Hawkes model for price and trades high-frequency dynamics 3 1 0 Emmanuel Bacry∗ Jean-Franc¸ois Muzy† 2 December 11, 2013 n a J 7 Abstract ] R We introduce a multivariate Hawkes process that accounts for the T dynamicsofmarketpricesthroughtheimpactofmarketorderarrivalsat . microstructural level. Ourmodel is a point process mainly characterized n by 4 kernels associated with respectively thetrade arrival self-excitation, i f the price changes mean reversion the impact of trade arrivals on price - q variationsandthefeedbackofpricechangesontradingactivity. Itallows [ one to account for both stylized facts of market prices microstructure (including random time arrival of price moves, discrete price grid, high 1 frequency mean reversion, correlation functions behavior at various time v scales)andthestylizedfactsofmarketimpact(mainlytheconcave-square- 5 root-like/relaxation characteristic shape of themarket impact of a meta- 3 1 order). Moreover, it allows one to estimate the entire market impact 1 profile from anonymous market data. Weshow that these kernels can be . estimated from the empirical conditional mean intensities. We provide 1 numerical examples, application to real data and comparisons to former 0 approaches. 3 1 : v 1 Introduction i X Market impact modeling (i.e, the influence of market orders on forthcoming r a prices) is a longstanding problem in market microstructure literature and is obviously of great interest for practitioners (see e.g., [6] for a recent review). Even if there are various ways to define and estimate the impact associated with an order or a meta-order1, a large number of empirical results have been obtained recently. The theory of market price formation and the relationship between the order flow and price changes has made significant progress during the last decade [7]. Many empirical studies have provided evidence that the ∗CMAP, UMR 7641 CNRS, Ecole Polytechnique, 91128 Palaiseau, France ([email protected]) †SPE,UMR6134CNRS,Universit´edeCorse,20250Corte,France([email protected]) 1Oneusuallyreferstoameta-orderasasetoforderscorrespondingtoafragmentationof asinglelargevolumeorderinseveralsuccessivesmallerexecutions 1 2 price impact has, to many respects, some universal properties and is the main source of price variations. This corroborates the picture of an “endogenous” nature of price fluctuations that contrasts with the classical scenario according to which an “exogenous” flow of information leads the prices towards a “true” fondamental value [7]. We will not review in details all these results but simply recall the point of viewofBouchaudetal[6]. These authorsproposea modelofpricefluctuations by generalizing Kyle’s pioneering approach [18] according to which the price is written(uptoanoiseterm)astheresultoftheimpactofalltrades. Ifnstands for the trading time, Bouchaud et al. model is written as follows [8, 7]: p = G(n j)ξ +η (1) n j j − j n X≤ where η is a white noise while ξ = ε f(v ) with ε =+1 (resp. ε = 1) if a j j j j j j − trade occurs at the best ask (resp. at best bid) and f(v ) describes the volume j dependence of a single trade impact. The function G(n j) accounts for the − temporal dependence of a market order impact. Evenifmodelslike(1)representarealbreakthroughintheunderstandingof price dynamics they have many drawbacks. First, the nature and the status of thenoiseη isnotwelldefined. Moreimportantly,thesemodelsinvolvediscrete j events (through the trading or event time) only defined at a microstructural level though they intend to representsome coarse versionof prices (indeed, the price p in the previous equation can take arbitrary continuous values, more- n over, calibrating its volatility at any scale involves some additional parameter, ...). Moreover, these models cannot account for real time (i.e “physical time”) dynamics or real time aggregation properties of price fluctuations. In that re- spect, they are not that easy to be used in real high frequency applications such as optimized execution. To make it short, though being defined at finest timescales,theycannotaccountforthemainmicrostructurepropertiesofprice