General compound hawkes processes in limit order books

S iščuk, Ana olij; Hu man, Aiden A icle Gene al compound hawkes p ocesses in limi o de books Risks P o ided in Coope a ion wi h: 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, h ps://doi.o g/10.3390/ isks8010028 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/257983 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. 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I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ isks 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 . 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