# Literature Review

Most of the studies about tick size present in the literature are case studies of the impact of a reduction of tick size on market quality, i. e. on microstructural quantities like the narrowing of the bid-ask spread (Loistl et al. 2004) or liquidity provision (Goldstein et al. 2000; Ahn et al. 2007). The part of literature more related with our work is composed by papers that have revealed how the investors actually use the price resolution allowed by the tick size. We focus also on statistical properties of price fluctuations (Onnela et al. 2009; Munnix and Schafer 2010; La Spada et al. 2011; Gopikrishnan 1999; Plerou et al. 1999) and on the connection between bid – ask spread and midprice dynamics (Dayri and Rosenbaum 2013; Wyart et al. 2008; Robert and Rosenbaum 2011).

The concept of price clustering is known in the literature for daily price time series. It appears that instead of making full use of the available price spectrum, investors stick to a subset of it and use coarser prices instead. There are at least two alternative explanations for this: natural clustering (Harris 1991; Osborne 1962) or collusion (Christie and Schultz 1994; Christie et al. 1994). Harris (1991) studied the frequency distribution of the integer portion of CSRP daily closing price stocks for the years 1963 to 1987, including NYSE, AMEX and NASDAQ stocks. In this case the minimum ticks size ranged from $1/8 to $1/16, and the tick size was smaller for stocks with lower prices. He argued that stock price clustering is pervasive and that clustering distributions from the mid-nineteenth century appear very similar to those observed in the late twentieth century. Clustering increases with price level and volatility and occurs if traders use discrete price sets to simplify their negotiations. He claimed also that clustering must affect price changes distributions and bid/ask quote distributions. Collusion instead refers to the idea that market makers quote prices only in certain fractions in order to increase bid-ask spreads. Christie and Schultz (1994), and Christie et al. (1994) show that many NASDAQ stocks exhibit a paucity of odd-eights quotes and quote prices mainly in even-eights. Bessembinder (2000, 1997, 1999, 2003) provides empirical evidence on relations between trade execution costs and price rounding practices on the NYSE and NASDAQ. His results indicate that higher execution costs are associated with the rounding of quotations and trade prices, and finds that the effect of clustering on trading costs decreases as the tick size decreases.

Onnela et al. (2009) study the effect of changes in tick size, enabled by the decimalization process, on asset log-returns. They analyze a set of NYSE and TSE (Toronto Stock Exchange) cross-listed stocks that were traded under different tick sizes. The data were daily closing prices from Jan-1-1990 to Jun-30-2003. They show that investors do not use all price fractions uniformly as allowed for by the tick size, leading to a clustering of prices on certain fractions, a phenomenon that could potentially affect the way returns are distributed. This phenomenon persists after decimalization. They observed that approximately 57 % of cases exhibit a price clustering such that the effective tick size deviates from the nominal tick size. In this study the tick-to-price ratio, i. e. a measure of the effective tick size, appears to be indicative of the zero returns frequency. They conjectured that large effective ticks lead to a distortion of the shape of return distribution, and this effect should be particularly strong when the price of stock is low, i. e. when tick-to-price ratio is high.

Munnix and Schafer (2010) demonstrate that the tick size has a large impact on the structure of financial return distributions. They analyze a basket of stocks from the S&P 500 index ranging from 1 min to 1 day frequency during the first half of 2007. They find returns clustering at 1 min frequency but do not connect their statistical properties to an effective tick size. They observe that the discrete distribution of price changes could lead to think that the transition from integer price changes to relative price changes, i. e. returns, remove the discretization from the distribution. A closer analysis instead reveals that the discretization effect are still visible when considering returns. They argue that the discretization affects returns on any time scale. They perform an approximate analysis that reveals a sort of mapping between the discrete distribution of price changes and the distribution of returns. They decompose the set of returns according to the absolute price changes, i. e. one value of price change corresponds to a specific set of returns. Their computations lead to the conclusion that the width of this sets are proportional to the absolute value of price changes, while the distance between their centers remains almost constant. In this way the sets of values of returns are increasingly overlapping for larger values of absolute price changes. From their viewpoint the discretization is only visible for small absolute price changes, i. e. one could see an unusual distortion of return distribution near its center. Moreover they find that the shape of the distribution of normalized returns compared to the underlying normalized price changes are quite similar for time scales ranging from 5 min to 1 day. According to Munnix and Schafer (2010) the meaning of clustering is that the distribution of returns is defined on specific sets of the real line, i. e. we do not have a smooth distribution like the Gaussian or Levy distributions. This effect is less and less visible if we have a large number of possible different values for price changes, because we have the overlap of the different sets.

Wyart et al. (2008) use a theoretical framework to obtain a linear relation between the bid-ask spread and the instantaneous impact of market orders and then use this relation to justify a strong empirical correlation between the spread and the volatility per trade. They test this on empirical data and find good agreement with the predicted bound for small tick electronic markets. The case of large tick stocks is different since in this case the spread is nearly always one tick, with very large volumes at both the bid and the ask, leading to a spread that is substantially larger than that predicted for small tick stocks.

Curato and Lillo (2013) develop a statistical model in order to reproduce the statistical properties of the discrete process of price changes for a large tick asset. Large tick assets display a dynamics in which price changes and spread are strongly coupled. They introduce a Markov-switching modeling approach that describes this coupling and the dynamics of spread and return in transaction time. The latent Markov process is the transition process between spreads. Montecarlo simulations of this model reproduce remarkably well the statistical properties of time series representing stocks on NASDAQ market.

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