Clustering of order arrivals, price impact and trade path optimisation Patrick Hewlett* May 6, 2006 Abstract We fit a bivariate Hawkes process to arrival data for buy and sell trades in FX markets. The model can be used to predict future imbalance of buy and sell trades conditional on history of recent trade arrivals. We derive formulae for the raw price impact of a trade as a function of time assuming that trade arrivals are governed by a Hawkes process and that the price is a martingale, and show that the price impact of a series of trades is given by superposition of their individual price impacts. We use these formulae to parameterise a model for optimal liquidation strategies. 1 Introduction It is a well known feature of many financial markets that trading activity tends to cluster in time, and that trades of the same sign tend to cluster together in the se- quence of buys and sells. Such clustering can be modelled with a multivariate point process. Liquidity providers1 (���market makers���) in the FX market are well aware of this clustering, and anecdotal evidence suggests that they pay close attention to the pattern of arrivals of buy and sell orders when setting prices. In this paper we focus on a model in which order arrivals are governed by a special class of point process, the Hawkes self-exciting process in this case, the mathematical solution of the market maker���s problem has a particularly tractable form. We also see that the Hawkes process can be fitted quite successfully to empirical order arrivals data. In contrast to liquidity providers, liquidity demanders (���traders���) often split large trades into multiple tranches over a period of minutes or hours, in order to alleviate market impact costs. One of their concerns is that market makers will *OCIAM, 24-27 St Giles, Oxford OX1 3LB tel 01865 276502 email: hewlett@maths.ox.ac.uk. Thanks are due to Sam Howison and Giovanni Pilliterri for helpful comments on this version, and to Clive Bowsher for advice on Hawkes processes. Financial support from Deutsche Bank and the EPSRC is gratefully acknowledged. 1The FX market is arranged as an electronic limit order book, so all participants have the opportunity to act as market makers. In what follows we assume that there is a single market maker who sets competitive prices. 1

guess from their pattern of trading that they are in the process of executing a large trade, and penalise them accordingly through less favourable prices. This activity is formulated in our model as the market makers trying to predict future trading from past trading. We assume that market makers form their expectations of the timing and direction of future trading based on a Hawkes process model, and that they set prices competitively based on these expectations. The trader is then faced with the usual dilemma: trade too quickly and suffer severe market impact costs trade too slowly and run the risk of adverse price movements before the trade is completed. Whilst models describing this dilemma are well developed, methods of parameterising them are not. This work offers a method of parameterisation which only requires data on the trade arrival process, which is readily available to FX market participants. It should be emphasised that we do not attempt to offer a game-theoretic so- lution for the behaviour of the trader and market-maker. Instead, we take the empirical clustering of trades as given, compute how the market-maker ought to re- act to trading patterns assuming that the clustering is well described by a Hawkes process, and finally compute how the trader ought to behave given the market- maker���s (empirically) rational response to trading. This allows us to build a prac- tically applicable model of the trading environment without needing to model the heterogeneity of traders and their motivations. This paper is organised as follows. Section 2 reviews past work on trade arrival processes and multiple-tranche trading optimisation. In Section 3 we present the model for the market-marker���s problem and discuss the mathematical properties of Hawkes processes that allow a solution to be developed. In Section 4 we describe the dataset and fit the Hawkes model. In Section 5 we describe the trader���s problem and compute solutions using the parameters obtained above. Section 6 concludes. 2 Literature Review An introduction to the mathematical theory of point processes is given in [8]. Hawkes processes, first introduced by [23], are a particularly tractable type of point processes for our purposes, because closed form expressions exist for the expected future number of arrivals of each type and current intensity given the observed history of the process. Estimation is also relatively straightforward (see e.g. [28]). The state of the art in estimation and model validation for multivariate Hawkes processes in finance is summarised in [6]. There is a growing literature on the application of point processes to high fre- quency financial data. Important early work focuses on modelling inter-event du- ration ([13], [14]) and the effect of duration on price impact and trade sign auto- correlation ([10]). [12] extend these models to allow for the censoring of quotes after a trade by intervening trades. We are concerned here with predicting future trades given the pattern of past trades. These duration-based models are somewhat intractable for this purpose in a multivariate context. In high frequency finance, rounding of times to the nearest second is common. A half-way house between true point process modelling and discrete time mod- elling is obtained by binning data into discrete intervals (which may or may not correspond to the measurement frequency). [22] and [9] approach modelling of trades and quotes in this spirit. Whilst we believe that we could have developed

our model along these lines, we felt that the advantages of using ���true��� continuous time processes outweighed possible disadvantages2. The use of Hawkes processes in particular for modelling order arrivals and mi- crostructure scale price movements is investigated by [7], [6], [25], [3]. Elsewhere in the mathematical finance literature, Hawkes processes have been advocated for modelling of credit contagion [18] and clustering of extreme price moves [26]. Much thinking on the optimal way to split a large order into tranches over time is based on ideas advanced in [4], [2]. Despite (or perhaps because of) the practical relevance of such models, there is little published on how to parameterise them. [1] details one strategy for parametrisation, which requires as data the ex-post cost of various trading strategies. By contrast, the model we present in Section 5 could in principle be parameterised using only data on the times and directions of order arrivals. 3 A self-exciting model of trade arrivals Intensity-based approaches focus on arrival intensities for the counting processes Nt(i). Arrival intensity ��ti) ( conditional on a filtration Ft is defined by ��ti)|Ft ( = lim ��t���0 1 ��t E(Nt(+��t i) - Nt(i)|Ft) . (1) In the case of purely self-exciting processes, the intensity is a functional of past arrivals3. For a linear self-exciting process, we have ��ti) ( := ��(i) + summationdisplayintegraldisplayI j=1 ut hij(t - u) dNuj) ( . (2) Here ��i can be understood as the ���base��� intensity of arrivals of type i (the intensity if there have been no past arrivals of any type), and hij the propagator of an arrival of type j onto the intensity of arrivals of type i in the future. We will be interested in parametric forms for g ��� in particular we consider the case where g is a sum of exponentials: hij(s) = summationdisplayK k=1 ��ije-��ijs , (3) so that ��ti) ( = ��i + summationdisplaysummationdisplayintegraldisplayIK k=1 j=1 ut ��ije-��ij(t-u) dNuj) ( . (4) This specification is labelled a Hawkes-E(K) process in [6]. Other forms for h have been advocated in the finance literature (e.g. power law, Laguerre polynomial). The advantage of the exponential specification is that the likelihood function can be 2In particular, there is well-developed mathematical machinery for continuous time point processes, whereas the models of [9] (���CBin models���) are less developed whilst the mathematical foundations of point processes may appear to present a barrier to practitioners, the relevant properties can safely be understood intuitively in this context. Discrete sampling does create a few problems in the busiest FX markets (e.g. USD/EUR) where the probability of several events in one second is significant. 3in contrast to doubly stochastic processes where there is an unobserved component driving the intensity.