# How Tick Size Affects the High Frequency Scaling of Stock Return Distributions

Gianbiagio Curato and Fabrizio Lillo

Abstract We study the high frequency scaling of the distributions of returns for stocks traded at NASDAQ market as a function of the tick-to-price ratio. The tick-to-price ratio is a measure of an effective tick size. We find dramatic differences between distributions for assets with large and small tick-to-price ratio. The presence of returns clustering is evident for large tick size assets. The statistical differences between large and small tick size assets appear to reduce at higher time scales of observation. A possible way to explain returns dynamics for large tick size assets is the coupling of returns with bid-ask spread dynamics. A simple Markov – switching model is able to reproduce the properties of the distribution of returns for large tick size assets.

Keywords Bid-ask spread • Markov-switching models • Returns clustering • Returns distribution • Scaling • Tick size

In financial markets, the price of an order cannot assume arbitrary values but it can be placed on a grid of values fixed by the exchange. The tick size is the smallest interval between two prices, i. e. the grid step, and it is measured in the currency of the asset (Ascioglu et al. 2010). It is institutionally mandated and sets a limit on how finely prices may be specified. All price information is discretized by the tick size. Historically, the tick size of most securities has been consecutively reduced, resulting in tick sizes of 1/100th or smaller. This process is often referred to as decimalization (Gibson et al. 2003; He and Wu 2004; Chung et al. 2004;

G. Curato

Scuola Normale Superiore, Pisa, Italy e-mail: gianbiagio. curato@sns. it

F. Lillo (H)

Dipartimento di Fisica e Chimica, University of Palermo, Palermo, Italy

Scuola Normale Superiore, Pisa, Italy

e-mail: fabrizio. lillo@unipa. it; fabrizio. lillo@sns. it © Springer International Publishing Switzerland 2015

A. K. Bera et al. (eds.), Financial Econometrics and Empirical Market

Microstructure, DOI 10.1007/978-3-319-09946-0_____ 6

Loistl et al. 2004; U. S. Securities and Exchange Commission 2012). The currenttick size for stocks traded in US stock exchanges, such as the New York Stock Exchange (NYSE) or the National Association of Securities Dealers Automated Quotations (NASDAQ), is typically $0.01. An argument for maintaining the tick size is that it serves to maintain a minimum level of profits for market makers and thus guarantees the provision of liquidity (MacKinnon and Nemiroff 2004; Huang and Stoll 2001; Bollen and Busse 2006), but a too large tick size increases the transaction cost to investors by increasing the bid-ask spread. It is controversial whether a smaller tick size generally improves market quality.

Tick size can affect prices in a direct way on different time scales, starting from the microstructural scale to the daily scale. In this study we analyze the midprice process, i. e. the dynamics of midpoint between bid and ask quotes, in transaction time and in continuous time. We want to study the scaling of the distributional properties of price fluctuations at different time scales, starting from the smallest time scale, e. g. price changes and log-returns caused by 1 transaction. In this way we can see the connection between high frequency dynamics of prices, i. e. 1 s or 1 min dynamics, and low frequency dynamics, i. e. 1 h dynamics. The basic observation is that at the smallest time scale the distributions of returns are very far from Gaussian or Levy stable distributions, that are instead used to model price fluctuations at higher time scales (Bouchaud and Potters 2009; Hautsch 2012; Dacorogna et al. 2001). The return distribution at the smallest time scales strongly depends on the value of the tick-to-price ratio. We have large or small effective tick size assets if this ratio is high or small. As it is known in the literature, the value of the tick size is not the best indicator for understanding and describing the high frequency dynamics of prices. The tick-to-price ratio is one of the definitions of the notion of an effective tick size, introduced in order to account and quantify the different behavior of price fluctuations. Another useful definition is based on the bid-ask spread. In this case the measure is given by the frequency the spread is equal to one tick and we have a large tick size if the spread is almost always equal to one tick. Usually these measures of the effective tick produce the same ranking between different securities.

The key observation is that for large tick assets the price changes are clustered on the grid of the possible integer values that they could assume. Specifically, we find that even price changes are more populated than odd values. This property is found to hold from small to high time scales. Instead for small tick size assets the clustering of price changes is not present. The high frequency dynamics of price for a large tick asset is characterized by the presence of clustering. A similar property has been reported in literature (Harris 1991; Onnela et al. 2009) for daily closing price series. The presence of clustering affects also the distribution of returns for large effective tick size assets, instead this effect is negligible for small tick asset.

We want to quantify empirically the distortion of the shape of distributions of price changes and returns as a function of the effective tick size, measured by the tick-to-price ratio or by the frequency of bid-ask is equal to 1 tick. We expect that, after a certain time scale of aggregation, the shape of distributions becomes independent from the effective tick size of the asset. On one hand the distortion can be characterized by measuring how far the distributions are from the Gaussian, and on the other hand by fitting a microstructural model, developed for large tick assets, on our data in order to reproduce the statistical properties of price changes and returns at different time scales.

We start in Sect. 2 by reviewing the effect of tick size on the market microstructure and the statistical properties of price fluctuations. In Sect. 3 we study the influence of the effective tick size on the return distributions for four assets traded on the NASDAQ market. In Sect. 4 we fit a recently introduced microstructural model (Curato and Lillo 2013) on data of a large tick asset in order to reproduce the statistical properties that we have measured. We summarize the results in section “Conclusions”.

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