# Large Tick Assets: Implicit Spread and Optimal Tick Size

**By Khalil Dayri, Antares Technologies and Mathieu Rosenbaum, Laboratory of Probability and Random Models, University Pierre and Marie Curie (Paris 6)**

This paper is based on the article [1].

**Abstract**

We provide a framework linking microstructural properties of an asset to the tick value of the exchange. In particular, we bring to light a quantity, referred to as implicit spread, playing the role of spread for large tick assets, for which the effective spread is almost always equal to one tick. The relevance of this new parameter is shown both empirically and theoretically. This implicit spread allows us to quantify the tick sizes of large tick assets, to anticipate the consequences of a change in the tick value, and to define a notion of optimal tick size. In particular, our results allow us to forecast the behaviour of relevant market quantities after a change in the tick value and to give a way to modify it in order to reach an optimal tick size. Thus, we provide a crucial tool for regulators and trading platforms in the context of high frequency trading.

**1) Tick value, tick size and spread**

On a given market, the tick value of an asset is the smallest interval between two prices. It is a well-defined quantity, measured in euros, dollars, etc. However, when it comes to actual trading, the tick value is given little consideration. What is important is the so-called tick size. A trader considers that an asset has a small tick size when he “feels” it to be negligible, in other words, when he is not averse to price variations of the order of a single tick. In general then, the trader’s perception of the tick size is qualitative and empirical, and depends on many parameters such as the tick value, the price, the usual amounts traded in the asset and even his own trading strategy. Thus, the tick size is basically a subjective and ill-defined quantity. Nevertheless, we can still distinguish between small and large tick assets. Indeed, an asset is usually said to have a large tick when its bid-ask spread is almost always equal to one tick.

This work focuses on large tick assets and addresses the following questions:

- For small tick assets, the spread is a good proxy for the tick size. In the case of large tick assets, for which the spread is essentially equal to one tick, how to quantify the tick size?
- There exist some special relationships between the spread and some other market quantities. However, they are not valid for large tick assets since the spread is mechanically bounded from below by the tick value. How to extend these studies in the large tick case?
- When the tick value changes, what happens to the microstructure of the asset?
- Can we define an optimal tick value?

**2) The spread-volatility relationship and its consequences**

In general, for small tick assets, over a given time period, the average spread is proportional to the volatility per trade defined by σ / √ M, where σ² and M stand respectively for the cumulated price variance and the number of trades during the considered time period. From a theoretical point of view, this relationship can be well understood using a market makers / market takers dichotomy, see [2, 4]. Empirically, it is impressively well satisfied on data, see [2]. However, this relationship does not hold for large tick assets. Indeed, in this case, the spread is almost always equal to one tick and is therefore artificially bounded from below.

**Implicit spread**

We introduce a notion of implicit spread, playing the role of spread for large tick assets, for which the effective spread is almost always equal to one tick. This parameter arises from the model with uncertainty zones, see [3]. In this model, there is an underlying latent price, called efficient price, representing at any time some average opinion of market participants about the value of the asset. Depending on the position of the efficient price in the bid-ask spread, market orders are buy orders only, sell orders only, or can be of both types. The implicit spread is defined as the size of the interval where both buy and sell market orders can occur. Furthermore, it is shown to be equal to 2ηα, where α is the tick value and η is the microstructure parameter of the asset which summarizes all its microstructural features (high frequency volatility, correlations of the returns, …) see [3]. The parameter η lies between 0 et 1/2 and the larger η, the less intense the microstructure effects are. Furthermore , η can be very easily estimated from market data as follows:

η = Nc / 2Na,

where Nc is the number of continuations on the considered time period, that is the number of (last traded) price moves whose direction is the same as the one of the preceding move, and Na is the number of alternations, that is the number of price moves whose direction is opposite to the one of the preceding move.

On various large tick assets, listed on different exchanges, we show that the relationship between spread and volatility per trade still holds very well, provided that the conventional spread is replaced by the implicit spread 2ηα, see Figure 1.