variations related to their discrete nature: prices live on tick grids and jump at discrete random times. Our aim in this paper is mainly to define continuous time version of the market impact price model discussed previously. For that purpose, point pro- cesses [9] provide a natural framework. Let us not that point processes have been involvedin many studies in high frequency finance fromthe famous ‘zero- intelligence” order-book models [24, 11] to models for trade [16, 4] or book events [21, 10] irregular arrivals. In a recent series of papers [3, 2, 1], we have shownthat self-excitedpoint(Hawkes)processescanbe pertinentto model the microstructure of the price and in particular to reproduce the shape of the sig- nature plot and the Epps effect. Our goal is to extend this framework in order to account for the market impact of market orders. In that respect, the main ideas proposed in refs. [8, 13, 19] can be reconsidered within the more realistic framework of point processes where correlations and impact are interpreted as cross and self excitations mechanisms acting on the conditional intensities of Poisson processes. This allows us to make a step towards the definition of a 3 faithful model of price microstructure that accounts for most recent empirical findings on the liquidity dynamical properties and to uncover new features. The paper is organized as follows. In Section 2 we show how market order impactcanbe naturallyaccountedwithinthe classofmultivariate Hawkespro- cesses. Ourmainmodelforpricemicrostructureandmarketimpactispresented and its stability is studied. Numerical simulations of the model are presented. In Section 3, the microstructure of price and market order flows are studied throughthe covariancematrix ofthe process. Our resultsare illustratedonnu- mericalsimulations. Anextensionofthemodelthataccountsforlabeledagents is defined in Section 4 and results on market impact are presented in Section 5 including an explanation on how the newly defined framework allows one to estimate the marketimpactprofilefromnonlabeleddata. Section6 showshow kernels defining the dynamics of trade occurrences and price variations can be nonparametrically estimated. All the theoretical results obtained in the previ- ousSections areillustratedinSection7whenappliedonvarioushighfrequency future data. It allows one to reveal the different dynamics involved in price movements, market order flows and market impact. Intraday seasonalities are shown to be taken care in a particularly simple way. We also discuss, on a semi-qualitative ground, how the market efficiency can be compatible with the observed long-range correlation in supply and demand without any parameter adjustment. Conclusionandprospects arereportedinSection 8while technical computations are provided in Appendices. 2 Hawkes based model for market microstructure 2.1 Definition of the model As recalled in the introduction, in order to define a realistic microstructure price model while accounting for the impact of market orders, the framework of multivariate self-excited point processes is well suited. A natural approach is to associate a point process to each set of events one wants to describe. We choose to consider all market order events and all mid-price change events. Let us point out that we will not take into account the volumes associated toeachmarketorders. Thoughthiscanbebasicallydonewithintheframework of marked the point processes, it would necessitate cumbersome notations and make the estimation much more difficult. This issue be discussed briefly in Section 8 and addressed in a forthcoming work. 2.1.1 Market orders and price changes as a 4 dimensionalpoint pro- cess Thearrivalsofthemarketordersarerepresentedbyatwodimensionalcounting process T = Tt− (2) t T+ (cid:18) t (cid:19) 4 representing cumulated number of market orders arrived before time t at the best ask (Tt+) and at the best bid (Tt−). Each time there is a market order, either T+ or T jumps up by 1. We suppose that the trade process2 T is a − counting process that is fully defined by λT the conditional