Non-parametric Estimation of Quadratic Hawkes Processes for Order Book Events
NNon-parametric Estimation of Quadratic Hawkes Processesfor Order Book Events
Antoine Fosset , Jean-Philippe Bouchaud , and Michael Benzaquen ∗ Ladhyx, UMR CNRS 7646, Ecole polytechnique, 91128 Palaiseau Cedex, France Chair of Econophysics & Complex Systems, Ecole polytechnique, 91128 Palaiseau Cedex, France Capital Fund Management, 23-25, Rue de l’Université 75007 Paris, France
May 13, 2020
Abstract
We propose an actionable calibration procedure for general Quadratic Hawkes models of orderbook events (market orders, limit orders, cancellations). One of the main features of such modelsis to encode not only the influence of past events on future events but also, crucially, the influenceof past price changes on such events. We show that the empirically calibrated quadratic kernel iswell described by a diagonal contribution (that captures past realised volatility), plus a rank-one“Zumbach” contribution (that captures the effect of past trends). We find that the Zumbach kernelis a power-law of time, as are all other feedback kernels. As in many previous studies, the rate oftruly exogenous events is found to be a small fraction of the total event rate. These two featuressuggest that the system is close to a critical point – in the sense that stronger feedback kernels wouldlead to instabilities.
Contents ∗ Email address for correspondence: [email protected] a r X i v : . [ q -f i n . T R ] M a y Introduction
The accumulation of empirical clues over the past few years provides mounting evidence that most ofmarket volatility is of endogenous nature [ ] . This obviously does not mean that significant news,such as the very recent Covid-19 crisis, do not impact financial markets, but rather that these onlyaccount for a small fraction of large price moves. Think for example of the S&P500 flash crash ofMay 6th, 2010 [ ] , see also [ ] , which has not been triggered by any outstanding piece of news.Furthermore, while one may argue that in some cases large drops are exogenously triggered, theiramplification is often due to endogenous mechanisms [ ] .The behaviorally supported idea that agents tend to overreact, especially during crises, has driventhe market modeling community to fall back on self-exciting processes, better known as Hawkes pro-cesses [ ] . The latter have proven to be extremely efficient to tackle the intricate dynamics of the orderflow and other self-excited effects in financial markets [ ] . Nonetheless, linear Hawkes processesare unable to account for an empirical finding essential to our eyes to tackle endogenous instabilities:the Zumbach effect [ ] . The latter states that past price trends increase future activity, regardlessof their sign. Quadratic Hawkes processes (Q-Hawkes), inspired by quadratic ARCH processes [
28, 31 ] ,were recently introduced to circumvent this issue [
29, 32 ] , and have proven key to understand fat-tailsin the distribution of returns, as well as spread, volatility and liquidity dynamics [ ] .In our recent paper [ ] we indeed argued that price or spread jumps could be the result of endoge-nous feedback loops that trigger liquidity seizures, see also [ ] . In particular, we empirically showedthat Zumbach-like effects exist in order book data, i.e. past trends and volatility tend to promote futureactivity, and in particular cancellations that diminish liquidity and fragilise the system, possibly leadingto a liquidity crisis. Combining Q-Hawkes processes with a stylized order book model [
34, 35 ] revealedan interesting scenario with a second order phase transition between a stable regime for weak feed-back and an unstable regime for strong feedback, in which liquidity crises arise with high probability.However, for such a scheme to be relevant for financial markets, the system must sit very close to theinstability threshold (perhaps as the result of “self-organised criticality”). As an alternative scenario,we also proposed a non-linear Hawkes process which exhibits liquidity crises as occasional “activated”events, separating locally stable periods of normal activity.In the present paper, we calibrate on real market data a version of the generalized Q-Hawkes pro-cess proposed in our recent work [ ] . We provide convincing evidence for the price / liquidity feedbackmechanism described above and quantify its implications. In section 2 we briefly recall the ingredientsof the model and present the non-parametric calibration procedure, inspired by the methods intro-duced by Bacry et al. [ ] . We apply such