20110223-QFWTokyoSlides

8/4/2019 20110223-QFWTokyoSlides

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Some Applications of Hawkes Processesfor Order Book Modelling

Ioane Muni TokeEcole Centrale Paris

Chair of Quantitative Finance

First Workshop on Quantitative Finance and Economics

International Christian University, Tokyo (Mitaka)February 23rd, 2011

Ioane Muni Toke (ECP) Hawkes Processes in Finance February 23rd, 2011 1 / 60
http://find/http://goback/


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Table of contents

1 Introduction : Modeling with Point processes

2 Self- and mutually exciting Hawkes processes

3 A simple model for buy and sell intensities

4 Some statistical findings about the order flows

5 An order book model with Hawkes processes



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Introduction : Modeling with Point processes

Table of contents


2 Self- and mutually exciting Hawkes processesDefinition and stationarity conditionSimulation of a multivariate Hawkes processMaximum-likelihood estimationGoodness of fit






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Modeling Financial Data with Point Processes

(Ultra-) High frequency data i.e. tick-by-tick data

(Engle 2000) often cited as seminal paper (1996)ACD models by (Engle & Russell 1997)

Empirical fits on durations : Weibull, . . .

Tractability in multivariate settings

0

1

2

3

4

t1 t2 t3 t4

Time

t2

Counting process NEvents t

Figure: Illustration of a simple point process: events, durations, counter


I d i M d li i h P i


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Intensity process

Intensity

Let N be a point process adapted to a filtration Ft. The left-continuousFt-intensity process is defined as

(t|Ft) = limh0

E

N(t + h) N(t)

h

Ft

, (1)

or equivalently

(t|Ft) = limh0

1

hP [N(t + h) N(t) > 0|Ft] . (2)


S lf d t ll iti H k


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Self- and mutually exciting Hawkes processes

Table of contents







Self and mutually exciting Hawkes processes Definition and stationarity condition


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Self- and mutually exciting Hawkes processes Definition and stationarity condition

Table of contents









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Multidimensional Hawkes processes

Let M

N. Let

{(tmi )i

}m=1,...,M be a M-dimensional point process.

Nt = (N1t , . . . , NMt ) denotes the associated counting process.

Definition

A multidimensional Hawkes process is defined with intensitiesm, m = 1, . . . , M given by :

m(t) = m0 (t) +

Mn=1

t0

Pj=1

mnj emnj (ts)dNns , (3)

i.e. in its simplest version with P = 1 and m

0 (t) constant :

m(t) = m0 +Mn=1

t0

mnemn(ts)dNns =

m0 +

Mn=1

tni


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Stationarity condition (I)

Well take here P = 1 to simplify the notations. Rewriting equation (4)

using vectorial notation, we have :

(t) = 0 +

t

G(t s)dNs, (5)

where

G(t) =

mnemn

t1R+ (t)m,n=1,...,M

. (6)

Assuming stationarity gives E [(t)] = constant vector, and thusstationary intensities must satisfy :

= 0 + Et

G(t s)dNs = 0 + E t

G(t s)(s)ds , (7)

i.e.

= I

0

G(u)du1

0 (8)




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Stationarity condition (II)

Stationarity of a multivariate Hawkes process

A sufficient condition for a multivariate Hawkes process to be linear is thatthe spectral radius of the matrix

=

0G(u)du =

mn

mnm,n=1,...,M (9)be strictly smaller than 1.

We recall that the spectral radius of the matrix G is defined as :

(G

) = maxaS(G) |a|, (10)

where S(G) denotes the set of all eigenvalues of G.Note that this result can also be seen as a particular (linear) case of(Bremaud 1996, Theorem 7) which deals with general non-linear Hawkes

processes.Ioane Muni Toke (ECP) Hawkes Processes in Finance February 23rd, 2011 10 / 60

Self- and mutually exciting Hawkes processes Simulation of a multivariate Hawkes process


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y g p p

Table of contents









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y g p p

Thinning procedure

Lewis & Shedler (1979) proposes a thinning procedure that allows thesimulation of a point process with bounded intensity.

