S iščuk, Ana olij; Hu man, Aiden
A icle
Gene al compound hawkes p ocesses in limi o de books
Risks
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MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: S iščuk, Ana olij; Hu man, Aiden (2020) : Gene al compound hawkes p ocesses
in limi o de books, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 8, Iss. 1, pp. 1-25,
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A icle
Gene al Compound Hawkes P ocesses in Limi
O de Books
Ana oliy Swishchuk *,† and Aiden Hu man †
Depa men o Ma hema ics and S a is ics, Facul y o Science, Calga y, AL T2N1N4, Canada;
aiden.hu man@ucalga y.ca
*Co espondence: aswish@ucalga y.ca; Tel.: +1-(403)-220-3274
† These au ho s con ibu ed equally o his wo k.
Recei ed: 29 Janua y 2020; Accep ed: 11 Ma ch 2020; Published: 14 Ma ch 2020


Abs ac :
In his pape , we s udy a ious new Hawkes p ocesses. Speci ically, we cons uc gene al
compound Hawkes p ocesses and in es iga e hei p ope ies in limi o de books. Wi h ega d
o hese gene al compound Hawkes p ocesses, we p o e a Law o La ge Numbe s (LLN) and a
Func ional Cen al Limi Theo ems (FCLT) o se e al speci ic a ia ions. We apply se e al o hese
FCLTs o limi o de books o s udy he link be ween p ice ola ili y and o de low, whe e he
ola ili y in mid-p ice changes is exp essed in e ms o pa ame e s desc ibing he a i al a es and
mid-p ice p ocess.
Keywo ds:
Hawkes p ocesses; gene al compound Hawkes p ocesses; limi o de books; unc ional
cen al limi heo ems; LOBSTER da a
1. In oduc ion
The Hawkes p ocess (HP) is named a e i s c ea o , Hawkes (1971); Hawkes and Oakes (1974).
The HP is a simple poin p ocess equipped wi h a sel -exci ing p ope y, clus e ing e ec and long
un memo y. Th ough i s dependence on he his o y o he p ocess, he HP cap u es he empo al and
c oss sec ional dependence o he e en a i al p ocess as well as he ’sel -exci ing’ p ope y obse ed
in ou empi ical da a on limi o de books. Sel -exci ing poin p ocesses ha e ecen ly been applied o
high equency da a o p ice changes Bac y e al. (2011) o o de a i al imes Emb ech s e al. (2011).
HPs ha e seen hei applica ion in many a eas, like gene ics Ca s ensen (2010), occu ence o c ime
Mohle e al. (2011), bank de aul s Azizpou e al. (2018) and ea hquakes Oga a (1988).
Poin p ocesses gained a signi ican amoun o a en ion in s a is ics du ing he 1950s and
1960s. Cox (1955) in oduced he no ion o a doubly s ochas ic Poisson p ocess (called he Cox
p ocess now) and Ba le (1963) in es iga ed s a is ical me hods o poin p ocesses based on hei
powe spec al densi ies. Lewis (1964) o mula ed a poin p ocess model ( o compu e powe ailu e
pa e ns) which was a s ep in he di ec ion o he HP. A nice in oduc ion o he heo y o poin
p ocesses can be ound in Daley and Ve e-Jones (2003). The i s ype o poin p ocess in he
con ex o ma ke mic os uc u e is he au o eg essi e condi ional du a ion (ACD) model in oduced
by Engle and Russell (1998).
A ecen applica ion o HP is in inancial analysis, in pa icula limi o de books. In his pape ,
we s udy a ious new Hawkes p ocesses, namely gene al compound Hawkes p ocesses o model
he p ice p ocess in limi o de books. We p o e a Law o La ges Numbe s (LLN) and a Func ional
Cen al Limi Theo em (FCLT) o speci ic cases o hese p ocesses. Se e al o hese FCLTs a e
applied o limi o de books whe e we use asymp o ic me hods o s udy he link be ween p ice
ola ili y and o de low in ou models. The ola ili y o he p ice changes is exp essed in e ms
o pa ame e s desc ibing he a i al a es and p ice changes. We also p esen some nume ical
Risks 2020,8, 28; doi:10.3390/ isks8010028 www.mdpi.com/jou nal/ isks
Risks 2020,8, 28 2 o 25
examples. The gene al compound Hawkes p ocess was i s in oduced in Swishchuk (2017) o
model he isk p ocess in insu ance and s udied in de ail he e. In he pape Swishchuk e al. (2019),
we ob ained unc ional CLTs and LLNs o gene al compound Hawkes p ocesses wi h dependen
o de s and egime-swi ching compound Hawkes p ocesses. Bowshe (2007) was he i s one who
applied he HP o inancial da a modelling.
Ca ea e al. (2014)
applied HP o model ma ke o de
a i als.
Filimono e al. (2014)
and
Filimono and So ne e (2012)
applied he HPs o es ima e he
pe cen age o p ice changes caused by endogenous sel -gene a ed ac i i y a he han by he exogenous
impac o news o no el in o ma ion. Bauwens and Hau sch (2009) used a i e-dimensional HP o
es ima e mul i a ia e ola ili y be ween i e s ocks, based on p ice in ensi ies. Hewle (2006) used
he ins an aneous jump in he in ensi y caused by he occu ence o an e en o quali y he ma ke
impac o ha e en , aking in o accoun he cascading e ec o seconda y e en s causing u he
e en s. Hewle (2006) also used he Hawkes model o de i e op imal p icing s a egies o ma ke
make s and op imal ading s a egies o in es o s gi en ha he a ional ma ke make s ha e he
his o ic ading da a. La ge (2007) applied a Hawkes model o he pu pose o in es iga ing ma ke
impac , wi h a speci ic in e es in o de book esiliency. Speci ically, he conside ed limi o de s, ma ke
o de s and cancella ions on bo h he buy and sell side, and u he ca ego izes hese e en s based on
hei le el o agg ession, esul ing in a en-dimensional Hawkes p ocess. O he econome ic models
based on ma ked poin p ocesses wi h s ochas ic in ensi y include au o eg essi e condi ional in ensi y
(ACI) models wi h he in ensi y depending on i s his o y. Hasb ouck (1999) in oduced a mul i a ia e
poin p ocess o model he di e en e en s o an o de book bu did no pa ame ize he in ensi y.