calibration to order book data on the EURO STOXX andBUND futures contracts. In section 3 we present an alternative method that needs fewer assumptionsto compute the overall effect of past price moves on future liquidity flow. We introduce a low rank(Zumbach-like) approximation that allows us to denoise the feedback kernels and separate the effectsof trend and volatility, and apply it to our futures contracts. In section 4, we focus on the liquidity flowand analyse spread time series in relation with adequate trend and volatility signals. Results appearto favour the “self-organized criticality” scenario over the metastable, “activated” scenario discussedabove and in [ ] . In section 5 we conclude. We present a simplified version of the Generalized Quadratic Hawkes process (GQ-Hawkes) intro-duced in [ ] , where the influence of the size of the queues on event rates is neglected. Consider a6-dimensional process N t = (cid:0) N C,b t , N LO,b t , N MO,b t , N MO,a t , N LO,a t , N C,a t (cid:1) counting six types of order bookevents: limit orders (LO), cancellations (C), and market orders (MO), for both the bid (b) and ask (a)sides of the order book; we consider best quotes only. We further assume that the process N t is coupledto the past price process P t (cid:48) < t in the following way. Denoting λ t the intensity of the the 6-dimensional2rocess N t we let: λ t = α + (cid:90) t φ ( t − s ) d N s + (cid:90) t L ( t − s ) d P s + (cid:90) t (cid:90) t K ( t − s , t − u ) d P s d P u , (1)with φ , L and K causal decaying kernels. One can always choose K ( u , s ) = K ( s , u ) without loss ofgenerality. Note that φ is a 6 × L and K are 6-dimensional vectors.The intensity λ t is the sum of four different contributions, from left to right in the RHS of Eq. (1),one has the base rate α , the standard linear Hawkes contribution, followed by the linear and thequadratic contributions of price fluctuations. As pointed out in [
5, 29 ] , assuming that P t is a martingalemakes analytical calculations, and numerical calibration, much more congenial. Finally, assuming aswe shall do hereafter that a stationary state is reached allows us to replace the lower bound of theintegrals in Eq. (1) by −∞ . Here we introduce a non-parametric scheme to calibrate Eq. (1) to real market data. Our method is anextension of the second moment method introduced by Bacry et al. in [
36, 37 ] , see also [ ] . Before deriving the equations that will be used for the calibration, we introduce the following averagesand covariances: ∆ k d t : = (cid:69) (cid:2) ( d P t ) k (cid:3) , (2a) Λ i d t : = (cid:69) (cid:2) d N it (cid:3) , (2b) χ i jN N ( t − s ) d t ds : = Cov (cid:0) d N it , d N js (cid:1) − Λ j δ i j δ ( t − s ) d t ds , (2c) χ iN P ( t − s ) d t ds : = Cov (cid:0) d N it , d P s (cid:1) , (2d) χ iN P ( t − s ) d t ds : = Cov (cid:0) d N it , d P s (cid:1) , (2e) χ iN PP ( t − s , t − x ) d t dsd x : = Cov (cid:0) d N it , d P x d P s (cid:1) , (2f) χ P P ( t − s ) d t ds : = Cov (cid:0) d P t , d P s (cid:1) − ∆ δ ( t − s ) dsd t , (2g)where we have assumed for simplicity that the jumps of P and N are not simultaneous. Note that whileprice jumps can only occur if one order book event triggers them, the relative frequency of the latteris so much larger that this approximation is fully justified. Combining Eq. (1) with Eqs (2) yields thefollowing set of equations for the first and second moments of the processes. Introducing the notations || f || = (cid:82) (cid:82) f ( t ) d t and K d ( t ) : = K ( t , t ) the diagonal part of K , one obtains for t , x > t (cid:54) = x : Λ i = α i + (cid:88) k || φ ik || Λ k + || K i d || ∆ , (3a) χ i jN N ( t ) = Λ j φ i j ( t ) + (cid:90) (cid:82) + (cid:150)(cid:88) k φ ik ( s ) χ k jN N ( t − s ) + L i ( s ) χ jN P ( s − t ) + K i d ( s ) χ jN P ( s − t ) (cid:153) ds , + (cid:90) [ t , + ∞ [ K i ( s , u ) χ jN PP ( s − t , u − t ) { s (cid:54) = u } duds , (3b) χ iN P ( t ) = (cid:90) (cid:82) + (cid:88) k φ ik ( s ) χ kN P ( t − s ) ds + L i ( t ) ∆ + K i d ( t ) ∆ , (3c) χ iN P ( t ) = (cid:90) (cid:82) + (cid:88) k φ ik ( s ) χ kN P ( t − s ) ds + L i ( t ) ∆ + K i d ( t ) ∆ + (cid:90) (cid:82) + χ P P ( t − s ) K i d ( s ) ds , (3d) χ iN PP ( t , x ) = (cid:90) (cid:82) + (cid:88) k φ ik ( s ) χ kN PP ( t − s , x − s ) ds + ∆ K i ( t , x ) . (3e)3rovided the number of events generated by price fluctuations is small compared to that generated bythe linear Hawkes contribution, i.e. (cid:80) i , k || φ ik || Λ k (cid:29) (cid:80) i || K i d || ∆ , Eq. (3b) conveniently simplifies to: χ i jN N ( t ) = Λ j