High-frequency data and limit order books
—
Lecture 4 - March 18th, 2022
Ioane Muni Toke
MICS Laboratory and Chair of Quantitative Finance
CentraleSupélec – Université Paris-Saclay, France
CentraleSupelec 3A Ingénieur Mathématiques et Data Science
CentraleSupelec MS Mathématiques Appliquées à la Finance
IP Paris M2 Statistics, Finance and Actuarial Science
Ioane Muni Toke (CentraleSupélec)
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Table of contents
One dimensional Hawkes processes
Multidimensional Hawkes processes
Hawkes processes as branching processes
Lab 4
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One dimensional Hawkes processes
Table of contents
One dimensional Hawkes processes
Multidimensional Hawkes processes
Hawkes processes as branching processes
Lab 4
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One dimensional Hawkes processes
One-dimensional Hawkes processes
Definition (Linear self-exciting process)
A counting process N is called a (linear, one-dimensional) Hawkes process if it is a counting
process with (stochastic) intensity:
Z t
X
λt = λ0 (t) +
ν(t − s)dNs = λ0 (t) +
ν(t − ti ),
0
0<ti <t
where λ0 : R 7→ R+ and ν : R+ 7→ R+ are deterministic functions.
I ν expresses the positive influence of the past events ti on the current value of the
intensity process.
I Hawkes processes are self-exciting processes.
I λ0 is the baseline intensity (λ ≥ λ0 ). From now on, λ0 > 0 constant.
I Original paper by [1]. Used in geology (earthquakes replicas), finance (see later in the
course), biology (neuron excitation, multidimensional case, see below).
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One dimensional Hawkes processes
Sample path, kernels
5
Intensity maximum λ*
Intensity λ
Events
I Exponential kernel commonly used:
4
ν(t) =
3
P
X
αj e −βj t 1R+ (t).
j=1
2
1
0
0
2
4
6
8
10
Time
Sample path of a 1D-Hawkes process with a single
exponential kernel (P = 1) and parameters
λ0 = 1.2, α1 = 0.6, β1 = 0.8.
Ioane Muni Toke (CentraleSupélec)
I Exponential kernels allow for efficient
computations, especially for
maximum-likelihood estimation (see
later), but do not exhibit long-memory
properties.
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One dimensional Hawkes processes
Stability of a Hawkes process
If the process has reached a stationary regime, the average intensity value is:
1−
λ
R ∞0
.
0 ν(x)dx
I The average number of points of a Hawkes process on a time interval of length T is thus
λ0 T
.
in a stationary regime
1 − kνk1
I If kνk1 < 1, the Hawkes process is sub-critical / stationary. If kνk1 = 1 (resp. > 1), the
process is called critical (resp. supercritical) (See discussion in finance later in the course).
I In the case of exponential kernels, the stability condition is written
P
X
αj
j=1
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βj
< 1.
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One dimensional Hawkes processes
Thinning for stochastic intensities
I [4] extends the thinning method for non-homogeneous Poisson processes introduced by
[2] to point processes with stochastic intensities.
Theorem (Thinning)
Consider point process N with intensity λt on an interval (0, T ]. Suppose we can find a
process λ∗t such that λt ≤ λ∗t (P-a.s.). Let t1∗ , t2∗ , . . . tn∗ ∈ (0, T ] be the points of the process
N ∗ with intensity λ∗ . For each of the points, attach a mark p = 1 with probability
λ(tj∗ )
λ∗ (tj∗ ) ,
else
p = 0. Then the points with marks p = 1 form a point process which is the same as N.
I If λ∗t can be constructed pathwise as a piecewise constant process, we can thus simulate
the point process N with simple exponential random variables.
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One dimensional Hawkes processes
Simulation algorithm for a one-dimensional Hawkes process
1. Initialization : Set t ← 0, λ∗ ← λ0 (0) and an empty list of events.
2. Main loop : While t < T :
1
(a) New event candidate : Generate U
U[0,1] and set t ← t − ∗ log U.
λ
If t ≥ T , Then return the list of events.
(b) Thinning : Generate D
U[0,1] .
λ(t)
If D ≤ ∗ ,
λ
Then New event:
PP add t to the list of events and update the maximum intensity
λ∗ ← λ(t) + j=1 αP ;
Else No new event: update the maximum intensity λ∗ ← λ(t).
3. Output: return the list of events.
It is crucial to observe that exponential kernels lead to easy update of the jump part of the
intensity, without having to parse all previous events.
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One dimensional Hawkes processes
Path of a Hawkes process and its intensity
I Simulation of a one-dimensional Hawkes process with parameters
P = 1, λ0 = 1.2, α1 = 0.6, β1 = 0.8.
12
5
*
*
Intensity maximum λ
Intensity λ
Events
Intensity maximum λ
Intensity λ
Events
10
4
8
3
6
2
4
1
2
0
0
0
10
20
30
40
50
Time
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60
70
80
90
100
0
2
4
6
8
10
Time
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One dimensional Hawkes processes
Log-likelihood of a point process I
Proposition (Log-likelihood of a point process)
The log-likelihood of a counting process (Nt )t≥0 with conditional intensity (λt )t≥0 observed
on [0, T ] is
Z T
Z T
log LT =
log λs dNs −
λs ds.