Basic thinning theorem

Consider a one-dimensional non-homogeneous Poisson process {N(t)}t0with rate function

(t), so that the number of points N

(T0) in a fixedinterval (0, T0] has a Poisson distribution with parameter

0 =T0

0 (s)ds. Let t1 , t

2 , . . . , t

N(T0)

be the points of the process in

the interval (0, T0]. Suppose that for 0 t T0, (t) (t).

For i = 1, 2, . . . , N(T0), delete the points ti with probability 1 (t

i )(ti )

.

Then the remaining points form a non-homogeneous Poisson process

{N(t)}t0 with rate function (t) in the interval (0, T0].




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y g

Simulation of a multivariate Hawkes process (I)

A general algorithm based on thinning is proposed by Ogata (1981).We use the following notation :

U[0,1] denotes the uniform distribution on the interval [0, 1],[0, T] is the time interval on which the process is to be simulated,

and we define

IK(t) =Kn=1

n(t) (11)

the sum of the intensities of the first K components of the multivariateprocess. IM(t) =M

n=1 n(t) is thus the total intensity of the multivariate

process and we set I0 = 0. The algorithm is then rewritten as follows.




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Simulation of a multivariate Hawkes process (II)

Algorithm - Initialization

1 Initialization : Set i 1, i1 1, . . . , iM 1 andI

IM(0) =

M

n=i

i0(0).

2 First event : Generate UU[0,1] and set s 1

ln U.

1 If s > T Then go to last step.2 Attribution Test : Generate DU[0,1] and set tn01 s where n0 is

such that In0

1(0)I

< D In0 (0)I

.

3 Set t1 tn01 .




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Simulation of a multivariate Hawkes process (III)

Algorithm - General routine3 General routine : Set in0 in0 + 1 and i i + 1.

1 Update maximum intensity: Set I IM(ti1) +M

n=1

Pj=1

nn0j .

2 New event : Generate UU[0,1] and set s s 1I

ln U.

Ifs > T,

Thengo to the last step.3 Attribution-Rejection test : Generate DU[0,1].

If D IM(s)

I,

Then set tn0in0 s where n0 is such thatIn01(s)

I< D I

n0 (s)

I, and

ti tn0in0 and go through the general routine again,Else update I IM(s) and try a new date at step (b) of the generalroutine.

4 Output: Retrieve the simulated process ({tni }i)n=1,...,M on [0, T].




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Sample paths of a bivariate Hawkes process (I)

We simulate a bivariate Hawkes process with P = 1 and the followingparameters:

10 = 0.1, 111 = 0.2,

111 = 1.0,

121 = 0.1,

121 = 1.0,

20 = 0.5, 211 = 0.5,

211 = 1.0,

221 = 0.1,

221 = 1.0, (12)




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Sample paths of a bivariate Hawkes process (II)

0

1

2

3

4

0 20 40 60 80 100

Time

Intensity maximum I*

Intensity 1

Intensity 2

Events t1

Events t2

Figure: Simulation of a two-dimensional Hawkes process with P = 1 andparameters given in equation (12).




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Sample paths of a bivariate Hawkes process (III)

0

1

2

3

20 22 24 26 28 30 32 34 36 38 40

Time

Intensity maximum I*

Intensity 1

Intensity 2

Events t1

Events t2

Figure: Simulation of a two-dimensional Hawkes process with P = 1 andparameters given in equation (12). (Zoom of the previous figure).


