Modelling order arrivals at price limits using Hawkes processes

ARTICLE IN PRESS

JID: FRL

[m3Gsc;August 25, 2016;13:31]

Finance Research Letters 0 0 0 (2016) 1–6

Contents lists available at ScienceDirect

Finance Research Letters journal homepage: www.elsevier.com/locate/frl

Modelling order arrivals at price limits using Hawkes processes Afshin Haghighi∗, Saeid Fallahpour, Reza Eyvazlu Faculty of Management, University of Tehran, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 29 June 2016 Revised 29 July 2016 Accepted 22 August 2016 Available online xxx JEL Classification: C32 C51 G14

a b s t r a c t Some financial market regulators utilize a price limit mechanism. A number of past studies show that the price limit mechanism has a considerable impact on investors’ behaviour. The altered mechanism per se, and its impact on investors’ behaviour, change the order flow dynamics at price limit hits. We have proposed a model using Hawkes processes to model order arrivals when market dynamics switch to price limit hits. Goodness of fit tests showed that the model appropriately captures order arrival dynamics of intraday data from the Tehran Securities Exchange (TSE), which is a volatile market with narrow banded price limits (±4). © 2016 Elsevier Inc. All rights reserved.

Keywords: Price limits Hawkes processes Order flow High frequency modelling

1. Introduction Modelling microstructure order flow is of great interest to researchers and practitioners. In particular, the former type of participant is interested in capturing the dynamics of orders and trades to understand the source of volatility and price formation. The latter may use it for practical utilization like optimal order execution. A number of different types of order flow models are proposed in the various studies (e.g., Cont et al., 2010; Farmer et al., 2005; Ng, 2008), but recently utilizing Hawkes processes in modelling high frequency dynamics of order flow is trending. Hawkes processes are a type of point processes that allow modelling in physical time, and interpret interrelation and correlation between events as cross- and self-exciting behaviour. These processes, first introduced by Hawkes (1971), have been adopted in financial applications (see Bacry et al., 2015 for review) especially order arrival modelling (Bacry and Muzy, 2014; Hewllet, 2006; Toke, 2011). Although order driven financial markets over the world have similar properties, some differences need to be considered in an order flow model. These differences could be in regulations and their effects on investors’ behaviour. In some financial markets, regulators establish limits on instruments’ prices in each trading session to decrease volatility caused by investors’ overreaction, and to calm the market by limiting irrational behaviours. The mechanism of trading would change after price limit hits.

∗

Corresponding author. E-mail address: [email protected] (A. Haghighi).

http://dx.doi.org/10.1016/j.frl.2016.08.012 1544-6123/© 2016 Elsevier Inc. All rights reserved.

Please cite this article as: A. Haghighi et al., Modelling order arrivals at price limits using Hawkes processes, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.012

ARTICLE IN PRESS

JID: FRL 2

[m3Gsc;August 25, 2016;13:31]

A. Haghighi et al. / Finance Research Letters 000 (2016) 1–6

Figure 1. Illustration of order placement at upper price limit. Mechanism of orders at lower price limit is similar.

The effectiveness of price limits is a controversial topic in the academic literature. Researchers have challenged this regulatory tool by empirical research. Kim and Rhee (1997) introduced three primary costs of price limits consisting of delayed price discovery, volatility spillover hypothesis and trading interference hypothesis. They studied hit events on the Tokyo Stock Exchange and compared stocks that hit price limits against those stocks that almost hit price limits. Many empirical studies also support their findings in different markets. Those works include Bildik and Gülay (2006); Chou et al. (2005); Henke and Voronkova (2005), and Kim and Limpaphayom (2000). The magnet effect of price limits is another topic that research studies have focused on. Price limits act as magnets when traders think a price limit hit is going to occur, they buy or sell before the limit hit occurrence and this behaviour leads prices to limits. Empirical studies supported this effect (Abad and Pascual, 2007; Cho et al., 2003; Hsieh et al., 2009; Yan Du et al., 2009). Hence altering the mechanism of trade and also changing investor behaviour in the case of price-limit hit occurrence necessitates switching to a new model when modelling high frequency order arrivals. In this paper, we propose a model using multivariate Hawkes process to address this need and apply the model on two stocks on the Tehran Securities Exchange. In Section 2, we explain the mechanism of price limits and the model. In Section 3, we explain the data, the numerical estimation method and goodness of fit computations of the model. Section 4 presents the results and discussion and Section 5 concludes the paper.

