Statistical Models for Large Tick Assets
In this section we present briefly the statistical models recently introduced by Curato and Lillo (2013) describing the high frequency dynamics of price changes for a large tick size asset in trade time. We want to show that these models are able to reproduce the phenomenon of clustering for logreturns and the scaling of hypercumulants Aq (n) in trade time.
The building blocks of these models are simple: the distribution of price changes caused by 1 transaction, i. e. Ap(i, и = 1), and the statistical properties of the dynamics of the bidask spread s (i). In our model we impose a coupling between the process of the price changes and of the spread in order to reproduce the price – change clustering.
We consider first a benchmark model, hereafter called i. i.d. model, in which this coupling is absent and where we use only the information contained in the distribution of Ap (i, n = 1).[5] Our empirical analysis indicates that for a large tick asset the distribution of Ap is mainly concentrated on Ap = 0. This observation allows us to limit the discrete set on which we define the distribution at the scale of 1 transaction, i. e. Ap є {—2, —1,0,1,2} in units of half tick size. In the i. i.d.
model the Ap(i) process is simply an i. i.d. process in which each observation has the distribution estimated from data. Numerical simulations and analytical considerations show that this model is unable to reproduce price change clustering at any scale, i. e. when we aggregate n values we recover a bell shaped distribution for Ap (i, n) = J2l=1AP (i).
Our solution to recover price changes clustering is to use the process of spread s(i). The key intuition behind our modeling approach is that for large tick assets the dynamics of midprice and of spread are intimately related and that the process of price changes is conditioned to the spread process. For large tick assets the spread can assume only few values. For example, for MSFT and CSCO spread size is only 1 or 2 ticks. The discreteness of midprice dynamics can be connected to the spread dynamics if we observe that when the spread is constant in time, price changes can assume only even values in units of half tick size. Instead when the spread changes, price changes can display only odd values. This effect is visible in Fig. 2 for MSFT stock where even values of price change are more populated than odd values, because spread changes are relatively rare. The dynamics of price changes is thus linked to dynamics of spread transitions. It is well known that spread process is autocorrelated in time (Ponzi et al. 2009; Plerou et al. 2005; Dayri and Rosenbaum 2013). In our models the spread process s (i) is represented by a stationary Markov(1) process:
P (s (i) = ks (i – 1) =j, s(i – 2) = l, •••) =P(s (i) = ks (i – 1) = j) = pjk,
(6)
where j, k,l є {1,2} are spread values and i є N is the trade time. The spread process is described by the transition matrix S є M (2,2):
S = (P11 P12)
P21 P22
where the normalization is given by Y? k=i Pjk = 1. For example for CSCO we estimate Pu = 0.97 and p21 = 0.58, i. e. the transitions in which the spread changes are not frequent. In this model the spread could assume 2 values so we could have 4 possible transitions t (i) between two subsequent transactions, that we could identify with an integer number from 1 to 4. For example, the transition s (i) = 1 ! s (i + 1) = 1 is described by the state t (i) = 1, etc. In this way we can derive a new Markov(1) process that describe the process t (i). At this point the mechanistic constraint imposed by a price grid, defined by the value of the tick size, allows us to couple the price changes Ap (i) with the process of transitions t (i). In this way we are able to define a Markovswitching model (Hamilton 2008) for price changes Ap (i) conditioned to the Markov process t (i) by the conditional probabilities:
where m 2 {1,2,3,4}. The estimation of such conditional probabilities enable us to simulate the process for price changes. In order to compute logreturns we follow a simple procedure. We generate the simulated series of price changes from the Markovswitching model calibrated from data, then we integrate it choosing as starting point the first midprice recorded on the measured series. At this point we have a synthetic discrete series of midprice on which we can compute logreturns r (i, 1) correspondent to 1 transaction. Then we aggregate individual transaction returns on nonoverlapping windows of width n to recover the process at a generic time scale n. This model is able to reproduce clustering for price changes and for logreturns. In fact as we can observe in Fig. 6 this model reproduces the returns clustering at different time scales. The clustering starts to disappear beyond the time scale of aggregation n = 512.
The Markovswitching model is not able to explain the empirically observed correlation of squared price changes, that is related to the presence of volatility clustering. Usually in financial econometrics an autoregressive conditional het – eroskedasticity model (ARCH) (Bera and Higgins 1993; Engle et al. 2008) can account for volatility clustering and nonGaussianity of returns. We do not make use of this class of models because they are defined by continuous stochastic variables. Instead, we have seen in Sect. 3.2 that the high frequency return distribution is characterized by the presence of discretization and clustering. For this reason we have chosen to define our model directly on discrete variables as price changes, but this choice prevents us from using models like ARCH. Therefore in Curato and Lillo (2013) we develop a second model based on an autoregressive switching model for price changes, which preserves the ideas of ARCH type models that past squared returns affect current return distribution. This means that the conditional
probabilities of Eq. (7) can depend not only on the last spread transition, but also on the recent past values of price changes. In the case in which regressors are defined only by past squared price changes, our model can be viewed as a higherorder double chain Markov model of order p (Berchtold 1999). For our purposes we remember only that we have fitted this model for an order p = 50 on our data. Here we use it to generate a series in order to study the scaling of hypercumulants Aq (n). We refer to our original article (Curato and Lillo 2013) for the details and definitions for this autoregressive model.
In order to fit our three models we split our daily time series in two series, the first displays low volatility instead the second displays high volatility. Here we focus on the low volatility series that starts at 10:30 and ends at 15:45, but our findings are the same for the series with high volatility. We compute the logreturns from the Montecarlo simulations of our models and then compute the normalized logreturn g(i, n) in trade time. The Montecarlo simulations generate 5961600 data points that correspond to 1 year of transactions for a mean duration time, i. e. the interval of time between two transaction, of 1 s and 6 h of trading activity each day. The sample for the stock CSCO instead covers 2 months of trading for a total of 275879 transactions. We can observe from the Figs. 7 and 8 that the proposed models converge to a Gaussian behavior. The Markovswitching model and the double chain
q q








Fig. 8 Loglinear plot of the scaling of hypercumulants Aq=і;2.5,з, з.5 (n) for the stock CSCO and the related simulated returns processes
Markov model, i. e. DCMM in the figures, reproduce slightly better the scaling of the hypercumulants respect to the simple i. i.d. model, although all models are very close to the empirical moments. We could observe a little difference in Fig. 8 for high values of n between simulation and real data. We think that this distortion rises from the reduced number of the data sample used to compute Aq for the stock CSCO with respect to the number of data points available for simulated data. The Markovswitching model results to be the simplest model able to reproduce at the same time the clustering of price changes and logreturns together with the correct scaling of hypercumulants toward a Gaussian behavior.
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