φ i j ( t ) + (cid:88) k (cid:90) (cid:82) + φ ik ( s ) χ k jN N ( t − s ) ds . (4)This approximation is relatively well supported by real data for short enough times (see below). It isessential at this stage as it allows us to decouple the estimation of the Hawkes kernel from that of L and K : one can first estimate φ from Eq. (4) and then compute L and K from Eqs. (3c), (3d) and (3e). Thebase rate is finally obtained from Eq. (3a). Note that while in principle an exact calibration of Eqs. (3)is possible, it does not perform well on real data – but see section 3 below. In section 2 we stressed that the point process P t needs to be a martingale for Eqs. (3) to be valid. Yet, itis well established that the mid-price in financial markets displays substantial mean-reversion at shorttimescales. To circumvent this issue we consider the volume weighted mid-price, sometimes called the micro-price , P micro t , known to be closer to a martingale at high frequency [
39, 40 ] . It is defined as: P micro t = v a t b t + v b t a t v a t + v b t , (5)where v b t , v a t denote the available volume at the best bid b t and ask a t respectively. To enforce furtherthe martingale property we use the so-called surprise price, that we shall henceforth denote by P t , andwhich consists in subtracting to the price its (linear) statistical predictability. Mathematically speaking,this reads: d P t = d P micro t − (cid:90) t − −∞ ρ P ( t − s ) d P micro s , (6)where ρ P ( t − s ) : = Cor (cid:0) d P micro t , d P micro s (cid:1) denotes the price auto-correlation function.We also note that the intensity of order book events exhibit an intraday U-shape, very much like thewell known U-shaped volatility pattern. Computing the total intensity of events Λ tot = (cid:80) i Λ i over 5-min bins and averaging over trading days, a U-shape is clearly visible. To avoid spurious effects relatedto these intraday seasonalities, we rescale time flow by this average pattern to enforce a constant rateof events in the new time variable.In order to estimate the kernels from real order book data, one must choose a time grid t Hn withweights w Hn for kernel φ , such that || φ || ≈ (cid:80) n φ ( t Hn ) w Hn . We decide to use quadrature points [ ] toensure a good approximation of the integrals with a minimal number of points. Further, given that weexpect power-law kernels, see e.g. [
18, 29, 37 ] , we choose a linear scale at short times that switchesto logarithmic at longer times. Finally, given that typical timescales are usually quite different (seebelow), we choose a different time grid t n , w n for the kernels L and K . See Appendix A for moredetails.Finally, the empirical covariances are usually very noisy, so we choose to smooth them using aconvenient fitting function in order to obtain better behaved kernels. Concerning the volatility covari-ance χ P P , it is found to behaves like a power law at large times so the chosen fitting function is A ( + t / B ) − C We also fit the logarithm of χ N P ( t ) , χ N P ( t ) by a polynomial in log t , and smooth theoff-diagonal kernel, see 3.2 for details. Plots of the “raw” kernels obtained without smoothing fits areprovided in Appendix. B. Apart from being more noisy, as expected, these raw kernels are very similarto the smoothed ones. More refined definitions of the micro-price, even closer to a martingale, are discussed in [ ] . For the EUROSTOXX, the fitting parameters are found to be A = × − $ s − , B =
81 s and C = i d C b i d L O b i d M O a s k M O a s k L O a s k C bid Cbid LObid MOask MOask LOask C 10 Figure 1:
Norms of the Hawkes kernel || φ i j || for the EURO STOXX futures contract between 2016 / /
12 and 2020 / / The calibration recipe then amounts to the following steps. • Compute the surprise price from the micro-price using Eqs (5) and (6). • Rescale time by the typical daily pattern of Λ tot = (cid:80) i Λ i . • Estimate ∆ k , Λ and the covariances χ P P , χ N N , χ N P , χ N P and χ N PP from the data using Eqs (2), • Use adequate fitting functions to smooth the empirical covariances (optional), • Discretise and solve Eq. (4) to obtain the Hawkes kernel φ , • Discretise and solve Eqs. (3c), (3d), and (3e) to obtain the kernels L and K , • Discretise and solve Eq. (3a) to obtain the base rate α .Further details on how to solve these equations in practice are provided in Appendix A. We now apply the calibration procedure presented above to the EURO STOXX futures contract in theperiod 2016 / /
12 to 2020 / /