2. Mechanism of price limits and the model 2.1. Mechanism of price limits There is a regulation on the Tehran Securities Exchange (TSE) such that prices are not allowed to go above upper price limits or below lower price limits. The price limits are arranged each day at ± 4% of the closing price of the last trading session for each individual stock during the empirical data period. When the market is volatile, the price moves to the price limits, after that, orders in the same direction of movement can only be placed at the price limit as limit orders. Limit orders at the price limits forms a queue of orders. These orders could match with opposite direction incoming market orders in the sequence of placement time. Please cite this article as: A. Haghighi et al., Modelling order arrivals at price limits using Hawkes processes, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.012

ARTICLE IN PRESS

JID: FRL

[m3Gsc;August 25, 2016;13:31]

A. Haghighi et al. / Finance Research Letters 000 (2016) 1–6

3

As illustrated in Fig. 1, when the market mechanism switches to the price limit, considering the direction of price movement as a gauge, there are three events that can occur: (1) same direction orders (limit orders at price limit) (2) cancellation of limit orders (3) opposite direction orders (market orders). These three events are studied in the proposed model. As we mentioned, when the price approaches the price limits, the magnet effect has been seen in empirical research studies. In order to consider this behaviour, we treat orders at one tick to limits as a same direction order event (event 1), i.e. market orders in the same direction of price movement at a distance of one tick to price limits. Similarly, opposite direction limit and market orders at price limits or at the distance of one tick to price limit are considered as an opposite direction order event (event 3). 2.2. The model Hawkes processes were first introduced by Hawkes (1971). A multivariate Hawkes process is a counting-process Nt such that the intensity vector can be written as:

λti = μi +

D  

φ i j (t − s )dNsj ,

(1)

i=1

where the quantity {μi }D is a vector of exogenous intensities, and {∅i j (t )}D is a matrix-valued decay kernel. i=1 i, j=1

The proposed model is a three-dimensional Hawkes process in which Nt1 , Nt2 , Nt3 accounts for same direction order, cancellation and opposite direction order events, respectively. The intensity function of each component is as below:

λti = μi +



φ i1 (t − s )dNs1 +



φ i2 (t − s )dNs2 +



φ i3 (t − s )dNs3 ,

(2)

where μi is a constant background rate for component i, which is related to exogenous source rather than endogenous interaction of order arrival dynamics. The intensity function of each component is conditional on past events. In other words each event that occurred before time t, has an impact on the intensity of the events at time t. This impact is measured by decay kernels which take into account the impact of each event on another i.e. φ i j (t − s ) takes into account the impact of event j that occurred at time s (s < t) on the intensity of event i at time t.  is a matrix illustration of decay kernels:



=

φ 11 φ 21 φ 31

φ 12 φ 22 φ 32

φ 13 φ 23 φ 33



.

(3)

φ 11 , φ 22 , φ 33 are self-exciting kernels and account for the impact of each event on itself and can interpret behaviours like herding and splitting. Since the orders that are close to price limits are also considered as event 1, φ 11 specifically can account for the magnet effect of price limits. Cross-exciting kernels measure the impact of the other events. For example φ 23 accounts for the impact of opposite direction orders on cancellation. Kernels of process are considered as exponential because of their advantages in computing the likelihood efficiently and the possibility of being directly simulated. The decay function is as below:

φ i j (t ) = α i j β i j e−β

ij

(t ) .

(4)

The α can be interpreted as the one setting the overall strength of impact, while the β controls the relaxation time of the perturbations induced from past to future events (Bacry et al., 2015). 3. Application to empirical data 3.1. Data We used intraday data for two of the most frequent price limit hit stocks on the Tehran Securities Exchange (TSE) with codes IKCO and PCOD. The intraday data were collected when the price limits hit during 20 March 2014 to 20 August 2014, and when trading at price limits lasted more than 30 min. Usually when price is locked at price limits, the volume of the order queue goes up, so investors become reluctant to place their orders at the end of a high volume queue. Consequently, the rate of order arrivals decreases substantially. Therefore, we considered the first 30 min events after price limit hits. The data consist of order book and trades data with a precision of 1 ms provided by Tehran Securities Exchange Technology Management Co. (TSETMC). Market order arrival times have been directly obtained using trades data. Limit orders and cancellation were extracted from quotes and trades data jointly. 3.2. Parameter estimation A maximum likelihood estimation procedure for self-exciting point processes has been developed by Ogata (1978) and Ozaki (1979). We follow the Toke and Pomponio (2012) computations according to our assumed decay kernels. Let all the Please cite this article as: A. Haghighi et al., Modelling order arrivals at price limits using Hawkes processes, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.012

ARTICLE IN PRESS

JID: FRL 4

[m3Gsc;August 25, 2016;13:31]

A. Haghighi et al. / Finance Research Letters 000 (2016) 1–6

events in each M dimension be ordered as {ti }i=1,...,N , the log-likelihood of a multivariate Hawkes process can be computed as below:





M 

ln L {ti }i=1,...,N =

lnLm ({ti } ),

(5)

m=1

where:

l nLm ({ti } ) =



T



(1 − λm (s ))ds +

0

T

l nλm (s )dNsm .