We no e ha B émaud and Massoulié (1996) gene alized he HP o i s nonlinea o m. In addi ion,
a unc ional cen al limi heo em o nonlinea Hawkes p ocesses was ob ained in Zheng e al. (2013).
The ’Hawkes di usion model’ in oduced in Aï -Sahalia e al. (2015) a emp ed o ex end p e ious
models o s ock p ices o include inancial con agion. Cha ez-Demoulin and McGill (2012) used
Hawkes p ocesses o model high- equency inancial da a. An applica ion o a ine poin p ocesses o
po olio c edi isk may be ound in E ais e al. (2010). Some applica ions o Hawkes p ocesses o
inancial da a a e also gi en in Emb ech s e al. (2011).
Cohen and Ellio (2013) de i ed an explici il e o Ma ko modula ed Hawkes
p ocesses. Vinko skaya (2014) conside ed a egime-swi ching Hawkes p ocess o model i s dependency
on he bid-ask sp ead in limi o de book. Regime-swi ching models o p icing o Eu opean and
Ame ican op ions we e conside ed in Bu ing on and Ellio (2002b) and Bu ing on and Ellio (2002a),
espec i ely. Semi-Ma ko p ocesses we e applied o limi o de books in
Swishchuk and Vado i (2017)
o model he mid-p ice. We also no e ha le el-1 limi o de books wi h ime dependen a i al
a es
λ( )
we e s udied in Chá ez-Casillas e al. (2019), including he asymp o ic dis ibu ion o he
p ice p ocess.
The pape by Bac y e al. (2015) p oposes an o e iew o he ecen academic li e a u e de o ed
o he applica ions o Hawkes p ocesses in inance. I is a nice su ey o applica ions o Hawkes
p ocesses in inance. In gene al, he main models in high- equency inance can be di ided in o
uni a ia e models, p ice models, impac models, o de -book models and some sys emic isk models,
models accoun ing o news, high-dimensional models and clus e ing wi h g aph models. The book
by Ca ea e al. (2015) de eloped models o algo i hmic ading such as me hods o execu ing la ge
o de s, ma ke making, ading pai s o collec ions o asse s, and execu ing in he da k pool. This book
also con ains a link om which se e al da ase s can be downloaded, along wi h MATLAB code o
assis in expe imen a ion wi h he da a.
A de ailed desc ip ion o he ma hema ical heo y o Hawkes p ocesses is gi en in
Linige (2009).
The pape by Laub e al. (2015) p o ides backg ound, in oduces he ield and his o ical de elopmen ,
and ouches upon all majo aspec s o Hawkes p ocesses. The esul s o he cu en pape
we e i s announced in Swishchuk and Hu man (2018). The main con ibu ion and no el y o
He and Swishchuk (2019)
pape consis s o conside ing di e en ypes o gene al compound Hawkes
p ocesses and hei di usi e limi s o model he mid-p ices o six di e en s ocks. Namely, EBAY, FB,
Risks 2020,8, 28 3 o 25
MU, PCAR, SMH, and CSCO we e used o quan i a i e and compa a i e analysis, and o de ining
he e o a es o es ima e he models i ing accu acy. The pape Swishchuk e al. (2017) deals wi h
compound and egime-swi ching Hawkes p ocesses o model he mid-p ice p ocesses in limi o de
books. Di usi e limi s we e used o bo h models o s udy he link be ween he p ice ola ili y and
he o de low. Nume ical examples we e p esen ed using CISCO da a da ed om 3 No embe o
7 No embe 2014. Thus, he p esen pape con ains no only new esul s wi h p oo s o di e en
gene al compound Hawkes p ocesses, bu also ano he da a se whe e hose esul s we e applied
compa ed wi h p e ious pape s.
The pape is o ganized as ollows. A de ini ion o a Hawkes p ocess and a desc ip ion o i s
p ope ies a e gi en in Sec ion 2. Law o La ge Numbe s (LLN) and Func ional Cen al Limi Theo ems
(FCLT) o a ious gene al compound Hawkes p ocesses, including nonlinea , in limi o de books
a e p o ed in Sec ion 3. Desc ip ions o he da a se and empi ical esul s a e p esen ed in Sec ion 4.
Sec ion 5gi es a quan i a i e analysis o ou esul s and Sec ion 6concludes he pape .
2. De ini ion and Some P ope ies o Hawkes P ocesses (HP)
2.1. De ini ion o Hawkes P ocesses (HPs)
In his sec ion, we gi e a ious de ini ions and some p ope ies o Hawkes p ocesses which can be
ound in he exis ing li e a u e (see, e.g., Hawkes (1971), Hawkes and Oakes (1974),
Emb ech s e al. (2011)
and Zheng e al. (2013), o name a ew). They include in pa icula one-dimensional and nonlinea
Hawkes p ocesses.
De ini ion 1
(Coun ing P ocess)
.
A coun ing p ocess is a s ochas ic p ocess
N( )
wi h
≥
0, whe e
N( )
akes posi i e in ege alues and sa is ies
N(
0
) =
0. I is almos su ely ini e and a igh -con inuous s ep
unc ion wi h inc emen s o size +1.
Deno e by
FN( )
,
≥
0, he his o y o he a i al up o ime ; ha is,
FN( )
,
≥
0, is a il a ion
(an inc easing sequence o σ-algeb as).
A coun ing p ocess
N( )
can be in e p e ed as a cumula i e coun o he numbe o a i als in o a sys em
up o he cu en ime
. The coun ing p ocess can also be cha ac e ized by he sequence o andom a i al imes
(T1
,
T2
,
. . . )
a which he coun ing p ocess
N( )
has jumped. The p ocess de ined by hese a i al imes is
called a poin p ocess (see Daley and Ve e-Jones 2003).
De ini ion 2
(Poin P ocess)
.
I a sequence o andom a iables
(T1
,
T2
,
. . . )
, aking alues in
[
0,
∞)
has
P(
0
≤T1≤T2≤. . . ) =
1, and he numbe o poin s in a bounded egion is almos su ely ini e,
hen (T1,T2, . . . )is called a poin p ocess.