07. For this contract, the average time between two order book eventsis τ e ≈ s , two orders of magnitude below the average time between two price changes τ P ≈ s ,indicating that the range of the kernels L and K is likely to be greater than that of φ , and allowingone to choose discretisation time grids accordingly. We also apply the procedure to the BUND futurescontract but do not show all the (redundant) results for the sake of readability; summarising resultsare displayed in Fig. 5 and Tables 1, 2 and 3.As specified in section 2.2.2, we start with the calibration of the Hawkes kernel φ . The resultsare displayed in Fig. 1 for the norms of the kernels, and in Fig. 7 in the Appendix for the full time-dependence. The temporal decay of the kernels appears to be power law with exponent ≈ − [
18, 29, 37 ] .The calibration leads to a stable Hawkes process with spectral radius of || φ || (computed over 1000s)found to be ≈ ≈ [
10, 11 ] . The results showthat the expected bid-ask symmetry holds with a high level of accuracy (see [ ] ), such that one canaverage the kernels accordingly to improve the statistics without loss of information.Plugging the obtained Hawkes kernels into Eqs. (3c), (3d) and (3e) allows us to calibrate the kernels L and K , see Fig. 2. Again the expected bid-ask symmetry properties hold rather well: while the linearkernel L is anti-symmetric (the effect of the positive trend on the bid is the same as that of a negativetrend on the ask), the quadratic kernel K is bid-ask symmetric. We will therefore not distinguish furtherbid and ask events in the following.Figure 2(c) shows that the quadratic contribution cannot be reduced to the diagonal part K d only.Indeed, the off-diagonal contribution of the kernel is non-zero and rather long-ranged. The decay ofthe diagonal contribution is a power law with exponent ≈ −
1. Such a decay is very slow and meansthat || K d || is logarithmically sensitive to long timescales, for which we do not have much information5 t L (a) t d i a g K - 1 (b) MOLOC10 t x (c) LO t C t MO K Figure 2:
Kernels resulting from the non-parametric calibration on the EURO STOXX futures contract between 2016 / / / /
07. (a) Linear kernels L . Note that the sign is such that an up (resp. down) trend increases all the event ratesat the bid (resp. ask) at short times. (b) Diagonal of quadratic kernels K d . (c) Full quadratic kernels K ( t , x ) . since we only use data belonging to the same trading day to avoid the thorny discussion of overnighteffects and how to treat them.Finally, while the Hawkes and price feedback effects are difficult to compare as they do not oper-ate on the same timescales, one can argue that the approximation presented at the end of Sec. 2.2.1is well supported by data: considering a cut-off of 1000 seconds to compute the norms, one finds: (cid:80) i || K i d || ∆ / (cid:80) i , k || φ ik || Λ k ≈ (cid:80) i || K i d || ∆ , which must be compared to the to-tal activity (cid:80) i Λ i . The ratio of these two quantities is found to be 5% for the EURO STOXX and 7% forthe BUND (see Table 3 for more details). Although not dominant, this feedback is clearly not negligible.Together with the standard Hawkes contribution, this means that the exogenous contribution α to thetotal activity is only 19% of the total for the EURO STOXX (17% for the BUND). Note that this fractionis expected to decreases further as the upper cut-off of the slowly decaying kernels is extended beyond1000 seconds (see e.g. [ ] ). Here we present a framework which improves the above calibration in a threefold manner. As we shallsee, (i) it allows to circumvent the approximation given in Eq. (4) which, we recall, is not perfectlysatisfied by real data, (ii) it helps cleaning further the noisy off-diagonal contribution of the quadratickernel, and (iii) it gives a more relevant measure of the global effect of price fluctuations on event rateswith no longer having to consider, nor calibrate, the Hawkes contribution.
Using the resolvent method, see [
37, 41 ] , one can rewrite Eq. (1) as: λ t = ( I − || φ || ) − α + (cid:90) t −∞ (cid:82) ( t − s ) d M s + (cid:90) t −∞ ¯ L ( t − s ) d P s + (cid:90) t −∞ (cid:90) t −∞ ¯ K ( t − s , t − u ) d P s d P u , (7)with M a martingale satisfying d M t = d N t − λ t d t , (cid:82) = (cid:80) n ≥ φ ∗ n the resolvent, ¯ L = L + (cid:82) ∗ L and ¯ K ( t , s ) = K ( t , s ) + (cid:82) + ∞ (cid:82) ( u ) K ( t − u , s − u ) du . The kernels ¯ L and ¯ K account for the overall feedback6 t L (a) t d i a g K - 1 (b) MOLOC10 t x (c) LO t C t MO K Figure 3:
Effective kernels resulting from the simplified calibration on the EURO STOXX futures contract between2016 / /
