄2 = limt→+∞
1t
∑i
∫ t
0λi((Fi−∆i(g))(Xs))2ds ≡
∑i
∫λi((Fi−∆i(g))(X))2Π(dx).
where ∆i(g)(X) denotes the jump of the process g(X) when the processNi jumps and the limit order book is in the state X .
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 30/43
Plan
1 Stylized facts of limit order booksA word on dataArrival times of ordersCancellation of ordersIntraday seasonalityCompetitive liquidity
2 Mathematical modelling of limit order booksZero-intelligence models
Stability and long-time dynamicsLarge-scale limit of the price process
Modelling dependenciesHawkes processesApplication to order book modellingStability and long-time dynamics
3 Simulation with Hawkes processes
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 31/43
Hawkes processes
Zero-intelligence models fail to capture the dependencies betweenvarious types of orders:
clustering of market orders;
interplay between liquidity taking and providing;
leverage effect,
as has been demonstrated in many contributions, see e.g. Muni Toke(2011)Muni Toke and Pomponio (2012).
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 32/43
Hawkes processes
Hawkes processes provide an ad hoc tool to describe the mutualexcitations of the arrivals of different types of orders.In D dimensions, the process Nj
s has a stochastic intensity λjt such that
λjt = λ
j0 +
D∑p=1
∫ t
0φjp(t − s)dNp
s . (2.1)
A simplifying choice is the exponential kernel
φjp(s) = αjp exp(−βjps) (2.2)
leading to markovian processes. A classical result states that the processis stationary iff the spectral radius of the matrix
[αjp
βjp] (2.3)
is < 1, see Brémaud and Massoulié (1996).
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Introducing dependence in the order flow
As an example, Muni Toke (2011) study variations of the model below
A better order book model
λM(t) = λM0 +
∫ t
0αMMe−βMM(t−s)dNM(s),
λL (t) = λL0 +
∫ t
0αLL e−βLL (t−s)dNL (s)
+
∫ t
0αML e−βML (t−s)dNM(s).
MM and LL effect for clustering of orders
ML effect as observed on data
No global LM effect observed on data
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Infinitesimal generator
LF(−→a ;−→b ;−→µ ) = λM+
(F([ai − (q − A(i − 1))+]+; JM+
(−→b );−→µ + ∆M+
(−→µ ))− F)
+K∑
i=1
λL+
i (F(ai + q; JL+
i (−→b );−→µ + ∆L+
i (−→µ ))− F)
+K∑
i=1
λC+
i ai(F(ai − q; JC+
i (−→b )
)− F)
+ λM−(F
(JM−(−→a ); [bi + (q − B(i − 1))+]−;−→µ + ∆M−(−→µ )
)− F
)+
K∑i=1
λL−i (F
(JL−i (−→a ); bi − q;−→µ + ∆L−i (−→µ )
)− F)
+K∑
i=1
λC−i |bi |(f
(JC−i (−→a ); bi + q
)− f)
−
D∑i,j=1
βijµij∂F∂µij
. (2.4)
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Large-time behaviour
Results similar to those obtained in the zero-intelligence case holdAbergel and Jedidi (2015)
Large-time behaviour for Hawkes-process driven LOB
Under the usual stationarity conditions for the intensities, thereexists a Lyapunov function V =
∑|ai |+
∑|bi |+
∑Ukλk and the LOB
converges exponentially to its stationary distribution Π
The rescaled, (deterministically) centered price converges to aWiener process
The proofs rely on the same decomposition as in the Poisson arrivalcase, thanks to the fact that the proportional cancellation rate remainsbounded away from zeroSome extra care is required to prove that the solution to the Poissonequation is in L2(Π(dx))
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 36/43
Plan
1 Stylized facts of limit order booksA word on dataArrival times of ordersCancellation of ordersIntraday seasonalityCompetitive liquidity
2 Mathematical modelling of limit order booksZero-intelligence models
Stability and long-time dynamicsLarge-scale limit of the price process
Modelling dependenciesHawkes processesApplication to order book modellingStability and long-time dynamics