De ini ion 3 (Condi ional In ensi y Func ion).Conside a coun ing p ocess N( )wi h associa ed his o ies
FN( ), ≥0. I a non-nega i e unc ion λ( )exis s such ha
λ( ) = lim
h→0
E[N( +h)−N( )| FN( )]
h(1)
Then, i is called he condi ional in ensi y unc ion o
N( )
(see Laub e al. 2015). We no e ha o iginally his
unc ion was called he haza d unc ion (see Cox 1955).
De ini ion 4
(One-dimensional Hawkes P ocess)
.
The one-dimensional Hawkes p ocess
(see Laub e al. 2015;
Hawkes and Oakes 1974) is a poin p ocess
N( )
which is cha ac e ized by i s in ensi y
λ( )
wi h espec o i s
na u al il a ion:
λ( ) = λ+Z
0µ( −s)dN(s)(2)
whe e λ>0, and he esponse unc ion µ( )is a posi i e unc ion ha sa is ies R∞
0µ(s)ds <1.
Risks 2020,8, 28 4 o 25
The cons an
λ
is called he backg ound in ensi y and he unc ion
µ( )
is some imes called he
exci a ion unc ion. To a oid he i ial case o a homogeneous Poisson p ocess, we assume
µ( )6=
0.
Thus, he Hawkes p ocess is a non-Ma ko ian ex ension o he Poisson p ocess.
Wi h espec o he De ini ions o λ( )in 3and N( )in 4, i ollows ha
P(N( +h)−N( ) = m| FN( )) = 






λ( )h+o(h)m=1
o(h)m>1
1−λ( )h+o(h)m=0
The in e p e a ion o Equa ion
(2)
is ha he e en s occu acco ding o an in ensi y wi h a
backg ound in ensi y
λ
which inc eases by
µ(
0
)
a each new e en , e en ually decaying back o he
backg ound in ensi y alue acco ding o he e olu ion o he unc ion
µ( )
. Choosing
µ(
0
)>
0 leads
o a jol in he in ensi y a each new e en , and his ea u e is o en called he sel -exci ing ea u e.
In o he wo ds,
i an a i al causes he condi ional in ensi y unc ion
λ( )
in Equa ions
(1)
and
(2)
o
inc ease, hen he p ocess is called sel -exci ing.
We would like o men ion ha he condi ional in ensi y unc ion
λ( )
in Equa ions
(1)
and
(2)
can
be associa ed wi h he compensa o Λ( )o he coun ing p ocess N( ), ha is,
Λ( ) = Z
0λ(s)ds (3)
We no e ha
Λ( )
is he unique non-dec easing,
FN( )
,
≥
0, p edic able unc ion, wi h
Λ(
0
) =
0
such ha
N( ) = M( ) + Λ( )a.s.,
whe e
M( )
is an
FN( )
,
≥
0, local ma ingale (exis ence o which is gua an eed by he Doob–Meye
decomposi ion).
A common choice o he unc ion
µ( )
in Equa ion
(2)
is he one o exponen ial decay
(see Hawkes 1971)
µ( ) = αe−β (4)
wi h pa ame e s
α
,
β>
0. In his case, he Hawkes p ocess is called he Hawkes p ocess wi h
exponen ially decaying in ensi y.
In he case o Equa ion (4), Equa ion (2) becomes
λ( ) = λ+Z
0αe−β( −s)dN(s)(5)
We no e ha , in he case o Equa ion
(4)
, he p ocess
(N( )
,
λ( ))
is a con inuous- ime Ma ko
p ocess, which is no he case o a gene al choice o exci a ion unc ion in Equa ion (1).
Wi h some ini i ial condi ion
λ(
0
) = λ0
, he condi ional in ensi y
λ( )
in Equa ion
(5)
wi h
exponen ial decay in Equa ion (4) sa is ies he SDE
dλ( ) = β(λ−λ( ))d +αdN( ), ≥0 (6)
which can be sol ed using s ochas ic calculus as
λ( ) = e−β (λ0−λ) + λ+Z
0αe−β( −s)dN(s), (7)
which is an ex ension o Equa ion (5).

Risks 2020,8, 28 5 o 25
Ano he choice o µ( )is a powe law unc ion
λ( ) = λ+Z
0
k
(c+ ( −s))pdN(s)(8)
wi h posi i e pa ame e s
(c
,
k
,
p)
. This powe law o m o
λ( )
in Equa ion
(8)
was applied in
he geological model called Omo i’s law, and used o p edic he a e o a e shocks caused by
an ea hquake.
De ini ion 5
(D-dimensional Hawkes P ocess)
.
The D-dimensional Hawkes p ocess (see Emb ech s e al. 2011)
is a poin p ocess ~
N( ) = (Ni( ))D
i=1which is cha ac e ized by i s in ensi y ec o ~
λ( ) = (λi( ))D
i=1such ha :
λi( ) = λi+Z
0µij( −s)dNj(s)(9)
whe e λi>0, and M( ) = (µij( )) is a ma ix- alued ke nel such ha :
1. i is componen -wise non-nega i e: (µij( )) ≥0 o each 1≤i,j≤D
2. i is componen -wise L1-in eg able
In ma ix-con olu ion o m, Equa ion (9)can be w i en as
~
λ( ) = ~
λ+M∗d~
N( )(10)
whe e~
λ( ) = (λi)D
i=1.
De ini ion 6
(Nonlinea Hawkes P ocess)
.
The nonlinea Hawkes p ocess (see, e.g., Zheng e al. 2013) is
de ined by he in ensi y unc ion in he ollowing o m:
λ( ) = hλ+Z
0µ( −s)dN(s)(11)
whe e
h(·)
is a nonlinea unc ion wi h suppo in
R+
. Typical examples o
h(·)
a e
h(x) = 1x∈R+
and
h(x) = ex.
Rema k 1.
Many o he gene aliza ions o Hawkes p ocesses ha e been p oposed. They include mixed
di usion–Hawkes models E ais e al. (2010), Hawkes models wi h sho noise exogenous e en s
Dassios and Zhao (2011),
and Hawkes p ocesses wi h gene a ion dependen ke nels Meh dad and Zhu (2014),
o name a ew.
2.2. Compound Hawkes P ocesses
In his sec ion, we de ine nonlinea compound Hawkes p ocess wi h
N
-s a e dependen o de s.
The dependen o de s means he dependency o bo h, he ype o a book e en and i s co esponding
in e -a i al imes, on he ype o he p e ious book e en . We also conside special cases o his
gene al compound Hawkes p ocess.