12 and 2020 / /
07. (a) Linear kernels ¯ L . Note that the sign is such that an up (resp. down) trend increasesall the event rates at the bid (resp. ask) at short times. (b) Diagonal of quadratic kernels ¯ K d . (c) Full quadratic kernels ¯ K ( t , x ) . effect of P t , including all subsequent Hawkes self-excited events that are induced by price fluctuations.The remarkable property of such kernels is that they solve a much simpler set of equations: χ iN P ( t ) = ¯ L i ( t ) ∆ + ¯ K i d ( t ) ∆ (8a) χ iN P ( t ) = ¯ L i ( t ) ∆ + ¯ K i d ( t ) ∆ + (cid:90) (cid:82) χ P P ( t − s ) ¯ K i d ( s ) ds (8b) χ iN PP ( t , x ) = K i ( x , t ) ∆ , (8c)where we have again enforced that ¯ K is symmetric. The results obtained from the inversion of Eqs (8)for the EURO STOXX futures contract are displayed in Fig. 3. These lead to similar, though slightlycleaner, conclusions to Fig. 2. In particular, the values of (cid:80) i || ¯ K i d || ∆ are compatible with those obtainedabove (taking into account the 1 − || φ || factor, see Table 3). Here we further dissect the results of the calibration presented in the previous section, with the objectivein particular of separating the contributions of trend and of volatility to the quadratic feedback. Ameaningful approximation for the quadratic kernel ¯ K was introduced in [ ] , as the sum of a purelydiagonal matrix and a rank-one contribution: ¯ K i ( t − s , t − u ) : = ¯ K i d ψ i ( t − s ) { s = u } + ¯ K i Z i ( t − s ) Z i ( t − u ) . (9)The first term on the right hand side of Eq. (9) reflects feedback of past volatility on current order bookevents. Its contribution in Eq. (7) can indeed by written as: (cid:2) σ i ( t ) (cid:3) : = (cid:90) t ψ i ( t − s ) ( d P s ) , (10) The the slight abuse of notation here since the diagonal part of ¯ K ( s ) is in fact ¯ K d ψ ( s ) + ¯ K Z ( s ) . t Z - 1 (a) t - 1 (b) MOLOC
Figure 4:
Zumbach approximation of the effective kernel ¯ K on the EURO STOXX futures contract between 2016 / /
12 and2020 / /
07. (a) Zumbach kernel Z , (b) Volatility kernel ψ . Both kernels are normalised such that || ψ || = || Z || =
1, with acut-off in the time integrals at 1000 secs.
The second term is in turn a reflection of the effect of past trends, as measured in Eq. (7) by [ µ i ( t )] ,where: µ i ( t ) : = (cid:90) t Z i ( t − s ) d P s . (11)This last term is reminiscent of the so-called Zumbach effect: past trends, regardless of their sign, leadto an increase in future activity. An altenative interpretation is that [ µ i ( t )] is a local measure of a low-frequency volatility, to be contrasted with [ σ i ( t )] which is a local measure of high-frequency volatility.Note that the kernels ψ and Z are normalised: (cid:90) ψ i ( s ) ds = (cid:90) Z i ( s ) ds =
1, (12)such that the overall strength of the volatility contribution is ¯ K d while that of the trend contribution is¯ K .While in practice such an approximation is of course not perfect, one can check that including higherrank contributions is unessential as the latter do not carry much additional signal. The rank-one kernelis obtained by minimizing (cid:115) (cid:0) ¯ K i ( s , u ) − ¯ K i Z i ( s ) Z i ( u ) (cid:1) { u (cid:54) = s } dsdu , which consists in finding the firsteigenvector of a well chosen linear map, see [ ] for more details. The ψ contribution is then obtainedby taking the diagonal of ¯ K i and subtracting ¯ K i Z i ( t ) . Figure 4 displays the kernels φ and Z as functionof time for the EUROSTOXX futures contract. As one can see, while the volatility kernel decays roughlyas 1 / t , although some curvature can be observed. The Zumbach counterpart decays as 1 / t , regardlessof event types (by that justifying the choice made in Fosset et al. [ ] , where the same functional formfor all event types was assumed). So far we have focused on the impact of past price moves one event rates. Here we wish to go on stepfurther and estimate the effect of past price changes on liquidity, i.e. volume weighted events. For thisone needs to consider order volumes. The average volumes are given in Tab. 1 for the different types oforders. Assuming bid / ask symmetry (consistent with the empirical results), Fig. 5 displays the amount V C,b V LO,b V MO,b V MO,a V LO,a V C,a
EUROSTOXX 10.1 9.2 7.2 8.2 9.2 10.0BUND 4.5 4.8 4.4 4.2 4.8 4.5
Table 1:
Average order volumes (in shares). of shares per second that can be attributed to the quadratic effect (both volatility and Zumbach) for8 r K K d EUROSTOXX K Tr K K d BUND K × V t o t a l c o n t r i b u t i o n o f o r d e r s (a) Tr K K d EUROSTOXX K Tr K K d BUND K × V c o n t r i b u t i o n i n % o f o r d e r t y p e s (b) C MO LO E U R O S T O XX B U N D E U R O S T O XX B U N D q u a d r a t i c li q u i d i t y f l u x : J K (c) c o n t r i b u t i o n i n % t o J K K K d Figure 5:
Average quadratic contribution on the EURO STOXX and BUND futures contracts between 2016 / /
12 and2020 / /
07. (a) ∆ (cid:80) i V i || ¯ K i || and its decomposition into ∆ (cid:80) i V i ¯ K i d and ∆ (cid:80) i V i ¯ K i . (b) Contributions of each ordertype to the latter quantities. (c) Overall contribution of the quadratic effect to the liquidity flow J ¯ K (in shares per second),and relative contribution of the volatility and Zumbach terms. each event type, namely ¯ K i d V i ∆ and ¯ K i V i ∆ where ¯ K i d , ¯ K i are obtained as explained in the previoussection, V i are given in Tab. 1, and ∆ is defined in Eq. (2a). Introducing the overall average quadratic liquidity flux as: J ¯ K : = (cid:0) || ¯ K LO || V LO − || ¯ K C || V C − || ¯ K MO || V MO (cid:1) ∆ , (13)one consistently finds that the quadratic (price) feedback has an overall negative effect on liquidity J ¯ K <
0, most of it associated to volatility, see Fig. 5(c). In other terms, the quadratic feedback tendsto decrease liquidity on average. Figure 5(b) shows that both the volatility and Zumbach terms havean average negative impact on liquidity (i.e. the green bars represent less than 50% of the total contri-bution). The Zumbach term is responsible for non-trivial long-range liquidity anomalies. In particular,Blanc et al. [ ] showed that the price process resulting from a quadratic Hawkes process follows isdiffusive with fat tailed stochastic diffusivity at large times, which can be attributed to the Zumbacheffect, rather than its volatility counterpart (see also the discussion in [ ] ). In any case, we believethat the quadratic feedback of price trends on order book events is a crucial ingredient to understandliquidity crises. In the next section we provide a direct test of this hypothesis. With the aim of making contact with our previous work [ ] , we now focus on the analysis of spreaddynamics. Since the EUROSTOXX futures is a large tick contract (the spread is equal to one over 99%of the time and seldom higher than two), we characterize the dynamics of liquidity using an effectivespread S eff t which is defined as follows. Calling v a t ( x ) ( resp v b t ( x ) ) the ask ( resp bid) volume at pricelevel x , we construct cumulative volumes as Q a t ( x ) = (cid:80) n ≤ x v a t ( n ) and Q b t ( x ) = (cid:80) n ≥ x v b t ( n ) . We thenchoose the average volume at best V best as a reference volume, and define: S eff t : = (cid:0) Q a t (cid:1) − ( V best ) − (cid:0) Q b t (cid:1) − ( V best ) , (14)where (cid:0) Q a / b t (cid:1) − denotes the inverse function of Q a / b t . The effective spread is a natural proxy for liquidityin the close vicinity of the midprice: when the liquidity is close to its average, the effective spreadcoincides with the regular spread; but when liquidity is low, it can be much larger as aggregatingthe volume of several queues is needed to recover the reference volume V best . Figure 6(a) displaysthe survival function of the effective spreads, revealing that (cid:80) ( S eff ) ∼ ( S eff ) − . This power-law tail isinteresting for the following reason: the effective spread can be seen as a proxy for the size of latentprice jumps, i.e. the jumps that are likely to happen if an aggressive market order hits the market. The normalisation of all kernels is computed with a time cut-of at 1000 seconds. Note that the linear terms give no net contribution, i.e. V LO || ¯ L LO || − V C || ¯ L C || − V MO || ¯ L MO || ≈
0, which explains why wefocus on the quadratic term). In other words, the trend has almost no linear effect on the liquidity flux at large time scales. Changing the reference volume to 2 V best or V best / S eff s u r v i v a l f un c t i o n - 4 (a) C o r (( + ) , S e ff ()) (b) C o r (( + ) , S e ff ()) (c) C o r (( + ) , S e ff ()) (d) Figure 6: (a) Survival function of the effective spread, showing that (cid:80) ( S eff > S ) ∼ S − (b) Correlations between effectivespread and past square trends (for τ <
0) and future square trends (for τ > (cid:84) : the ratio square trend over square volatility. EUROSTOXX futures contracts between 2016 / /