intensity vector of the process T at t. t t • The price is represented in the same way. Let X represents a proxy of the t price at high-frequency (e.g., mid-price). As in refs. [3, 2, 1] we write Xt =Nt+−Nt− (3) whereNt+ (resp. Nt−)representsthenumberofupward(resp. downward)price jumps at time t 3. Thus, each time the price goes up (resp. down), N+ (resp. N ) jumps up by 1. We set − N = Nt− . (4) t N+ (cid:18) t (cid:19) As for the trade process, N is fully defined by t λN the conditional intensity vector of the process N at t. t t • The 4 dimensional counting process is then naturally defined as Tt− T T+ Pt =(cid:18) Ntt (cid:19)= Ntt−  . (5)  N+   t    as well as its associated conditional intensity vector λT− t λT λT+ λ = t = t  . (6) t (cid:18) λNt (cid:19)  λλNN−t+   t    2.1.2 The Model Basically, the model consists in considering that the 4-dimensional counting process P is a Hawkes process [14, 15]. The structure of a Hawkes process t allows one to take into account the influence of any component in P on any t 2In the following, a ”trade” will refer to the execution of a given market order (which might involve several counterparts). Thus the process T will be referred indifferently to as themarketorderarrivalprocessorthetradearrivalprocess. 3Letus pointout that, inthis model,wedo nottake intoaccount the sizeof the upward or downward jumps of the price. We just take into account the direction of the price move, +1(reps. −1)foranyupward(reps. downward)jumps 5 component of λ . In its general form the model is represented by the following t equation4 λ =M +Φ⋆dP . (7) t t where Φ is a 4 4 matrix whose elements are causal positive functions (by t causal we mean f×unctions supported by R+). Moreover, we used the “matrix convolution” notation, B⋆dP = B dP , t t s s R − Z whereB dP refersto the regularmatrix product. M accountsforthe exoge- t s s − nous intensity of the trades, it has the form µ µ M =  (8) 0  0      since, by symmetry we assume that the exogenous intensity of T+ and T are − equal while mid-price jumps are only caused by the endogenous dynamics. The matrix Φ or any of its sub-matrices (or element) are often referred to as Hawkes kernels. Each element describes the influence of a component over anothercomponent. Thus,itisnaturaltodecomposethe4 4kernelΦ infour t × 2 2 matrices in the following way × ΦT ΦF Φ = t t (9) t ΦI ΦN (cid:18) t t (cid:19) where ΦT (influence of T on λT) : accounts for the trade correlations (e.g., • splitting, herding, ...). ΦI (influence of T on λN) : accounts for the impact of a single trade on • the price ΦN (influence ofN on λN) : accounts for the influence of pastchanges in • priceonfuture changesinprice(due tocancelandlimitordersonly, since changesin price due to marketordersare explicitly takeninto accountby ΦI) ΦF (influence of N on λT) : accounts for feedback influence of the price • moves on the trades. If we account for the obvious symmetries between bid-ask sides for trades and up-down directions for price jumps these matrices are naturally written as: φT,s φT,c φI,s φI,c ΦT = t t , ΦI = t t (10) t φT,c φT,s t φI,c φI,s (cid:18) t t (cid:19) (cid:18) t t (cid:19) 4Let us point out that Section 4 will introduce a generalization of this model including labeledtrades. 6 and φN,s φN,c φF,s φF,c ΦN = t t , ΦF = t t (11) t φN,c φN,s t φF,c φF,s (cid:18) t t (cid:19) (cid:18) t t (cid:19) where all φ?,? arecausalfunctions andthe upperscripts s andc stand for“self” t and “cross” influences of the Poisson rates (we use the same convention which wasinitiallyintroducedin[3]forφN). Thus,forinstance,ontheonehand,φI,s accounts for the influence of the past buying (resp. selling) market orders on the intensity ofthe future upward(reps. downward)price jumps. Onthe other hand, φI,c quantifies the influence of the past buying (resp. selling) market orders on the intensity of the future downward (resp. upward) price jumps. Remark 1 : All these 2 2 matrices commute since they diagonalize in × the same basis (independently of t). Their eigenvalues are the sum (resp. the difference) of the self term with the cross term. This property will be used all along the paper. Most of the computations will be made after diagonalizing all the matrices. 