De ini ion 7
(Nonlinea Compound Hawkes P ocess wi h
n
-s a e Dependen O de s (NLCHPnSDO)
in Limi O de Books).Conside he p ice p ocess S
S =S0+
N( )
∑
k=1
a(Xk)(12)
Risks 2020,8, 28 6 o 25
whe e
Xk
is a con inuous ime
n
-s a e Ma ko chain,
a(x)
is a con inuous and bounded unc ion on he s a e
space
X:={
1, 2, ...,
n}
,
N( )
is he nonlinea Hawkes p ocess (see, e.g., Zheng e al. (2013) de ined by he
in ensi y unc ion in he ollowing o m (see Equa ion (11)):
λ( ) = hλ+Z
0µ( −s)dN(s)
whe e
h(·)
is a nonlinea inc easing unc ion wi h suppo in
R+
. We no e ha in B émaud and Massoulié (1996)
i was shown ha , i
h(·)
is
α
-Lipschi z (see B émaud and Massoulié 1996) such ha
α||h||L1<
1, hen he e exis s a
unique s a iona y and e godic Hawkes p ocess sa is ying he dynamics o Equa ion
(11)
. We shall e e o he p ocess
in Equa ion (12)as a Nonlinea Compound Hawkes P ocess wi h n-S a e Dependen O de s (NLCHPnSDO).
This nonlinea compound Hawkes p ocess will be he ounda ion o ou s udies h oughou his
pape . In he ollowing subsec ion, we will in oduce ou speci ic examples, which will be used o
ou empi ical in es iga ions o he mid-p ice p ocesses.
2.3. Gene al Compound Hawkes P ocesses
De ini ion 8
(Gene al Compound Hawkes P ocess Wi h N-s a e Dependen O de s (GCHPnSDO))
.
Suppose ha
Xk
is an e godic con inuous- ime Ma ko chain, independen o
N( )
, wi h s a e space
X={1, 2, ..., n}, N( )
is a one-dimensional Hawkes p ocess de ined in De ini ion 4and
a(x)
is any bounded
and con inuous unc ion on
X
. We de ine he Gene al Compound Hawkes p ocess wi h N-s a e Dependen
O de s (GCHPnSDO) by he ollowing p ocess:
S =S0+
N( )
∑
k=1
a(Xk)(13)
No e ha his p ocess can be eco e ed om Equa ion (12)by le ing h(x) = x.
De ini ion 9
(Gene al Compound Hawkes P ocess wi h Two-S a e Dependen O de s (GCHP2SDO))
.
Suppose ha
Xk
is an e godic con inuous ime Ma ko chain, independen o
N( )
, wi h wo s a es
{
1, 2
}
.
Then, Equa ion (13)becomes
S =S0+
N( )
∑
k=1
a(Xk)(14)
whe e
a(Xk)
akes only he alues
a(
1
)
and
a(
2
)
. O cou se, we can iew his as a special case o he n-s a e case,
whe e
n=
2. This model was used in Swishchuk e al. (2017) o he mid-p ice p ocess in limi o de books wi h
non- ixed ick δand wo- alued p ice changes.
De ini ion 10
(Gene al Compound Hawkes P ocess wi h Dependen O de s (GCHPDO))
.
Suppose ha
Xk∈ {−δ,δ}and ha a(x) = x, hen S in Equa ion (13)becomes
S =S0+
N( )
∑
k=1
Xk(15)
This ype o p ocess can be a model o he mid-p ice
= (bidp ice +askp ice)/
2in limi o de books, whe e
δ
is a ixed ick size and
N( )
is he numbe o o de a i als up o ime
. We shall call his p ocess a Gene al
Compound Hawkes P ocess wi h Dependen O de s (GCHPDO). This is a gene aliza ion o he p e ious p ocess,
ob ained by le ing a(1) = −δand a(2) = δ.
Ha ing de ined se e al modi ica ions o Hawkes p ocesses, we now p o e di usion limi
heo ems and LLNs o each p ice p ocess in he ollowing sec ion. These di usion p ocesses will
be used o ou explo a ion o he applicabili y o his model o eal wo ld limi o de book da a.
Risks 2020,8, 28 7 o 25
Since o de a i als and cancella ions a e e y equen , wi h many o he applica ions and p ac ical
needs occu ing a he millisecond ime scale (e.g., o de liquida ions, o de execu ions, e c.), we a e
in e es ed in he dynamics ha occu on a la ge ime scale, such as ens o second o minu es. The e o e,
in he ollowing sec ion, we conside he scale
n
ins ead o
, whe e
is in milliseconds and
n
can be
1000, 10,000, making n la ge wi h espec o .
3. Di usion Limi s and LLNs o GCHP
3.1. Di usion Limi and LLN o NLCHPnSDO
We conside he mid-p ice p ocess S de ined in De ini ion 7, namely
S =S0+
N( )
∑
k=1
a(Xk)
whe e
Xk
is a con inuous ime
N
-s a e Ma ko chain and a(x) is a con inuous bounded unc ion on
he s a e space
X={
1, 2, ...,
n}
.
N
is he numbe o p ice changes up o ime
, desc ibed by he
nonlinea Hawkes p ocess gi en in Equa ion (11).
Theo em 1
(Di usion Limi o NLCHPnSDO)
.
Le
Xk
be an e godic Ma ko chain wi h
N
-s a es
{1, 2, ..., n}and wi h e godic p obabili ies (π∗
1,π∗
2, ..., π∗
n). Le also S be as de ined in De ini ion 7, hen
Sn −N(n )·ˆ
a∗
√n
n→∞
−−−→ ˆ
σ∗qE[N[0,1]]W (16)
whe e
W
is a s anda d Wiene p ocess and
E[N[
0,1
]]
is he mean o he numbe o a i als on a uni in e al
unde he s a iona y and e godic measu e. Fu he mo e,
0<ˆ
µ:=Z∞
0µ(s)ds <1and Z∞
0sµ(s)ds <∞(17)
(ˆ
σ∗)2:=∑
i∈X
π∗
i (i)
(i) = b(i)2+∑
j∈X
(g(j)−g(i))2P(i,j)−2b(i)∑
j∈X
(g(j)−g(i))P(i,j)
b= (b(1),b(2),..., b(n))0
b(i):=a(Xi)−a∗:=a(i)−a∗
g:= (P+Π∗−I)−1b
ˆ
a∗:=∑
i∈X
π∗
ia(Xi)
(18)
P is he ansi ion p obabili y ma ix o
Xk
, i.e.,
P(i
,
j) = P(Xk+1=j|Xk=i)
.