3 Simulation with Hawkes processes
Institute for Pure and Applied Mathematics Hawkes-process-driven LOB 37/43
Hawkes model
In Muni Toke (2011), the flow of limit and market orders are modelled byHawkes processes NL and NM with intensities λ and µ defined as follows:
µM(t) = µM0 +
∫ t
0αMMe−βMM(t−s)dNM
s ,
λL (t) = λL0 +
∫ t
0αLMe−βLM(t−s)dNM
s +
∫ t
0αLL e−βLL (t−s)dNL
s
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Numerical results on the order book: fitting
Fitting and simulation
Model µ0 αMM βMM λ0 αLM βLM αLL βLL
HP 0.22 - - 1.69 - - - -LM 0.22 - - 0.79 5.8 1.8 - -MM 0.09 1.7 6.0 1.69 - - - -
MM LL 0.09 1.7 6.0 0.60 - - 1.7 6.0MM LM 0.12 1.7 6.0 0.82 5.8 1.8 - -
MM LL LM 0.12 1.7 5.8 0.02 5.8 1.8 1.7 6.0Common parameters: mP
1 = 2.7, νP1 = 2.0, sP
1 = 0.9V1 = 275,mV
2 = 380λC = 1.35, δ = 0.015
Table: Estimated values of parameters used for simulations.
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Impact on arrival times
Figure: Empirical density function of the distribution of the interval times between a marketorder 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
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Impact on the bid-ask spread
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.05 0.1 0.15 0.2 0.25 0.3
Empirical BNPP.PAHomogeneous Poisson
Hawkes MMHawkes MM+LM
10-6
10-5
10-4
10-3
10-2
10-1
100
0 0.05 0.1 0.15 0.2
Figure: Empirical density function of the distribution of the bid-ask spread forthree simulations, namely HP, MM, MM+LM, compared to empirical measures. Ininset, same data using a semi-log scale. X-axis is scaled in euro (1 tick is 0.01euro).
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.05 0.1 0.15 0.2
Empirical BNPP.PAHomogeneous Poisson
Hawkes MM+LLHawkes MM+LL+LM
10-6
10-5
10-4
10-3
10-2
10-1
100
0 0.05 0.1 0.15 0.2
Figure: Empirical density function of the distribution of the bid-ask spread threesimulations, 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).
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References I
Abergel, F. and Jedidi, A. (2013). A mathematical approach to order bookmodelling. International Journal of Theoretical and Applied Finance,16(5).
Abergel, F. and Jedidi, A. (2015). Long time behaviour of a Hawkesprocess-based limit order book.
Anane, M., Abergel, F., and Lu, X. F. (2015). Mathematical modeling ofthe order book: New approach of predicting single-stock returns.
Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkesprocesses. Ann. Probab., 24:1563–1588.
Ethier, S. N. and Kurtz, T. G. (2005). Markov processes characterizationand convergence. Wiley.
Glynn, P. W. and Meyn, S. P. (1996). A Liapounov bound for solutions ofthe Poisson equation. Annals of Probability, 24(2):916–931.
Horst, U. and Paulsen, M. (2015). A law of large numbers for limit orderbooks.
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References II
Meyn, S. and Tweedie, R. (1993). Stability of markovian processes III:Foster-lyapunov criteria for continuous-time processes. Advances inApplied Probability, 25:518–548.
Muni Toke, I. (2011). Market Making in an order book model and itsimpact on the spread. In Abergel, F., Chakrabarti, B. K., Chakraborti,A., and Mitra, M., editors, Econophysics of Order-driven Markets,pages 49–64. Springer.
Muni Toke, I. and Pomponio, F. (2012). Modelling trade-throughs in alimited order book using Hawkes processes. Economics: TheOpen-Access, Open-Assessment E-Journal, 6.
Smith, E., Farmer, J. D., Gillemot, L., and Krishnamurthy, S. (2003).Statistical theory of the continuous double auction. QuantitativeFinance, 3(6):481–514.
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