Self- and mutually exciting Hawkes processes Maximum-likelihood estimation


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Table of contents


2 Self- and mutually exciting Hawkes processesDefinition and stationarity conditionSimulation of a multivariate Hawkes process

Maximum-likelihood estimationGoodness of fit







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Computation of the log-likelihood function (I)

The log-likelihood of a multidimensional Hawkes process can be computedas the sum of the likelihood of each coordinate, i.e. is written:

ln L({ti}i=1,...,N) =M

m=1 ln Lm

({ti}), (13)

where each term is defined by:

ln Lm

({ti}) = T

0 (1 m

(s)) ds +T

0 ln m

(s)dNm

(s). (14)




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Computation of the log-likelihood function (II)

In the case of a multidimensional Hawkes process, denoting {ti}i=1,...,Nthe ordered pool of all events {{tmi }m=1,...,M}, this log-likelihood can becomputed as:

ln Lm({ti}) = T m(0, T) (15)

+Ni=1

zmi ln

m0 (ti) +

Mn=1

Pj=1

tnk


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Computation of the log-likelihood function (III)

This can be computed in a recursive way. We observe that:

Rmnj (l) =tnk


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Computation of the log-likelihood function (IV)

The final expression of the log-likelihood may be written:

Log-likelihood of a multivariate Hawkes process

ln Lm({ti}) = T Ni=1

Mn=1

Pj=1

mn

jmnj

1 e

mn

j (Tti)

+

tml

ln

m0 (tml ) +

M

n=1P

j=1mnj R

mnj (l)

, (17)

where Rmnj (l) is defined with equation (16) and Rmnj (0) = 0.




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Properties of the maximum-likelihood estimator

Ogata (1978) shows that for a stationary one-dimensional Hawkes process

with constant 0 and P = 1, the maximum-likelihood estimatorT = (0, 1, 1)) is

consistent, i.e. converges in probability to the true values = (0, 1, 1)) as T :

> 0, limTP[|T | > ] = 0. (18)asymptotically normal, i.e.

TT N(0, I

1()) (19)

where I1() =

E

1

i

j

i,j

.

asymptotically efficient, i.e. asymptotically reaches the lower boundof the variance.




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Numerical estimation of a simulated process

T 10 111

111

121

121

20

211

211

221

221

100 0.614 0.510 0.369 1.482 1.710 0.518 0.337 0.600 1.605 2.595(0.372) (0.268) (0.269) (1.216) (3.172) (0.272) (0.206) (0.365) (2.051) (6.586)

500 0.516 0.505 0.268 1.043 0.865 0.518 0.264 0.507 0.814 1.048(0.112) (0.085) (0.080) (0.214) (0.479) (0.120) (0.080) (0.084) (0.278) (0.221)

1000 0.507 0.492 0.254 1.018 0.761 0.513 0.255 0.488 0.794 1.003(0.079) (0.054) (0.052) (0.122) (0.203) (0.092) (0.052) (0.061) (0.387) (0.152)

0.500 0.500 0.250 1.000 0.750 0.500 0.250 0.500 0.750 1.000

Table: Maximum likelihood estimation of a two-dimensional Hawkes process onsimulated data. Each estimation is the average result computed on 100 samplesof length [0, T]. Standard deviations are given in parentheses.


Self- and mutually exciting Hawkes processes Goodness of fit


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Table of contents










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Time change theorem and goodness of fit

Time change theorem

Let (Nt) be a one-dimensional point process on R+ such that0 (s)ds = . Let t be the stopping time defined by

t

0

(s)ds = . (20)

Then the process N() = N(t) is an homogeneous Poisson process withconstant intensity = 1.

(Multivariate componentwise) Corollary

The durations mi mi1 = m(tmi1, tmi ) =tmitmi1

m(s)ds are exponentially

distributed with parameter 1. See e.g. (Bowsher 2007).