Π∗
deno es he ma ix
o s a iona y dis ibu ions o P and g(j)is he j h en y o g.
P oo . F om Equa ion (12), we ha e
Sn =S0+
N(n )
∑
k=1
a(Xk)(19)
and
Sn =S0+
N(n )
∑
i=1
(a(Xk)−ˆ
a∗) + N(n )ˆ
a∗. (20)
Risks 2020,8, 28 8 o 25
The e o e,
Sn −N(n )ˆ
a∗
√n=S0+∑N(n )
k=1(a(Xk)−ˆ
a∗)
√n. (21)
As long as S0
√n
n→∞
−−−→ 0, we need only ind he limi o
a(Xk)−ˆ
a∗
√n
when n→+∞. Conside he ollowing sums
ˆ
R∗
n:=
n
∑
k=1
(a(Xk)−ˆ
a∗)(22)
and ˆ
U∗
n( ):=n−1/2[(1−(n −bn c))] ˆ
R∗
bn c+ (n −bn c)ˆ
R∗
bn c+1](23)
whe e
b·c
is he loo unc ion. Following he ma ingale me hod om Vado i and Swishchuk (2015),
we ha e he ollowing weak con e gence in he Sko okhod opology (see Ge oo 1967):
ˆ
U∗
n( )n→∞
−−−→ ˆ
σ∗W( )(24)
We no e ha he esul s om B émaud and Massoulié (1996) imply by he e godic heo em ha
N
→∞
−−→ E[N[0,1]] (25)
o Nn
n
n→∞
−−−→ E[N[0,1]]. (26)
Using he change o ime →Nn /n, we ind ha
ˆ
U∗
n(Nn /n)n→∞
−−−→ ˆ
σ∗W( E[N[0,1]]) (27)
o ˆ
U∗
n(Nn /n)n→∞
−−−→ ˆ
σ∗qE[N[0,1]]W( )(28)
The esul now ollows om Equa ions (19)–(21).
Lemma 1
(LLN o NLCHPnSDO)
.
The p ocess
Sn
in Equa ion
(19)
sa is ies he ollowing weak con e gence
in he Sko okhod opology (see Ge oo 1967):
Sn
n
n→∞
−−−→ ˆ
a∗E[N[0,1]] (29)
whe e ˆ
a∗is de ined in Equa ion (18), espec i ely.
P oo . F om Equa ion (12), we ha e
Sn
n=S0
n+
N(n )
∑
k=1
a(Xk)
n(30)
Risks 2020,8, 28 15 o 25
In his case,
−δ
will be s a e one, and
δ
wi h be s a e wo. This esul s in he ansi ion ma ix
P
gi en below:
P="pdd 1−pdd
1−puu puu #
A e de e mining ou pa ame e s and ansi ion p obabili ies, we calcula e
a∗
and
σ
in Table 4
oge he wi h puu and pdd.
Table 4.
P o ided abo e a e he alues o
s∗
,
σ
as well as he p obabili ies o an upwa d/downwa d
mo emen gi en an upwa d/downwa d mo emen o each o he i e s ocks in ques ion.
pdd puu σa∗
AAPL 0.4956 0.4933 0.0049 −1.1463 ×10−5
AMZN 0.4635 0.4576 0.0046 −2.7373 ×10−5
GOOG 0.4769 0.4461 0.0046 −1.4301 ×10−4
MSFT 0.6269 0.5827 0.0062 −2.7956 ×10−4
INTC 0.6106 0.5588 0.0059 −3.1185 ×10−4
P o ided hese alues, he asymp o ic me hod om Sec ion 3and i s Co olla ies can be used o
s udy he link be ween o de low in ou model and p ice ola ili y. The es ima ed ola ili y o p ice
changes is exp essed in e ms o pa ame e s desc ibing a i al a e o limi o de s. Allowing us o
es ou claim ha ou model accu a ely desc ibes he mid-p ice p ocess. We ecall ha he mid-p ice
= (bidp ice +askp ice)/2, and he ollowing di usion limi
Sn −N(n )a∗
√n
n→∞
−−−→ σsλ
1−α/βW( ). (48)
I he da a sa is y ou p oposed model, hen, a e conside ing la ge windows o ime (5 min,
10 min, 20 min), we would expec o see he empi ical and heo e ical s anda d de ia ions o ollow
each o he closely. To es his, we compa e he equi alen p ocess, cons uc ed by mul iplying he
LHS and RHS by
√n
. Then, cu ing ou da a in o disjoin windows o size
n
, speci ically
[in
,
(i+
1
)n]
wi h
=
1 and by se ing he le bound as ou s a ing ime, we can calcula e
Sn −N(n )a∗
o each
indi idual window and gi e a gene alized o mula o his below:
S∗
i=S(i+1)n −Sin −(N((i+1)n )−N(in ))a∗(49)
This gi es a collec ion o alues
{S∗
i}
o e which we compu e he s anda d de ia ion. I ou
model is accu a e, we would would expec ha
s d{S∗
i} ≈ √n σsλ
1−α/β, whe e =1. (50)
We plo he empi ical s anda d de ia ion agains he heo e ical one o a ious window sizes
s a ing a 10 s and inc easing in s eps o 10 s un il we each 20 min; his is illus a ed in Figu e 3.
Se e al impo an ema ks should be made a his poin . I is clea ha , while he model accu a ely
p edic s he o e all end o MSFT and INTC, we se e ely unde es ima e he a iabili y in he
mid-p ice p ocess o APPL, AMZN and GOOG. Fu he mo e, as he window size inc eases, he
o e all sp ead in he da a inc eases. We a ibu e his o he dec easing sample size imposed on us
as we inc ease he window size. Fo example, when we conside a 20 min window, we can only
cons uc 27 disjoin windows in he 9 h ading day o cing us o deal wi h he p oblem o p edic ing a
’popula ion’ s anda d de ia ion om a inc easingly small sample. We emedy his by using a a iance
s abilizing ans o ma ion in la e sec ions. Speci ically, a popula me hod o a Poisson p ocess is o

Risks 2020,8, 28 16 o 25
ake he squa e oo o ou empi ical and heo e ical s anda d de ia ions. This makes i possible o
quali a i ely iew he o e all end in he da a, gaining a clea e idea o goodness o i om he e.