0
0
I Example: In the case of a time-homogeneous Poisson process, we have for an observed
sample 0 < t1 < . . . < tn < T :
log LT = n log λ − λT .
The maximum-likelihood estimator of the intensity is thus
n
λ̂MLE = ,
T
as expected.
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One dimensional Hawkes processes
Log-likelihood of a Hawkes process I
I In the case of a Hawkes process with P exponential kernels, the log-likelihood of an
observation 0 < t1 < . . . < tn < T is:


n
P X
i−1
X
X
log LT =
log λ0 (ti ) +
αj e −βj (ti −tk )  − Λ(0, T ).
i=1
j=1 k=1
I The double sum can be bypassed by a recursive computation ([4])


n
P
X
X
log LT =
ln λ0 (ti ) +
αj Rj (i) − Λ(0, T ),
i=1
j=1
where Rj (1) = 0 and Rj (i) = e −βj (ti −ti−1 ) (1 + Rj (i − 1)), for all j = 1, . . . , P.
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One dimensional Hawkes processes
Log-likelihood of a Hawkes process II
Proposition (Log-likelihood of a Hawkes process, exponential kernels)
The log-likelihood of a Hawkes process (Nt )t≥0 with P exponential kernels observed on [0, T ]
is


Z T
n X
P
n
P
X
X
X
αj log LT = −
λ0 (s)ds −
1 − e −βj (T −ti ) +
log λ0 (ti ) +
αj Rj (i) .
β
j
0
i=1 j=1
i=1
j=1
I Maximum-likelihood estimator numerically available.
I Analytical gradients available for better optimization.
I In the case of non-exponential kernel, the absence of recursive evaluations may lead to
prohibitive computational times.
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One dimensional Hawkes processes
Maximum-likelihood estimation
I [3]: MLE estimator θ̂T = (λ̂0 , α̂j , β̂j )) is
I consistent, i.e. converges in probability to the true values θ∗ = (λ0 , αj , βj )) as T → ∞ :
∀ > 0,
lim P[|θ̂T − θ∗ | > ] = 0.
T →∞
I asymptotically normal, i.e.
√ T θ̂T − θ∗ → N (0, I −1 (θ∗ ))
h
∂λ
where I (θ) = E λ1 ∂θ
i
∂λ
∂θj
i
.
i,j
I asymptotically efficient, i.e. asymptotically reaches the lower bound of the variance of an
estimate.
Ioane Muni Toke (CentraleSupélec)
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One dimensional Hawkes processes
Change of time
I The “change of time” theorem proved for the non-homogeneous Poisson process can be
extended in the general case.
Theorem (Change of time of a point process)
Let (Ñt )t≥0 be a point process with intensity (λt )t≥0 . Define for each t the stopping time
τ (t) such that
Z τ (t)
λs ds = t
0
Then Nt = Ñτ (t) is a time-homogeneous Poisson process with intensity 1.
I Consequence: If the t̃i ’s are the points of Ñ, then the quantities
exponentially distributed with parameter 1.
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R t̃i
t̃i−1
λs ds are
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One dimensional Hawkes processes
Goodness-of-fit
I For a one-dimensional Hawkes process with
exponential kernels:
Z ti
Λ(ti−1 , ti ) =
λ0 (s)ds
8
7
ti−1
Empirical quantiles
6
5
+
4
P
X
αj j=1
3
2
βj
1 − e −βj (ti −ti−1 ) Aj (i − 1),
where Aj (1) = 1 and
Aj (i − 1) = 1 + e −βj (ti−1 −ti−2 ) Aj (i − 2).
1
0
0
1
2
3
4
5
Theoretical quantiles
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7
8
I Quantile plots available as a visual indication of
goodness-of-fit
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Multidimensional Hawkes processes
Table of contents
One dimensional Hawkes processes
Multidimensional Hawkes processes
Hawkes processes as branching processes
Lab 4
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Multidimensional Hawkes processes
Multidimensional Hawkes processes
Definition (Multidimensional Hawkes processes)
A M-dimensional Hawkes process is a point process Nt = (Nt1 , . . . , NtM ) with intensities
(λm
t )t≥0 , m = 1, . . . , M such that:
m
λ (t) =
λm
0 (t)
+
M Z
X
n=1
t
gmn (t − s)dNsn ,
0
where λ0 : [0, ∞) → R+ and gmn : [0, ∞) → R+ , m, n = 1, . . . , M are deterministic functions.
R
I We may write using a vector notation: λ(t) = λ0 + 0t G(t − s) · dNs .
mn
I In the case of exponential kernels, G(t) = αmn e −β t
.
m,n=1,...,M
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Multidimensional Hawkes processes
Stability of multidimensional Hawkes processes
I If the process is in a stationary state, the average intensities µt ∈ RM should satisfy:
Z
µ= I−
−1
∞
G(u)du
λ0
0
R
I If the spectral radius of the matrix Γ = 0∞ G(u)du is strictly smaller than 1, then the
M-dimensional Hawkes process is stable.
mn I In the exponential kernel case, Γ = βαmn .
m,n=1,...,M
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Multidimensional Hawkes processes
Simulation of multidimensional Hawkes processes
I The algorithm is formally identical to one in one dimension.
