In this chapter, we present empirical studies that shed some new light on the dependency structure of order arrival times, in particular, on the mutual and self-excitations of limit and market orders and, to a lesser extent, of cancellation orders. Examples for various assets and markets (mostly equities, but also bond and index futures) are provided. These empirical studies lay the ground for the advanced mathematical models studied in Chapters 8 and 9.
Re-introducing Physical Time
As seen in Chapter 2, the Poisson hypothesis for the arrival times of orders of different kinds does not stand under careful scrutiny. However, the study of arrival times of orders in an order book has not been a primary focus in the first attempts at order book modelling. Toy models leave this dimension aside when trying to understand the complex dynamics of an order book. In many order-driven market models (Cont and Bouchaud, 2000; Lux and Marchesi, 2000; Alfi et al. 2009a), and in some order book models as well (Preis et al., 2006), a time step in the model is an arbitrary unit of time during which many events may happen. We may call that clock aggregated time. In most order book models (Challet and Stinchcombe, 2001; Mike and Farmer, 2008), one order is simulated per time step with given probabilities, i.e. these models use the clock known as event time. In the simple case where these probabilities are time-homogeneous and independent of the state of the model, such a time treatment is equivalent to the assumption that order flows are homogeneous Poisson processes. A likely reason for the use of event time in order book modelling – leaving aside the fact that models can be sufficiently complicated without 30 Limit Order Books adding another dimension – is that many puzzling empirical observations can be made in event time (e.g. autocorrelation of the signs of limit and market orders) or in aggregated time (e.g. volatility clustering) (see Chapter 2).
However, it is clear that physical time has to be taken into account for the modelling of a realistic order book model. For example, market activity varies widely, and intraday seasonality is often observed as a well known U-shaped pattern. Even for a short time scale model – a few minutes, a few hours – durations of orders (i.e. time intervals between orders) are very broadly distributed.