Figu e 3.
Each igu e compa es he empi ical s anda d de ia ion o a ixed window size o he
heo e ical s anda d de ia ion. We ha e plo ed an empi ical s anda d de ia ion o all
n
om 10 s
o 20 min in s ep sizes o 10 s. Each empi ical s anda d de ia ion co esponds o a single poin in he
sca e plo , and he plo ed cu e co esponds o he p edic ed heo e ical alue.
4.3. Gene al Compound Hawkes P ocess wi h Two Dependen O de s
As o now, we ha e conside ed a ixed
δ
ela ed o he ading ick size. Howe e , i we conside
he mid-p ice changes o APPL, AMZN and GOOG he assump ion o a ixed ick size is iola ed.
In ac , we obse e ha app oxima ely 61%, 53% and 71% o all mid-p ice changes a e la ge han hal
Risks 2020,8, 28 17 o 25
a ick size, which is opposed o wha we obse e o MSFT and INTC whe e all mid-p ice changes
occu a he hal ick size, we illus a e his o AAPL, AMZN and GOOG in Figu e 4.
Figu e 4.
We can see clea ly ha he change in he mid-p ice is o en la ge han a hal ick. These
mid-p ice changes make up a signi ican po ion o he ac ual da a, con adic ing he assump ion
needed o he CHPDO model ha he mid-p ice changes occu on a e age a a hal ick size.
I is clea in Figu e 4 ha addi ional conside a ions need o be made. A simple way o include
he a iabili y in mid-p ice mo emen s in ou model is o in oduce
a(Xi)
as desc ibed in De ini ion 9.
I is o cou se necessa y o de e mine he alues o
a(·)
o each s a e o ou Ma ko chain. A nai e
me hod is o ake he mean o he downwa d and upwa d mid-p ice mo emen s and assign hem o
a(1)and a(2), espec i ely. We p o ide hese alues in Table 5.
Table 5. a(i)
is he a e age o he upwa d o downwa d mid-p ice mo emen s. Following ou p e ious
con en ion, he i s s a e will be associa ed wi h he mean o all downwa d mid-p ice mo emen s and
he second s a e will be associa ed wi h he mean o all upwa d mid-p ice mo emen s.
a(1)a(2)
AAPL −0.0172 0.0170
AMZN −0.0134 0.0133
GOOG −0.0302 0.0308
In his s ep, we ha e only endea ou ed o be e ealize he ac ual p ice mo emen s in ou da a.
The e o e, when we obse e a downwa d mid-p ice mo emen , we con inue o assign i o s a e one
and simila ly o an upwa d p ice mo emen we con inue o assign i o s a e wo. I ollows ha ou
ansi ion ma ix will emain he same. Then, using hese new s a e alues, we ecalcula e
a∗
and
σ
,
Risks 2020,8, 28 18 o 25
p o iding hem in Table 6. The e ec o hese changes is in es iga ed in Figu e 5. No e ha in Figu e 5
we ha e used he a iance s abilizing ans o ma ion discussed ea lie in o de o be e isualize he
o e all end in ou da a.
Table 6.
Abo e, we ha e he alues o
a∗
,
σ
, as well as he p obabili ies o an upwa ds/downwa ds
mo emen , gi en an upwa ds/downwa ds mo emen o he h ee s ocks o in e es .
pdd puu σa∗
AAPL 0.4956 0.4933 0.0169 −1.5624 ×10−4
AMZN 0.4635 0.4576 0.0123 −1.0475 ×10−4
GOOG 0.4769 0.4461 0.0282 −5.5095 ×10−4
No ice ha he e is a signi ican quali a i e imp o emen in he i s o AAPL and GOOG in
Figu e 5, bu he a iabili y in mid-p ice mo emen s o AMZN is s ill clea ly unde es ima ed by ou
model. The unexplained a iance may be cap u ed by in es iga ing an
N
-s a e Ma ko chain since
he addi ional ansi ion p obabili ies could explain he a iabili y missing in he 2-s a e case.
AAPL AMZN
GOOG
Figu e 5.
A compa ison o he empi ical s anda d de ia ion o a ixed window size
n
o he heo e ical
s anda d de ia ion o AAPL, AMZN and GOOG using he 2-s a e dependen o de model. We ha e
plo ed he empi ical s anda d de ia ion o all
n
om 10 s o 20 min in s ep sizes o 10 s. Each empi ical
s anda d de ia ion co esponds o a single poin in he sca e plo and he plo ed cu e co esponds o
he p edic ed heo e ical alue. Visually, he e is a signi ican imp o emen o all s ocks, al hough he
heo e ical s anda d de ia ion o AMZN is s ill unde es ima ing he empi ical a iabili y.
Risks 2020,8, 28 19 o 25
4.4. Gene al Compound Hawkes P ocess wi h NDependen O de s
We ecall he
N
-s a e model desc ibed in De ini ion 8. The immedia e ques ion becomes how
bes o choose he s a e alues. We modi y he quan ile based app oach om Swishchuk e al. (2017).
A e calcula ing he mid-p ice changes, we sepa a e he da a in o upwa d and downwa d p ice
mo emen s. Then, we calcula e e enly dis ibu ed quan iles o bo h da a se s. Depending on he da a,
se e al quan iles may be iden ical, we ejec any duplica es. We hus ob ain a lis o bounds which we
comple e by adding he minimum obse ed alue i necessa y.
To de e mine he s a e alues
a(Xi)
, we ake he a e age o all mid-p ice changes loca ed be ween
wo neighbou ing bounda y alues. Fu he mo e, we assign a mid-p ice change o s a e
i
i i is g ea e
han o equal o he
(i−
1
)
h bounda y and s ic ly less han he
i
h bounda y. An excep ion is made
o he la ges uppe bound whe e equali y is pe mi ed a bo h ends.
As we could no cap u e he ull a iabili y o mid-p ice changes o AMZN in he p e ious
me hod, we in es iga e i o his case. Fu he mo e, o ac abili y, we only conside 14 bounda y
alues om which we ob ain a 12-s a e Ma ko chain. Ins ead o p o iding he ansi ion ma ix, we
p o ide he e godic p obabili ies o he ansi ion ma ix and he associa ed s a es in Table 7.