12 and 2020 / / Hence, one expects the distribution of effective spread is not far from the distribution of price returns r , which is well known to decay as (cid:80) ( r ) ∼ r − .Let us now study the relation between effective spread, square volatility σ and square trend µ ,as defined in Eqs. (10) and (11). Figures 6(b), (c) and (d) display the correlation functions C µ ( τ ) : = Cor (cid:2) µ ( t + τ ) , S eff ( t ) (cid:3) , C σ ( τ ) : = Cor (cid:2) σ ( t + τ ) , S eff ( t ) (cid:3) and C (cid:84) ( τ ) : = Cor (cid:2) (cid:84) ( t + τ ) , S eff ( t ) (cid:3) respec-tively, with (cid:84) = µ /σ . Note that a causal positive impact of past trends on future spreads shouldtranslate as a strong contribution to C µ ( τ ) for negative τ . Interestingly, this is compatible with Fig. 6(b),which confirms in a model-free fashion that the Zumbach-like coupling is important: past square trendsincrease future effective spread, or equivalently decrease future liquidity. While also slightly asymmet-ric, the volatility / spread correlation C σ ( τ ) does not reveal such a level of asymmetry (see Fig. 6(c)).Fig. 6(d) shows an even more pronounced asymmetry when we rescale the trend by the local volatility: (cid:84) is a proxy of the autocorrelation of returns, independently of their amplitude. In this sense, it is abetter signature of trend behaviour, as the volatility aspect of recent price changes is discarded. In this work, we have proposed several actionable procedures to calibrate general Quadratic Hawkesmodels for order book events (market orders, limit orders, cancellations). One of the main featuresof such models is to encode not only the influence of past events on future events but also, crucially,the influence of past price changes on such events. We have shown that the empirically calibratedquadratic kernel (describing the part of the feedback that is independent of the sign of past returns) iswell described by the shape postulated in [
5, 29, 32 ] , namely: • a diagonal contribution that captures past realised volatility, and • a rank-one contribution that captures the effect of past trends.The latter contribution can be interpreted as the microstructural origin of the Zumbach effect: pasttrends, independently of their sign, tend to reduce the liquidity present in the order book, and thereforeincrease future volatility. As we have shown in our companion paper [ ] , such coupling can in fact bestrong enough to destabilise the order book and lead to liquidity crises.One of the perhaps unexpected result of our calibration is that the Zumbach kernel is found to bea power-law of time for the futures contracts studied here, and not an exponential as was found in [ ] for US stock prices. Hence, all Hawkes kernels in our study are found to be power-laws of time.Furthermore, as in many previous studies [
11, 18, 25 ] , the rate of truly exogenous events is found tobe much smaller than the total event rate, typically 1 / [
28, 29 ] ). Such a power-law is not compatible with the10lternative “activated” scenario proposed in [ ] , which would rather suggest a bimodal distributionwith a hump at large effective spreads. Hence, we favour at this stage the scenario of markets poisedclose to a point of instability, although the detailed mechanisms that lead to such a fine tuning are stillsomewhat obscure. We note that the near-criticality has also been argued to be crucial to understand the“rough” nature of volatility [
32, 43, 44 ] . We believe that understanding these mechanisms is probablyone of the most intellectually challenging (and exciting) issue for microstructure theorists. Acknowledgments
We thank Jonathan Donier, Iacopo Mastromatteo, José Moran, Mehdi Tomas, Stephen Hardiman andMathieu Rosenbaum for fruitful discussions. This research was conducted within the
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Availableat SSRN 3180582 , 2018. Appendix: Estimation procedure
Here we show how to practically estimate the kernels presented in section 2.2 from empirical data. First, wedetail the empirical estimators for averages and covariances, then focus on the time grids used for estimation,and finally discuss the numerical discretisation of Eqs. (3).
Covariance estimators