0

(6)

The partial log-likelihood can be written as: N 

lnLm ({ti } ) = T − m (0, T ) +



i zm ln

μm (ti ) +



m  

α mn β mn e−β ( mn j

ti −tkn

) .

(7)

n=1 tkn
i=1

T i is equal to 1 if the event t is of type m, 0 otherwise. The Where m (0, T ) = 0 λm (s )ds is the integrated intensity, and zm i log-likelihood could be computed in a recursive way:

R

mn

(l ) =





e

(

−β mn tlm −tkn

mn m n e−β (tl −tl−1 ) Rmn (l − 1 ) +

)=

e −β (

n tlm −tl−1

mn

tkn


mn m n e−β (tl −tk )

) (1 + Rmn (l − 1 ) )

lnL ({ti } ) = T −  (0, T ) − m

n  M 

α

mn

(1 − e

−β mn (T −ti )

)+

i=1 n=1

.

(8)

if m = n

And the final equation is as below: m

if m = n

m ≤t n


 ln

 m

μ tl m

+

m 

α β R (l ) . mn

mn mn

(9)

n=1

tlm

3.3. Goodness of fit Bowsher (2007) generalized stochastic time change theorem of point processes to multidimensional Hawkes processes

∞ and provided a framework for diagnostic tests. Let N be a point process on R+ such that 0 λ(s )ds = ∞ and let the tτ be stopping time defined by



0

tτ

λ(s )ds = τ .

(10)

˜ (t ) = N (tτ ) is a homogeneous Poisson process with constant intensity λ = 1. Hence, in a mulThen the process N tidimensional case, the durations τim − τim = m (tim , t m ) are exponentially distributed with parameter 1. To calculate −1 −1 i integrated intensity process between two consecutive events tim and tim of type m, following the numerical method in Toke −1 and Pomponio (2012), we computed them according to our assumed decay kernels:

   m tim−1 , tim =

tim

tim−1

+

λm (s )ds =

M 



tim

tim−1

μm ds +

M  

mn m n  mn m n α mn e−β (ti−1 −tk ) − e−β (ti −tk )

n=1 tkn
mn m n  α mn e−β (ti −tk ) .



(11)

n=1 tim−1 ≤tkn
That can be computed in a recursive way as follows:

Amn (i − 1 ) =



mn m n mn m m e−β (ti−1 −tk ) = e−β (ti−1 −ti−2 ) Amn (i − 2 ) +

tkn
tim

tim−1

mn m n e−β (ti−1 −tk ) ,

(12)

tim−2 ≤tkn
And we observe the final equation as:

   m tim−1 , tim =



μm ds +

M 



α mn

n=1

  mn m m Amn (i − 1 ) 1 − e−β (ti −ti−1 ) +



(1 − e

(

−β mn tim −tkn

)) .

(13)

tim−1 ≤tkn
For testing the goodness of fit of the model following Bowsher (2007), we should test that the durations (residual) are homogenous Poisson process, therefore we tested the hypotheses below: H1. durations are exponentially distributed; H2. durations are independent. We used the Kolmogorov-Smirnov test for H1 and Ljung–Box test up to the twentieth term for H2. Please cite this article as: A. Haghighi et al., Modelling order arrivals at price limits using Hawkes processes, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.012

ARTICLE IN PRESS

JID: FRL

[m3Gsc;August 25, 2016;13:31]

A. Haghighi et al. / Finance Research Letters 000 (2016) 1–6

5

Table 1 Parameter estimation by maximum likelihood method for the stocks: PKOD and IKCO. Estimated parameters PKOD