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Testing the simulated data (I)

The integrated intensity of the m-th coordinate of a multidimensional

Hawkes process between two consecutive events tm

i1 and tm

i of type m iscomputed as:

m(tmi1, tmi ) =

tmitmi1

m(s)ds

=

tmi

tmi1

m0 (s)ds +

tmi

tmi1

Mn=1

Pj=1

tnk


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Testing the simulated data (II)

m(tmi1, tmi ) =

tmi

tmi1

m0 (s)ds

+M

n=1

P

j=1

tnk


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Testing the simulated data (III)

We observe that

Amnj (i 1) =

tnk


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Testing the simulated data (IV)

The integrated density can thus be written i N:

m(tmi1, tmi ) =

tmi

tmi1

m0 (s)ds +M

n=1

P

j=1

mnj

mn

j1

emnj (t

mi t

mi1)

Amnj (i 1) +

tmi1t

nk


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Testing the simulated data (V)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

Emp

iricalquantiles

Theoretical quantiles

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

Emp

iricalquantiles

Theoretical quantiles

Figure: Quantile plots for one sample of simulated data of a two-dimensionalHawkes process with P = 1 and parameters given in equation (12). (Left) m = 0.(Right) m = 1.


A simple model for buy and sell intensities

T bl f t t


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Table of contents









A d l f b d s ll i t siti s (I)


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A model for buy and sell intensities (I)

Hewlett (2006) proposes to model the clustered arrivals of buy and selltrades using Hawkes processes. Using the exponent B for buy variablesand S for sell variables, the model is written :

B(t) = B0 +

t

0BBe

BB(tu)dNBu +

t

0BSe

BS(tu)dNSu ,(25)

S(t) = S0 + t

0SBe

SB(tu)dNBu + t

0SSe

SS(tu)dNSu . (26)



A model for buy and sell intensities (II)


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A model for buy and sell intensities (II)

Hewlett (2006) imposes some symmetry constraints, stating that mutualexcitation and self-excitation should be the same for both processes, whichis written :

B0 =

S0 = 0 (27)

SB = BS = cross (28)

SB = BS = cross (29)

SS = BB = self (30)

SS = BB = self (31)



Goodness of fit


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Goodness of fit

Hewlett (2006) fits this model on two-month data of EUR/PLNtransactions (no dates given): the Hawkes model is a much better fit ofthe empirical data than the Poisson model.

Figure: Quantile plots of integrated intensities for the Hawkes model (left) and aPoisson model (right) on EUR/PLN buy and sell data. Reproduced from(Hewlett 2006).



Numerical results


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Numerical results

The numerical values obtained are :

0 = 0.0033, cross = 0, self = 0.0169, self = 0.0286. (32)

In other words,the occurrence of a buy (resp. sell) order has an exciting effect on thestream of buy (resp. sell) orders, with a typical half-life of ln 2

self 24

seconds;

the zero value of cross tends to indicate that there is no influence ofbuy orders on sell orders, and conversely.



Test on our own data (I)


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Test on our own data (I)

We perform the fit of a bivariate Hawkes model on buy/sell market orderson the following data : BNPP.PA, Feb. 1st 2010 to Feb. 23rd, 2010 (14trading days), 10am-12am without symmetry constraints. Numericalresults are :

B0 = 0.080, BB = 3.230, BB = 13.304, BS = 0.276, BS = 6.193B0 = 0.086,

SB = 0.515, SB = 13.451, SS = 3.789, SS = 14.151

Confirmation of the very limited cross-excitation effect.Change of magnitude of parameters : difference in precision of data(second, millisecond)



Test on our own data (II)


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Test on our own data (II)

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7

Day 0Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day 8Day 9

Day 10Day 11Day 12

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7

Day 0Day 1Day 2Day 3Day 4Day 5Day 6Day 7Day 8Day 9

Day 10Day 11Day 12

Figure: Quantile plots of integrated intensities for a bivariate Hawkes model onbuy/sell market orders fitted on 13 trading days of the stock BNPP.PA (fromFeb. 1st 2010 to Feb. 22nd, 2010), 10am-12am each day.