2.1.3 The Impulsive impact kernel model or how to deal with simul- taneous jumps in the price and trade processes It is important to point out that a buying market order that eats up the whole volumesittingatbestaskresultsinan“instantaneous”changeinthemid-price. From our model point of view, it would mean that T and N have simultaneous jumps with a non zero probability. It is clearly not allowed as is by the model. However,from a numerical point of view, this can be simulated by just consid- ering that the jump in the price takes place within a very small time interval (e.g., of width 1ms which is the resolution level of our data) after the market order has arrived. There is no ambiguity in the “direction” of the causality : it is a market order that makes the price change and not the other way around. This would result in an impact kernel φI,s which is “impulsive”, i.e., localized around 0, actually close to a Dirac distribution δ . t From a practical numerical point of view, choosing φI,s to be a Dirac dis- tribution is fairly easy. It basically amounts to considering that it is a positive function of given L1 norm I and with a support ∆t of the order of a few mil- liseconds. Let us point out that it means that, the price increment between the moment of the trade and ∆t milliseconds afterwards, follows a Poisson law whose parameter is I. This actually allows price jumps (spread over a few milliseconds) of several ticks (greater than 1). Of course, from a mathematical point of view, this is not that simple. The limit ∆t 0 has to be defined properly. This will be rigorously defined and → extensivelydiscussedinafutureworkandisoutofthescopeofthispaper. The ”practical”approachdescribedaboveandthefactthatwecanformallyreplace, in all the computations, φI,s by a Iδ is far enough for our purpose. t It is clear that we expect to find an impulsive component in φI,s when estimating on real data. Though, a priori, we do expect also a non singular component that could have a large support (e.g., when the marker order eats 7 uponlyapartofthe volumesittingatbestask),wewillseeinestimationsthat mostoftheenergyofφI,s islocalizedaround0. Moreover,wewillfindthatφI,c is close to 0. All these remarks will lead us to study a particularly interesting case of the previously defined model for which φI,s =Iδ and φI,c =0. t • Consequently φI =Iδ I, where I refers to the identity matrix. t • This model will be referred in the following as the Impulsive Impact Kernel model. Beforemovingonandstudy the conditionsforourmodeltobe welldefined, we need to introduce some notations that will be used all along the paper. 2.1.4 Notations Let us introduce the following notations: Notations 1 If f is a function f refers to the Laplace transform of this func- t z tion, i.e., f b= e iztf dt. z − t Z δt refers to the Dirac distributiobn, consequently δ =1. z Moreover, we will use the convenient convention (which holds in the Laplace b domain) : δ ⋆δ =δ . t t t The L1 norm of f is referred to as: f = f dt. t || || | | Z Thus if t, f 0 then f = f dt=f . t t 0 ∀ ≥ || || Z We extendthese notations to matrix of functions. Thus, ifbFt is a matrix whose element are functions of t, let Fˆ denote the matrix whose elements are the z Laplace transform of the elements of F . Following this notation, we note t ΦT ΦF Φ = z z . (12) z ΦIz ΦNz ! b b b Notations 2 If M is a matrix, M∗ rebfers tbo the matrix M whose each element has been replaced by its conjugate and M refers to the hermitian conjugate † matrix of M. 8 Notations 3 Whenever λ is a stationary process, we will use the notation t ΛT ΛT Λ=E(λt)= ΛN   ΛN      where ΛT =E(λT+)=E(λT−) and ΛN =E(λN+)=E(λN−). t t t t Let us point out that the fact that the mean intensities are equal is due to the ± symmetries of the kernels in (10) and (11). Notations 4 We define the kernel’s imbalance : ∆φT =φT,s φT,c and ∆φT =φT,s φT,c • − − ∆φI =φI,s φI,c and ∆φI =φI,s φI,c • − b b − b ∆φF =φF,s φF,c and ∆φF =φF,s φF,c • − b b −b ∆φN =φN,s