Table 7.
Abo e, we ha e p o ided he s a e, associa ed e godic p obabili ies and s a e alues
a(i)
o
AMZN, gi en a 12-s a e Ma ko chain ha was ob ained om choosing a 16 quan ile me hod.
AMZN
iπ∗
ia(Xi)
1 0.0275 −0.0524
2 0.0281 −0.0318
3 0.0264 −0.0250
4 0.0382 −0.0200
5 0.0576 −0.0150
6 0.3249 −0.0064
7 0.2321 0.0050
8 0.0923 0.0100
9 0.0578 0.0150
10 0.0353 0.0200
11 0.0412 0.0271
12 0.0387 0.0476
In o de o compa e he wo s a e and
N
-s a e app oaches, we i s ake a quali a i e app oach
and plo he wo heo e ical and empi ical s anda d de ia ions agains each o he in Figu e 6. When we
compa e he mean squa ed esiduals, he 2-s a e model discussed be o e has mean squa ed e o 0.0208
while ou 12-s a e Ma ko chain b ings ha down o 0.0125. Conside ing an e en la ge Ma ko chain
wi h 24-s a es, we a e only able o ob ain a meage imp o emen o 0.0123, sugges ing ha he e is some
unde lying a iance in he mid-p ice p ocess no cap u ed by ou model. We in es iga e hese mo e
quan i a i e measu es in he ollowing subsec ion. Fi s , compa ing he models agains a nume ical
bes i , and hen in es iga ing he mean squa ed e o , we gain a be e quan i a i e unde s anding o
he o e all imp o emen ob ained om each model inc easing he numbe o quan iles. While we do
no p opose a me hod o selec ing an
N
-s a e model in gene al, a possible piece o c i e ia is o hal
when he e is no app eciable gain in he MSE o mo e in o ma ion (see Swishchuk e al. 2017, p. 17).
In ligh o his, in he ollowing sec ion, we use k- old c oss- alida ion o make a de e mina ion.
Risks 2020,8, 28 20 o 25
AMZN
Figu e 6.
We conside he N-s a e model o AMZN discussed p e iously in he pape . While he e is a
sligh imp o emen agains he o iginal i , he model s ill s uggles o pe ec ly p edic he a iabili y
in he mid-p ice changes o ou da a.
5. Quan i a i e Analysis
While ou model does isually appea o i he expec ed a iabili y in ou o he i e cases, i s ill
ails o cap u e he comple e dynamics o mid-p ice changes seen in ou AMZN da a. We in es iga e
he mean squa e e o o ou models wi h a a ying numbe o quan iles in Table 8. This gi es a good
indica ion as o whe he he
N
-s a e model is a be e i o ou da a. I we look closely a AAPL and
GOOG, we see ha he
N
-s a e case can s ill imp o e ou esul s om he 2-s a e case. Fo AAPL,
we cons uc ed a 17-s a e Ma ko chain by aking 16 quan iles on he downwa d mo emen s and
16 quan iles upwa d mo emen s. This esul ed in a mean squa ed e o o 0.0036 ha is app oxima ely
a 28% imp o emen o he 2-s a e case whe e he mean squa ed e o was 0.0050. E en mo e ex eme,
using a 25-s a e Ma ko chain o GOOG which was cons uc ed simila ly, we obse ed a mean
squa ed e o o 0.0046, which is a 60% imp o emen om he 2-s a e case wi h a mean squa ed e o
o 0.0115. We conclude wi h Table 8, which p o ides he mean esiduals o AAPL, AMZN and GOOG
wi h se e al Ma ko chains cons uc ed om a ious numbe s o quan iles. We also include mean
esiduals o INTC and MSFT o compa ison.
Ano he quan i a i e measu e o ou i would be o compa e hem o he one which bes
minimizes he esiduals. No ice ha each o ou models assumes ha he s anda d de ia ion
p opo ionally o he squa e oo o he ime s ep. The e o e, we can es ima e he bes possible
coe icien by minimizing
L2
-no m o equi alen ly he mean squa ed e o . We p o ide plo s o hese
hypo he ical bes i s agains he empi ical da a and heo e ical i s in Figu e 7. We also p o ide
he coe icien s o he heo e ical i s and eg ession in Table 9in o de o ha e a mo e quan i a i e
compa ison. They a e calcula ed using leas -squa e eg ession.

Risks 2020,8, 28 21 o 25
Table 8.
We lis he mean esiduals o se e al Ma ko chains wi h a ying numbe s o s a es. These
we e gene a ed using ou modi ied quan ile app oach choosing o s a wi h 2, 8, 16 o 32 quan iles.
We see ha , in gene al, he mean esidual dec eases o some lowe limi whe e we can no longe
pe o m any be e . Recall ha he only obse ed mid-p ice changes o INTC and MSFT we e o a hal
ick size, and any inc ease in he numbe o quan iles will esul in he same pe o mance.
CHPDO 2 8 16 32
AAPL 0.2679 0.0050 0.0036 0.0036 0.0036
AMZN 0.1122 0.0208 0.0131 0.0124 0.0123
GOOG 0.4036 0.0115 0.0048 0.0045 0.0047
INTC 1.7917 ×10−51.7917 ×10−51.7917 ×10−51.7917 ×10−51.7917 ×10−5
MSFT 1.0586 ×10−41.0586 ×10−41.0586 ×10−41.0586 ×10−41.0586 ×10−4
We no ice ha in each case he e o s a e close o, o unde i e pe cen , wi h AMZN being
he bigges o ende . This is consis en wi h he discussion p o ided h oughou ou analysis and
highligh s he gene al applicabili y o ou model.
Table 9.
The coe icien s calcula ed o AAPL, AMZN and GOOG a e gene a ed using a Ma ko chain
c ea ed by 16 quan iles on he upwa d and downwa d mo emen s, while he coe icien s o INTC and
MSFT a e ob ained om he CHPDO case.