Some statistical findings about the order flows

Table of contents


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Table of contents









Classifying orders according to their aggressiveness


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Classifying orders according to their aggressiveness

Large (2007) models streams of orders by Hawkes processes, extending themodel by Hewlett (2006) using a much finer description of orders.Following classical typologies used in microstructure, events occurring inan order book are classified in ten categories :

Type Description Aggressiveness

1 Market order that moves the ask Yes

2 Market order that moves the bid Yes3 Limit order that moves the ask Yes4 Limit order that moves the bid Yes

5 Market order that doesnt move the ask No6 Market order that doesnt move the bid No7 Limit order that doesnt move the ask No8 Limit order that doesnt move the bid No9 Cancellation at ask No

10 Cancellation at bid No



A 10-variate Hawkes model for aggressive orders


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A 10 variate Hawkes model for aggressive orders

Events of type 1 to 4 are Hawkes processes whose intensities depend onthe 10 different sorts of events, i.e. can be written for m = 1, . . . , 4:

m(t) = 0(t) +10n=1

t0

mnemn(tu)dNnu. (33)



Hawkes parameters for aggressive limit orders


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Hawkes parameters for aggressive limit orders

Figure: Representation of the influences on aggressive limit orders measured bythe fitting of a Hawkes model on the Barclays order book on January 2002. mn

are in abscissas, mn are in ordinates, the size of the discs are proportional to thenumber of observed events. Reproduced from (Large 2007).



Hawkes parameters for aggressive market orders


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p gg

Figure: Representation of the influences on aggressive market orders measured bythe fitting of a Hawkes model on the Barclays order book on January 2002. mn

are in abscissas, mn are in ordinates, the size of the discs are proportional to thenumber of observed events. Reproduced from (Large 2007).



Aggressive market orders on 2010 CAC40 data


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gg

0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00

0

.0

0.5

1.0

1.5

2.0

2.5

Hawkes Parameters For Aggressive Market Orders

Halflives

Intens

ityincrease

AMAM

AMAL

AMPM

AMPL

AMAM

AMAL

AMPM

AMPL

AMAMAMAL

AMPM

AMPL

AMAM

AMAL AMPM

AMPL

AMAM

AMAL

AMPM

AMPL

AMAM

AMAL

AMPMAMPL

AMAM

AMAL

AMPM

AMPL

AMAM

AMAL

AMPM

AMPL

AMAM

AMAL

AMAMPL

AMAM

AMAL

AMPM

AMPL

AMAM

AMPM

AMPL

AMAM

AMAL

AMPMAMPL

BNPP.PAUBIP.PA

LAGA.PAALSO.PAAXAF.PABOUY.PACARR.PAPEUP.PAAIRP.PAMICP.PASASY.PASGEF.PA

Figure: Hawkes parameters for aggressive market orders for various CAC40stocks. These values are computed using median-MLE estimation on 14 days oftrading (Feb.1st-Feb.23rd 2010), 10am-12pm.



Aggressive limit orders on 2010 CAC40 data


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gg

0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00

0

1

2

3

4

5

Hawkes Parameters For Aggressive Limit Orders

Halflives

Intens

ityincrease

ALAL

ALPM

ALPL

ALAM

ALAL

ALPM

ALPL

ALAM

ALAL

ALPM

ALPL

ALAL

ALPM

ALPL

ALAM

ALAL

ALPM

ALPL

ALAM

ALAL

ALPM

ALPL

ALAM

ALAL

ALPM

ALPL

ALAM

ALAL

ALPM

ALPL

ALAM

ALAL

ALPM

ALPL

ALAM

ALAL

ALPM

ALPL ALAL

ALPM

ALPL

ALAM

ALAL

ALPM

ALPL

BNPP.PAUBIP.PA


Figure: Hawkes parameters for aggressive limit orders for various CAC40 stocks.These values are computed using median-MLE estimation on 14 days of trading(Feb.1st-Feb.23rd 2010), 10am-12pm.