φN,c and ∆φN =φN,s φN,c • − b b −b Let us point out that since the kernels are all positive functions, one has, re- b b b placing ? by T, N, I or F : φ?,s = φ?,s and φ?,c = φ?,c , 0 || || 0 || || and consequently b b ∆Φ? =φ?,s φ?,c = φ?,s φ?,c 0 0 − 0 || ||−|| || 2.2 Stability conbditiobn - bStationarity of the price incre- ments TheprocessP definedinSection2.1.2iswelldefinedaslongasthematrixΦ is t t locallyintegrableonR+. Hawkes,inhis originalpapers[14,15],hasformalized the necessary and sufficient condition for the previously introduced model (7) tobe stable : the matrixmadeofthe L1 normofthe elementsofΦ shouldhave eigenvalues whose modulus are strictly smaller than 1. This condition can be expressed in terms of conditions on the L1 norm of the different kernels : Proposition 1 (Stability Condition) The hawkes process P is stable if and t only if the following condition holds : (H) The eigenvalues of the matrix Φ have a modulus strictly smaller than 1. 0 b 9 In that case, P has stationary increments and the process λ is strictly station- t t ary. Moreover (H) holds if and only if c+ <(1 a+)(1 b+) and a+, b+ <1, (13) − − where a+ =φT,s+φT,c, • 0 0 b+ =φN,s+φN,c and • b0 b0 c+ =(φF,s+φF,c)(φI,s+φI,c). • b 0 b0 0 0 Moreover (13) implies that b b b b ∆φT∆φN 1<∆φF∆φI <(1 ∆φT)(1 ∆φN), (14) 0 0 − 0 0 − 0 − 0 where we used Notations 4 b b b b b b The proof is in Section 9.1. Letuspointoutthatinthecasethereisnofeedbackofthepricejumpsonthe trades, i.e., ΦF = 0 (or c+ = 0), then the stability condition (13) is equivalent to a+ <1 and b+ <1, i.e., φT,s + φT,c <1 and φN,s + φN,c <1. || || || || || || || || The mean intensity vector is given by the following Proposition. Proposition 2 (Mean Intensity) We suppose that (H) holds (i.e., (13)). Then Λ=E(λt)=(I Φ0)−1M. (15) − This can be written as b ΛT =µ(I+D )(I ΦN)v (16) ΛT 0 − 0 (cid:18) (cid:19) and b b ΛN =µ(I+D )ΦIv (17) ΛN 0 0 (cid:18) (cid:19) 1 b b where v = and where D is defined by its Laplace transform 1 t (cid:18) (cid:19) Dz =((I−ΦTz)(I−ΦNz )−ΦFzΦIz)−1−I. (18) Proof : The proof is basically an adaptation of a proof previously presented in b b b b b [2]. Let the martingale dZ be defined as t dZ =dP λ dt. t t t − Using (7), we get λ =M +Φ⋆dP =M +Φ⋆dZ +Φ⋆λ dt. (19) t t t t 10 Thus (δI Φ)⋆λ =M +Φ⋆dZ . (20) t t − Consequently, λ =(δ I+Ψ)⋆M +(δ I+Ψ)⋆Φ⋆dZ , (21) t t t t where Ψ is defined by Ψ =Φ (I Φ ) 1. z z z − − Taking the expectation, we get (15). Moreover,we have b b b I ΦT ΦF I Φ= − − (22) − ΦI I ΦN ! − − b b b Using Remark 1 a the end of Section 2b.1.2, one cban easily check that I ΦN ΦF (I−Φ)−1 =(I+D) −ΦI I ΦT ! (23) − b b b b where Dt is defined by (18). The Equatiobns (16) andb (17) are direct conse- quences of this last equation combined with (15). In the following we will always consider that (H) holds, i.e., that (13) holds. 2.3 Numerical simulations In order to perform numerical simulations of Hawkes models, various methods have been proposed. We chose to use a thinning algorithm (as proposed, e.g., in [22]) that consists in generatingon [0,t ] anhomogeneousPoissonprocess max with an intensity M > supt [0,tmax](λTt±,λNt ±). A thinning procedure is then applied, each jump being acc∈epted or rejected according to the actual value of λTt± orλNt ±. Inordertoillustratethe4-dimensionalprocesswechosetodisplay only the price path Xt =Nt+−Nt− (24) and the cumulated trade process path as defined by Ut =Tt+−Tt−. (25) In Figure 1, we show an example of sample paths of both X and U on a t t few minutes time interval. All the involved kernels are exponentials. Some microstructure stylized facts of the price can be clearly identified directly on the plot : price moves arrive at random times, price moves on a discrete grid and is strongly mean reverting (see beginning of next section). In the large time limit, one can show that these processes converge to correlated Brownian motions (see [2] or Section 3.3). This is illustrated in Fig. 2 where the paths are represented over a wider time window (almost 2 hours). As discussed in Section 7.3, since we choose φT,c = 0 and φN,s = 0, the small time increments