Theo e ical Coe icien Reg ession Coe icien Pe cen E o
AAPL 0.02868 0.02828 1.42%
AMZN 0.01450 0.01831 20.8%
GOOG 0.02883 0.03023 4.63%
INTC 0.00186 0.00193 3.4%
MSFT 0.00231 0.00246 6.4%
C oss-Valida ion
We conclude his subsec ion using a me hod simila o k- old c oss- alida ion o model selec ion.
Gene ally, he da a a e shu led andomly and hen pa i ioned in o k se s o app oxima ely equal size.
F om hese k se s, one is selec ed o es ing he model while he emaining a e kep o aining i .
Fo ou da a, we a e unable o pe o m his shu ling as he e is an impo an o de ing imposed by he
a i al imes. Fo una ely, since we a e conside ing high- equency models o he mid-p ice changes,
we expec a minimal amoun o co ela ion in he ola ili y o he p ice p ocess o e la ge windows o
ime. The e o e, di iding he da a in o wel e 30 min windows selec ing hem one a a ime should
s ill be meaning ul.
The aining da a will need o be glued back oge he a e he hi y-minu e window is emo ed.
This is done by assuming he e en s ne e occu ed. We p o ide an example below o how one would
emo e he second hi y-minu e window in Tables 10 and 11.
This in oduces a small e o in o he a i al imes, bu only a a single poin in a da a se o se e al
housand a i al imes. Had we andomly emo ed poin s, o shu led he da a, we could in oduce
many e o s which could signi ican ly change he unde lying a i al p ocess. A his poin , we pe o m
he same i ing p ocedu e on he aining da a and a e he i ing is comple ed, we can compu e he
empi ical s anda d de ia ions on he es ing da a o ob ain a mean squa ed e o . This de e mines
how well he model was able o p edic he aining da a, calcula ing his o each window o ime and
a e aging p o ides a sco e o how well he model was able o make p edic ions.
We ecall ha , o MSFT and INTC, he only obse ed p ices changes we e a single ick, and i
is no possible o ou pe o m he o iginal CHPDO model. Ins ead, we conside AAPL, AMZN and
GOOG calcula ing sco es o CHPDO, GCHP2SDO and a ious GCHPnSDO models. Speci ically he
se o GCHPnSDO models a e cons uc ed using he quan ile me hod, allowing o 2–14 quan iles.
Risks 2020,8, 28 22 o 25
This was chosen only o make he eco ding mo e ac able; he esul ing sco es a e p o ided in
Table 12.
AAPL AMZN
GOOG INTC
MSFT
Figu e 7.
A quali a i e compa ison o he eg ession o he heo e ical model. Fo APPL, AMZN and
GOOG, we ha e used a Ma ko chain gene a ed om 16 quan iles aken on he upwa d mo emen s
and downwa d mo emen s. Fo INTC and MSFT, we ha e aken he CHPDO coe icien since a
di e en coe icien is no possible wi h he o he models.
Risks 2020,8, 28 23 o 25
Table 10.
An imagined sequence o p ice change e en s be o e he second hi y minu e window has
been emo ed.
Time (s) ··· 1789 1795 1803 ··· 3193 3601 3608 ···
E en ··· n−1 n n + 1 ··· n+k n+k+1 n+k+2 ···
Table 11.
An imagined sequence o p ice change e en s a e he second hi y minu e window has
been emo ed.
Time (s) ··· 1789 1795 1801 1808 ···
E en ··· n−1 n n+k+1 n+k+2 ···
Table 12.
A able o es ing sco es o a ious models. Compu a ions we e pe o med o up o he
40-quan ile case o each s ock, and his able ep esen s a sample o ha da a. We no e ha he
models s op ob aining app eciable pe o mance gains a a ound 4–7 quan iles, luc ua ing a ound he
same sco es.
AAPL AMZN GOOG
CHPDO 0.0348 0.0142 0.0436
GCHP2SDO 0.0049 0.0048 0.0044
2-Quan iles 0.0042 0.0041 0.0036
3-Quan iles 0.0041 0.0042 0.0035
4-Quan iles 0.0040 0.0040 0.0035
5-Quan iles 4.0341 ×10−33.9878 ×10−33.4593 ×10−3
6-Quan iles 4.0362 ×10−33.8971 ×10−33.4168 ×10−3
7-Quan iles 4.0184 ×10−33.8971 ×10−33.3977 ×10−3
8-Quan iles 4.0097 ×10−33.8656 ×10−33.3881 ×10−3
9-Quan iles 4.0026 ×10−33.8124 ×10−33.3976 ×10−3
10-Quan iles 3.9984 ×10−33.8124 ×10−33.425 ×10−3
11-Quan iles 3.9971 ×10−33.81594 ×10−33.3867 ×10−3
12-Quan iles 3.9826 ×10−33.81463 ×10−33.3906 ×10−3
13-Quan iles 3.9804 ×10−33.81517 ×10−33.3562 ×10−3
14-Quan iles 3.9725 ×10−33.81427 ×10−33.3549 ×10−3
O e all, he e is a meaning ul gain in pe o mance o he
N
-s a e models, especially when we
ake in o accoun he p e ious discussion. The c oss- alida ion we’ e pe o med p o ides a easonable
c i e ia o model ejec ion, simila o when we s op obse ing app eciable gains in he mean squa ed
e o . F om ou analysis, he bes model would be one cons uc ed om 4–7 quan iles, depending on
he s ock o in e es .
6. Conclusions and Fu u e Wo k
O e all, he
N
-s a e model ou pe o ms he o he s when he numbe o s a es is kep small.
I gene a es i s o ou ou o he i e da ase s ha a e easonable and p o iding educ ions in he
mean squa ed e o by upwa ds o 25%. While no able o cap u e he ull dynamics obse ed in
AMZN, i appea s o be a s ong candida e o a simple model o p ice dynamics obse ed in ou da a.
Fu he in es iga ion would be necessa y o de e mine wha causes he addi ional ola ili y obse ed
in AMZN, and po en ially implemen a mo e obus model which cap u es his. The po en ial use s o
ou models a e p ac i ione s wo king in he inancial indus y who wish o implemen ou esul s in
high- equency and algo i hmic ading combined wi h hei indus ial knowledge.
Au ho Con ibu ions:
Bo h au ho s ha e con ibu ed equally o he pape . All au ho s ha e ead and ag eed o
he published e sion o he manusc ip .
Funding: This esea ch was unded by NSERC G an No. RT732266.
Risks 2020,8, 28 24 o 25
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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