μ1

μ2

μ3

0.0177

0.0188

0.0076

α 11

α 12

α 13

α 21

α 22

α 23

α 31

α 32

α 33

0.8064

0.0 0 0 0

0.0 0 0 0

0.0223

0.0 0 0 0

0.0176

0.0091

0.0617

0.8522

0.0593

4.6401

0.2028

0.0110

7.6936

0.0930

0.0247

0.3989

0.2242

β 11

β 12

β 13

β 21

IKCO

β 22

β 23

β 31

μ1

μ2

μ3

0.0298

0.0179

0.0033

β 32

β 33

α 11

α 12

α 13

α 21

α 22

α 23

α 31

α 32

α 33

0.6178

0.0196

0.6670

0.0305

0.1408

0.0 0 0 0

0.0444

0.0380

0.6380

0.0712

0.0024

0.0075

0.0081

0.0802

0.2667

0.0298

0.0735

0.2967

β 11

β 12

β 13

β 21

β 22

β 23

β 31

β 32

β 33

Table 2 Hypothesis test results for Kolmogorov–Smirnov test compare the residual sample against the standard exponential distribution and represent the null hypothesis that the data are from the same distribution. The result H1 is 1 if the test rejects the null hypothesis at the 5% significance level, and 0 otherwise. The Ljung–Box test assesses the null hypothesis that a series of residuals exhibits no autocorrelation for 20 lags. The result H2 is 1 if the test rejects the null hypothesis at the 5% significance level, and 0 otherwise. Hypothesis test results for each dimension PKOD

IKCO

Kolmogorov–Smirnov

Ljung–Box

Kolmogorov–Smirnov

Ljung–Box

H1

p-value

Stat.

H2

P-Value

Stat.

H1

P-Value

Stat.

H2

P-Value

Stat.

0 0 0

0.1554 0.1575 0.0718

0.0559 0.1369 0.0855

0 0 0

0.0547 0.7219 0.8326

31.03 15.91 13.95

0 0 0

0.6680 0.5079 0.4845

0.0687 0.1572 0.0982

0 0 0

0.2987 0.4681 0.1108

22.81 19.83 27.94

4. Results and discussion Table 1 shows the results of the parameter estimation obtained by the maximum likelihood estimation method and numerical recursive algorithm presented in Section 3.2. As illustrated α 11 and α 33 are large for both stocks. The α 11 shows the self-exciting behaviour in same direction orders and can be interpreted as the magnet effect. It means buy (sell) order arrival at upper (lower) price limit or one tick to upper (lower) price limit would trigger more similar orders. Showing the self-exciting behaviour in opposite direction order arrivals, α 33 represents similar behaviour. The μ3 is very small for both stocks which means this component is more related to endogenous interactions specifically self-exciting behaviour. A small self-exciting behaviour in order cancellation can be seen in IKCO. Overall, the cross-exciting terms are small and the dominant behaviour is self-exciting of same and opposite direction orders. The information from the Hypothesis tests is illustrated in Table 2. We used a Kolmogorov-Smirnov two-sample test to compare the duration (residuals) against the standard exponential distribution and the null hypothesis is accepted for both stocks in all three dimensions. The p-value for PCOD is much better. The result of the Ljung–Box test shows acceptance of the null hypothesis of no auto-correlation between residuals. It is worth saying that we tested the model when we considered the full time of price limit hits instead of the first 30 min and the tests results rejected the null hypotheses. We think that may be due to the psychological effect of limit orders queue. When prices are approaching the price limits, they act as a magnet, but after hitting and when the volume of the queue goes up, the rate of incoming limit orders drops substantially. However, the model captured the dynamics of order arrival times when prices approach the price limits and when the queue of limit orders is forming for the first 30 min. In order to capture the rest of the dynamics one should take into account the volume of the limit orders queue. 5. Conclusion We proposed a model based on Hawkes processes for order arrival times when prices hit the price limits (and almost hit). Goodness of fit tests showed that the model appropriately fits the empirical data. A comprehensive order model could switch to the proposed model when prices approach the price limits. The data we used in our study were from the TSE (Tehran Securities Exchange), a volatile stock market with narrow banded price limits that trigger many price limit hits and necessitate consideration of price limits. As there are other markets Please cite this article as: A. Haghighi et al., Modelling order arrivals at price limits using Hawkes processes, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.012

JID: FRL 6

ARTICLE IN PRESS

[m3Gsc;August 25, 2016;13:31]