Passive market orders on 2010 CAC40 data


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0.02 0.05 0.10 0.20 0.50 1.00

0

1

2

3

4

5

Hawkes Parameters For Passive Market Orders

Halflives

Intensityincrease

PMAM

PMAL

PMPM

PMPL

PMAM

PMAL

PMPM

PMPL

PMAM

PMAL

PMPM

PMPL

PMAM

PMAL

PMPM

PMPL

PMAM

PMPM

PMPL

PMAM

PMAL

PMPM

PMPL

PMAM

PMAL

PMPM

PMPL

PMAM

PMALPMPL

PMAM

PMAL

PMPM

PMPL

PMAM

PMAL

PMPM

PMPL

PMAM

PMPM

PMPL

PMAM

PMAL

PMPM

PMPL

BNPP.PAUBIP.PA


Figure: Hawkes parameters for passive market orders for various CAC40 stocks.These values are computed using median-MLE estimation on 14 days of trading(Feb.1st-Feb.23rd 2010), 10am-12pm.



Empirical conclusions


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Previous figures can be used to draw some conclusions on the way order

book events influence each other. The main findings are the followings:

passive limit orders can be seen as background noise ;

aggressive LO are firstly influenced by aggressive MO (resiliency) ;

aggressive LO are secondly influenced by passive MO ;

aggressive LO are thirdly influenced by aggressive LO ;

aggressive MO are firstly influenced by passive MO ;

aggressive MO are secondly influenced by aggressive MO ;

aggressive MO are thirdly influenced by aggressive LO (rush toliquidity) ;

passive MO are influenced by passive and aggressive market orders,but not by LO.


An order book model with Hawkes processes

Table of contents


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The basic Zero-Intelligence Poisson model (HP)


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Liquidity provider

1 arrival of new limit orders: homogeneous Poisson process NL(L)2 arrival of cancellation of orders: homogeneous Poisson process

NC(C)

3 new limit orders placement: Students distribution with parameters

(P1 , m

P1 , s

P1 ) around the same side best quote

4 volume of new limit orders: exponential distribution E(1/mV1 );5 in case of a cancellation, orders are deleted with probability

Noise trader (liquidity taker)1 arrival of market orders: homogeneous Poisson process NM()

2 volume of market orders: exponential distribution with meanE(1/mV2 ).



Need for physical time in order book models


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0

0.05

0.1

0.15

0.2

0.25

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22

Event time spreadCalendar time

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 inevent time and in physical time.



Measures of inter arrival times


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(red) Inter arrival times of the counting process of all orders (limitorders and market orders mixed), i.e. the time step between any orderbook event (other than cancellation)

(green) Interval time between a market order and immediatelyfollowing limit order



Empirical evidence of market making


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0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5

AllOrders

MarketNextLimit

10-4

10-3

10-2

10-1

100

0.01 0.1 1 10

Figure: Empirical density function of the distribution of the time steps between

two consecutive orders (any type, market or limit) and empirical density functionof the distribution of the time steps between a market order and the immediatelyfollowing limit order. X-axis is scaled in seconds. In insets, same data using asemi-log scale. Studied assets : BNPP.PA (left). Reproduced from (MuniToke 2011).



Adding dependance between order flows (I)


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Liquidity provider

1 arrival of new limit orders: Hawkes process NL(L)2 arrival of cancellation of orders: homogeneous Poisson process

NC(C)

3 new limit orders placement: Students distribution with parameters

(P1 , m

P1 , s

P1 ) around the same side best quote

4 volume of new limit orders: exponential distribution E(1/mV1 );5 in case of a cancellation, orders are deleted with probability

Noise trader (liquidity taker)1 arrival of market orders: Hawkes process NM()

2 volume of market orders: exponential distribution with meanE(1/mV2 ).