P
I Let Ik (t) = kn=1 λn (t), k = 1, . . . , M.
∗ , and the next candidate event
I The total intensity is now IM (t), bounded above by IM
∗
must not be generated with IM .
I The thinning step is now:
Generate D
U[0,1] .
If D ≤
IM (s)
∗ ,
IM
Then set s as a point of the process n0 , where n0 is such that
go through the main loop again ;
∗ ← I (s) and generate a new candidate point.
Else update IM
M
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In0 −1 (s)
∗
IM
<D≤
In0 (s)
∗ ,
IM
and
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Multidimensional Hawkes processes
Log-likelihood of multidimensional Hawkes processes
I Log-likelihood of a multidimensional Hawkes process:
log LT =
M
X
log Lm
T,
m=1
with: log Lm
T =
RT
0
m
log λm
s dNs −
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RT
0
λm
s ds..
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Hawkes processes as branching processes
Table of contents
One dimensional Hawkes processes
Multidimensional Hawkes processes
Hawkes processes as branching processes
Lab 4
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Hawkes processes as branching processes
Branching representation of a Hawkes process I
Proposition (Branching representation of a Hawkes process)
Let T > 0. Let λ0 > 0 and ν : [0, ∞) → R+ a deterministic function. Consider the following
procedure:
(0)
1. Generate a time-homogeneous Poisson process {ti } with intensity λ0 on [0, T ].
(0)
(1)
2. For each ti , generate a non-homogeneous Poisson process {ti } with deterministic
(0)
(0)
intensity t 7→ ν(t − ti ) on [ti , T ].
(1)
3. Repeat operation 2. for each point of all the sets {ti }, then for each point of all the
(2)
sets {ti }, etc. until there is no new point generated in [0, T ].
Then the ordered setR of all the points generated in [0, T ] is a Hawkes process N on [0, T ] with
t
intensity λt = λ0 + 0 ν(t − s) dNs .
I Link with Galton-Watson processes
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Hawkes processes as branching processes
Branching representation of a Hawkes process II
I The average number of points (offspring) generated by one point (parent) is kνk1 . We
retrieve the three regimes (sub-critical, critical, supercritical) discussed above depending
on the value of kνk1 compared to 1..
I Provides a new algorithm for the simulation of Hawkes processes
I Immediate extension to the case of time-dependent baseline intensity and/or the case of a
multidimensional Hawkes process.
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Lab 4
Table of contents
One dimensional Hawkes processes
Multidimensional Hawkes processes
Hawkes processes as branching processes
Lab 4
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Lab 4
Lab 4 - Questions I
1. Implementation of fundamental routines. It is sufficient to write the following routines
for a single exponential kernel (P = 1).
1.1 Write a function that simulates one path of a Hawkes process.
Input: process parameters and horizon.
Output: time series of events.
1.2 Write a function that computes the log-likelihood of a Hawkes process.
Input: process parameters, horizon and time series of events.
Output: value of the log-likelihood.
1.3 Write a function that computes the MLE estimates of the parameters of a Hawkes process.
Input: optimization starting point, horizon and time series of events.
Output: values of estimated parameters.
1.4 Write a function that computes the intensity integrated between consecutive events for a
Hawkes process.
Input: process parameters, horizon and time series of events.
Output: series of integrated intensities (transformed durations).
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Lab 4
Lab 4 - Questions II
2. Testing the implementation.
2.1 Choose some values to define a stationary Hawkes process with one exponential kernel.
Simulate a realization of this Hawkes process with horizon T and compute the MLE
estimates of the parameters.
2.2 Compute the QQ-plot of the integrated intensities of the fitted process versus the theoretical
exponential distribution. Check visually the goodness-of-fit.
3. Convergence and performance.
3.1 For a given T , simulate multiple realizations of a Hawkes process (e.g., 100). Show that the
MLE estimates converge towards the true values when T increases, and that the order of
convergence in T is the one expected.
3.2 Plot the computational cost of your simulation and estimation in T is the one expected.
3.3 Add to your plots answering the two questions above the results obtained with an external
library such as https://pypi.org/project/hawkes/.
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Lab 4
References I
[1]
Alan G. Hawkes. “Spectra of some self-exciting and mutually exciting point processes”.
In: Biometrika 58.1 (Apr. 1971), pp. 83 –90.
[2]
P. A. W Lewis and G. S. Shedler. “Simulation of nonhomogeneous poisson processes by
thinning”. In: Naval Research Logistics Quarterly 26.3 (1979), pp. 403–413.
[3]
Yoshiko Ogata. “The asymptotic behaviour of maximum likelihood estimators for
stationary point processes”. In: Annals of the Institute of Statistical Mathematics 30.1
(1978), pp. 243–261.
[4]
Yosihiko Ogata. “On Lewis’ simulation method for point processes”. In: IEEE
Transactions on Information Theory 27.1 (Jan. 1981), pp. 23–31.
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