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 log-returns 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 bid-ask 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 mid-price 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 mid-price 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,


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 Markov-switching 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 log-returns we follow a simple procedure. We generate the simulated series of price changes from the Markov-switching model calibrated from data, then we integrate it choosing as starting point the first mid-price recorded on the measured series. At this point we have a synthetic discrete series of mid-price on which we can compute log-returns r (i, 1) correspondent to 1 transaction. Then we aggregate individual transaction returns on non-overlapping 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 log-returns. 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 Markov-switching 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 non-Gaussianity 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 higher-order 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 log-returns from the Montecarlo simulations of our models and then compute the normalized log-return 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 Markov-switching model and the double chain


Подпись: Fig. 7 Linear-log plot of the scaling of hypercumulants Aq of normalized returns g(i,n) for the stock CSCO and the correspondent simulated returns processes. The different style of lines indicates the different time scales n. The vertical dotted lines indicate the values of q, i.e. q = 1,2.5, 3, 3.5, for which we illustrate the time scale dependence Aq (n) in Fig. 8
q q















Fig. 8 Log-linear plot of the scaling of hypercumulants Aq=і;2.5,з, з.5 (n) for the stock CSCO and the related simulated returns processes

Подпись: Conclusions In this work we show empirically that the effective tick size affects strongly the statistical properties of distributions of mid-price changes and log-returns of stocks traded on a decimalized market like NASDAQ. The effect of clustering of price changes on even values of the price grid is particularly strong for stocks with a large effective tick size. The clustering of price changes persists at each time scale in presence of a large effective tick. This effect is absent in presence of a small effective tick. On the other hand, the

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 Markov-switching model results to be the simplest model able to reproduce at the same time the clustering of price changes and log-returns together with the correct scaling of hypercumulants toward a Gaussian behavior.


discrete nature of prices affects also the distribution of log-returns. The effect of discretization for log-returns disappears when the time scale of observation of the mid-price process is high enough, both for large effective tick than for small effective tick. The difference between the two kinds of stock is that for a large tick asset we have also the effect of returns clustering. For example, if we observe the price process in continuous time for a time scale around 15 min, the effect of discretization and clustering starts to disappear for a large effective tick, instead for small effective tick sizes the discretization disappears at shorter time scales, i. e. 30 s.

The analysis of the hypercumulants of distributions of log-returns indicates a converge toward a Gaussian behavior for stocks with large and small effective tick sizes. The presence of a large effective tick seems to slow down the convergence toward a Gaussian behavior, coherently with a progressive disappearance of clustering of returns. The analysis of convergence by means of the Hill estimator shows a crossover time scale after which large and small tick size assets behave in the same way. However our analysis characterize how the blind use of estimators could lead to errors in the determination of the tail exponents.

We develop statistical models in trade time for large tick size asset that are able to reproduce the presence of clustering for price changes and log-returns. We find that in order to reproduce the clustering effect we need a model in which the dynamics of mid-price changes is coupled with the dynamics of the bid-ask spread. A Markov-switching model, where the switching process is defined by the possible transitions between subsequent values of the spread, is able to reproduce the effect of clustering and the scaling of hypercumulants computed from empirical data. This simple high frequency statistical microstructural model, defined by quantities like bid and ask prices on a discrete price grid determined by the value of the tick size, is able to recover a Gaussian behavior for returns at the macroscopic time scale of the hours.


Acknowledgements Authors acknowledge partial support by the grant SNS11LILLB Price formation, agents heterogeneity, and market efficiency.




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