Adding dependence between order flows (II)


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Hawkes processes NL and NM

(t) = 0 +

t0

MMeMM(ts)dNMs

L(t) = L0 +t

0LMeLM(ts)dNMs +

t0

LLeLL(ts)dNLs

(34)

MM and LL effect for clustering of orders

LM effect as observed on data

no ML effect



Impact on arrival times (I)


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0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2

Empirical BNPPHomogeneous Poisson

Hawkes MM

10-4

10-3

10-2

10-1

100

2 4 6 8 10 12 14 16 18 20

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2


Hawkes LL

10-5

10-4

10-3

10-2

10-1

100

2 4 6 8 10 12 14 16 18 20

Figure: Empirical density function of the distribution of the interarrival times ofmarket orders (left) and limit orders (right) for three simulations, namely HP,MM, LL, compared to empirical measures. In inset, same data using a semi-logscale. Reproduced from (Muni Toke 2011).



Impact on arrival times (II)


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2


Hawkes MMHawkes MM LL

Hawkes MM LL LM

10-4

10-3

10-2

10-1

100

1 2 3 4 5 6 7 8 9 10

Figure: Empirical density function of the distribution of the interval timesbetween a market order and the following limit order for three simulations, namelyHP, MM+LL, MM+LL+LM, compared to empirical measures. In inset, samedata using a semi-log scale. Reproduced from (Muni Toke 2011).



Impact on the bid-ask spread (I)


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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.In inset, same data using a semi-log scale. X-axis is scaled in euro (1 tick is 0.01euro). Reproduced from (Muni Toke 2011).



Impact on the bid-ask spread (II)


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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 bid-ask spread for threesimulations, namely HP, MM+LL, MM+LL+LM. In inset, same data using asemi-log scale. X-axis is scaled in euro (1 tick is 0.01 euro). Reproduced from(Muni Toke 2011).


Conclusion

Conclusion


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Self- and mutually exciting processes (epidemic,earthquakes,. . . finance)

Exponential kernel allows easy manipulation (simulation, estimation)

Quite good fit on tested data (buy/sell, market/limit)See (Bowsher 2007) for a generalized econometric framework

Lots of possible models/strategies to be imagined

See (Bacry, Delattre, Hoffmann & Muzy 2011) for a microstructure

toy model addressing stylized facts


References

Bacry, E., Delattre, S., Hoffmann, M. & Muzy, J. F. (2011), Modelingmicrostructure noise with mutually exciting point processes,

Xi 1101 3422 1


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arXiv:1101.3422v1 .Bowsher, C. G. (2007), Modelling security market events in continuous

time: Intensity based, multivariate point process models, Journal of Econometrics 141(2), 876912.

Bremaud, P. (1996), Stability of nonlinear Hawkes processes, The Annalsof Probability 24(3), 15631588.

Engle, R. F. (2000), The econometrics of Ultra-High-Frequency data,

Econometrica 68(1), 122.Engle, R. F. & Russell, J. R. (1997), Forecasting the frequency of changes

in quoted foreign exchange prices with the autoregressive conditionalduration model, Journal of Empirical Finance 4(2-3), 187212.

Hewlett, P. (2006), Clustering of order arrivals, price impact and tradepath optimisation, in Workshop on Financial Modeling with Jumpprocesses, Ecole Polytechnique.

Large, J. (2007), Measuring the resiliency of an electronic limit orderbook, Journal of Financial Markets 10(1), 125.


References

Lewis, P. A. W. & Shedler, G. S. (1979), Simulation of nonhomogeneouspoisson processes by thinning, Naval Research Logistics Quarterly26(3) 403 413


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26(3), 403413.Muni Toke, I. (2011), Market making in an order book model and its

impact on the bid-ask spread, in F. Abergel, B. Chakrabarti,A. Chakraborti & M. Mitra, eds, Econophysics of Order-DrivenMarkets, New Economic Windows, Springer-Verlag Milan, pp. 4964.

Ogata, Y. (1978), The asymptotic behaviour of maximum likelihoodestimators for stationary point processes, Annals of the Institute of

Statistical Mathematics30(1), 243261.Ogata, Y. (1981), On Lewis simulation method for point processes,

IEEE Transactions on Information Theory 27(1), 2331.


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