We assume that we have a sample of events of type i that happen at times (cid:0) T in (cid:1) n , with i = P for the price process. Calling T the total length of observation, the estimators of the average intensitiesread: Λ i ≈ N iT T (15a) ∆ k ≈ T (cid:88) n (cid:128) ∆ T Pn (cid:138) k . (15b)For the covariance estimators, we use a classical approach for asynchronous data. Denoting ∆ t , ∆ x the timesteps associated with times t and x , one has: χ i jNN ( t ) ≈ T ∆ t (cid:88) n , p { T in − T jp ∈ [ t − ∆ t / t + ∆ t / ] } − Λ i Λ j (16a) χ iN P ( t ) ≈ T ∆ t (cid:88) n ∆ T Pp { T in − T Pp ∈ [ t − ∆ t / t + ∆ t / ] } (16b) χ iN P ( t ) ≈ T ∆ t (cid:88) n , p (cid:128) ∆ T Pp (cid:138) { T in − T Pp ∈ [ t − ∆ t / t + ∆ t / ] } − Λ i ∆ (16c) χ iN PP ( t , x ) ≈ T ∆ t ∆ x (cid:88) n , p , q ∆ T Pp ∆ T Pq { T in − T Pp ∈ [ t − ∆ t / t + ∆ t / ] , T in − T Pq ∈ [ x − ∆ x / x + ∆ x / ] } (16d) χ P P ( t ) ≈ T ∆ t (cid:88) n , p (cid:128) ∆ T Pn (cid:138) (cid:128) ∆ T Pp (cid:138) { T Pn − T Pp ∈ [ t − ∆ t / t + ∆ t / ] } − ∆ . (16e)Note that, as mentioned above, one can choose different time grids for the Hawkes and price contributions.Symmetry properties of the covariances enable us to estimate them only for positive times: • χ i jNN ( − t ) = χ jiNN ( t ) • χ iN P ( t ) = χ iN P ( t ) = t < • χ iN PP ( t , x ) = ( t , x ) < • χ P P ( − t ) = χ P P ( t ) .One can reasonably assume that the covariances are (cid:67) except in zero. Choice of time grids
A good choice of time grid to estimate the kernels is provided in [ ] . Indeed, quadra-ture points in log-scale are well suited to accurately account for long range behaviour in the norm of the kernels.Consistently, it is advised to have time intervals increasing at the same rate as the grid of points we use. Onthe other hand, taking disjoined intervals [ t − ∆ t / t + ∆ t / ] enables fast computations of the covariances. Toenforce all of this, we compute the differences between the quadrature points, sort them and take the cumulativesum. This gives the disjoined time intervals suited for fast computations. Then, with linear interpolation, weobtain the final values on the quadrature points. Discretisation
Equations (3) can be discretised in two different ways, using properties of the covariances andtime grids. To show how to approximate the integrals, we provide an example of discretisation of (cid:82) (cid:82) + f ( s ) ds foran arbitrary function f using the time grid ( t n ) . The two possibilities are: • The quadrature technique: (cid:82) (cid:82) + f ( s ) ds ≈ (cid:80) n f ( t n ) w n . • The piece-wise (cid:67) approximation: (cid:82) (cid:82) + f ( s ) ds ≈ (cid:80) n t n + − t n (cid:2) f (cid:0) t + n (cid:1) + f (cid:0) t − n + (cid:1)(cid:3) , with f ( x + ) = lim y → xy > x f ( y ) and f ( x − ) = lim y → xy < x f ( y ) .The first approximation is very efficient to compute Tr K or || φ || using ( t hn ) and ( w hn ) . The second handles verywell the behavior around zero and can be useful to solve Eq. (4). Additional plots and tables b i d C - 3/210 b i d L O b i d M O a s k M O a s k L O t bid C10 a s k C t bid LO 10 t bid MO 10 t ask MO 10 t ask LO 10 t ask C Figure 7:
Hawkes kernels for the EURO STOXX futures contract between 2016 / /
12 and 2020 / /
07 ( t in seconds). t L (a) t d i a g K - 1 (b) MOLOC10 t x (c) LO t C t MO K Figure 8:
Raw effective kernels resulting from the calibration on the EURO STOXX futures contract between 2016 / / / /
07, without any smoothing procedure – compare with Fig. 3. (a) Linear kernels ¯ L . (b) Diagonal of quadratickernels ¯ K d . (c) Full quadratic kernels ¯ K ( t , x ) . t Z - 1 (a) t - 1 (b) MOLOC
Figure 9:
Zumbach approximation of the effective kernel ¯ K on the EURO STOXX futures contract between 2016 / /
12 and2020 / /
07 – without any smoothing procedure – compare with Fig. 4. (a) Zumbach kernel Z , (b) Volatility kernel ψ . Bothkernels are normalised such that || ψ || = || Z || =
1, with a cut-off in the time integrals at 1000 secs. LO MO V Tr ¯ K ∆ EUROSTOXX 20.4 18.8 2.1BUND 5.7 6.6 1.7 V ¯ K ∆ EUROSTOXX 9.7 8.6 0.5BUND 3.5 4.6 0.7 V ¯ K d ∆ EUROSTOXX 10.7 10.1 1.6BUND 2.2 2.1 1.0
Table 2:
Quadratic, Zumbach and volatility contributions to the liquidity rate of events (in shares per second). t (s) C LO MO α i / Λ i
10 0.25 0.14 0.29100 0.24 0.14 0.271000 0.23 0.13 0.27 ∆ Tr K i / Λ i
10 0.04 0.03 0.03100 0.05 0.03 0.041000 0.06 0.04 0.05 ∆ Tr ¯ K i / Λ i
10 0.15 0.16 0.14100 0.23 0.23 0.211000 0.28 0.28 0.24 α i / (cid:80) j || φ i j || Λ j
10 0.35 0.17 0.42100 0.34 0.17 0.401000 0.32 0.16 0.40 ∆ K i / (cid:80) j || φ i j || Λ j
10 0.06 0.05 − − − ∆ K i d / (cid:80) j || φ i j || Λ j − − − ∆ ¯ K i / Λ i
10 0.13 0.13 0.05100 0.13 0.13 0.051000 0.13 0.13 0.05 ∆ ¯ K i d / Λ i
10 0.02 0.03 0.09100 0.10 0.11 0.161000 0.15 0.15 0.19
Table 3:
Different ratios between the quadratic contributions, base rates and Hawkes contributions, truncated at differenttime scales t ( s ) for the EURO STOXX futures contract between 2016 / /
12 and 2020 / /
07. The top three entries are themost important ones. For sake of simplicity, we have here approximated K as ( − || φ || ) ¯ K and K d as ( − || φ || ) ¯ K d ..