A. Haghighi et al. / Finance Research Letters 000 (2016) 1–6

that utilize price limits, further studies of the model could be accomplished on those markets, for example, the Shanghai Stock Exchange which has a similar mechanism. There are ways for enhancing the model like taking into account the volume of the queue of orders at price limits. New orders should be placed at the end of the queue so when the volume of the queue goes up the investors become reluctant to place new limit orders at price limits. Therefore, the rate of order arrival at high queue volume would be affected. Another way may be to take into account the volume of order using the marked point processes. Acknowledgements We thank the journal’s editors, and anonymous referee for their overall handling, overseeing, and reviewing of our paper. We also thank Bin Li for assistance and also Mahsa Baktashmotlagh, and Akihiro Omura for comments that greatly improved the manuscript, although any errors are our own and should not tarnish the reputations of these esteemed persons. References Abad, D., Pascual, R., 2007. On the magnet effect of price limits. Eur. Financ. Manage. 13, 833–852. doi:10.1111/j.1468-036X.2007.00399.x. Bacry, E., Mastromatteo, I., Muzy, J.-F., 2015. Hawkes processes in finance. Mark. Microstruct. Liq. 01. doi:10.1142/S238262661550 0 057. Bacry, E., Muzy, J.-F., 2014. Hawkes model for price and trades high-frequency dynamics. Quant. Financ. 1–20. doi:10.1080/14697688.2014.8970 0 0. Bildik, R., Gülay, G., 2006. Are price limits effective? Evidence from the Istanbul stock exchange. J. Financ. Res. 29, 383–403. doi:10.1111/j.1475-6803.2006. 00185.x. Bowsher, C.G., 2007. Modelling security market events in continuous time: Intensity based, multivariate point process models. J. Econom. 141, 876–912. doi:10.1016/j.jeconom.20 06.11.0 07. Cho, D.D., Russell, J., Tiao, G.C., Tsay, R., 2003. The magnet effect of price limits: Evidence from high-frequency data on Taiwan Stock Exchange. J. Empir. Financ. doi:10.1016/S0927-5398(02)0 0 024-5. Chou, P.H., Lin, M.C., Yu, M.T., 2005. Risk aversion and price limits in futures markets. Financ. Res. Lett. 2, 173–184. doi:10.1016/j.frl.20 05.05.0 02. Cont, R., Stoikov, S., Talreja, R., 2010. A stochastic model for order book dynamics. Oper. Res. 58, 549–563. doi:10.1287/opre.1090.0780. Farmer, J.D., Patelli, P., Zovko, I.I., 2005. The predictive power of zero intelligence in financial markets. Proc. Natl. Acad. Sci. 102, 2254–2259. doi:10.1073/ pnas.0409157102. Hawkes, A.G., 1971. Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90. doi:10.1093/biomet/58.1.83. Henke, H., Voronkova, S., 2005. Price limits on a call auction market: Evidence from the warsaw stock exchange. Int. Rev. Econ. Financ. 14, 439–453. doi:10.1016/j.iref.20 04.02.0 01. Hewlett, P., 2006. Clustering of order arrivals, price impact and trade path optimisation. Workshop on Financial. Modeling with Jump Processes. Hsieh, P.H., Kim, Y.H., Yang, J.J., 2009. The magnet effect of price limits: a logit approach. J. Empir. Financ. 16, 830–837. doi:10.1016/j.jempfin.20 09.06.0 02. Kim, K.A., Limpaphayom, P., 20 0 0. Characteristics of stocks that frequently hit price limits: Empirical evidence from Taiwan and Thailand. J. Financ. Mark. 3, 315–332. doi:10.1016/S1386-4181(0 0)0 0 0 09-4. Kim, K.K.A, Rhee, S.G., 1997. Price limit performance: evidence from the Tokyo stock exchange. J. Finance LII, 885–902. doi:10.2307/2329504. Ng, W.L., 2008. Modeling duration clusters with dynamic copulas. Financ. Res. Lett. 5, 96–103. doi:10.1016/j.frl.2008.01.001. Ogata, Y., 1978. The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Ann. Inst. Stat. Math. 30, 243–261. doi:10.1007/ BF02480216. Ozaki, T., 1979. Maximum likelihood estimation of Hawkes’ self-exciting point processes. Ann. Inst. Stat. Math. 31, 145–155. doi:10.1007/BF02480272. Toke, I.M., 2011. “Market Making{”} in an Order Book Model and Its Impact on the Spread. In: Econophysics of Order-Driven Markets, pp. 49–64. doi:10. 1007/978- 88- 470- 1766- 5_4. Toke, I.M., Pomponio, F., 2012. Modelling trades-through in a limit order book using Hawkes processes. Economics 6. doi:10.5018/economics-ejournal.ja. 2012-22. Yan Du, D., Liu, Q., Rhee, S.G., 2009. An analysis of the magnet effect under price limits∗ . Int. Rev. Financ. 9, 83–110. doi:10.1111/j.1468-2443.2009.01086.x.

Please cite this article as: A. Haghighi et al., Modelling order arrivals at price limits using Hawkes processes, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.08.012