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].
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.
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.
Analysis and interpretation
We offer an interpretation for our striking empirical relationship in the framework of the model with uncertainty zones. This is done through a simple equilibrium equation for the profit and loss of market makers and market takers. Indeed, we can show that the average ex post cost of a market order is
(α/2) –ηα
Then it can be proved that the average P and L per trade of the aggregate market makers is equal to
(α/2) -c (σ /√M) +φ.
where c is a constant of order 1 or 2 and φ> 0 corresponds to extra compensations of the market makers related to their inventory control. Thus, the profits of the market makers being the losses of the market takers, we derive
ηα =c(σ /√M)- φ.
In the classical approach, the ex post average cost of both limit orders and market orders is zero. In contrast to this, our relationship states that for large tick assets, market orders are costly whereas at the aggregate level, limit orders are profitable (but of course individual market makers do not easily get gains because of the large size of the best queues in the order book).
Explaining microstructure effects
A very well-known stylised fact of high frequency data from large tick assets is the systematically decreasing behavior of the so-called signature plot (the realised volatility over a given time period when the sampling frequency decreases). Many models try to reproduce this phenomenon , but very few explain it. Our approach enables to show that this decreasing behavior is equivalent to the inequality η ≤ 1/2. According to the preceding equations, this in fact means that market orders are costly whereas limit orders (at the aggregate level) are favorable. This asymmetry between both types of orders always holds for large tick assets. Indeed, one cannot have η> 1/2 since it would imply that market makers lose money. In that case, they would simply increase the spread to remedy this.
3) Forecasting the effects of a change in the tick value
The tick value issue
Fixing the tick value is an intricate problem. On the one hand, if the tick value is very small, some market participants do not hesitate changing marginally the prices of their limit orders in order to gain in priority. This leads to unstable order books where traffic is very high. Such an environment is very discouraging for traditional market makers for which it is very hard to set quotes. This can induce severe economic consequences, in particular for small or mid cap companies. Indeed, quoting them may not be worthwhile for classical market makers in such an unfavorable market. As a result, the quality of the liquidity on such stocks can be very low. Such a situation is also difficult to manage for the exchange, which has to deal with overloaded platforms. On the other hand, a tick value which is too large prevents the price from moving freely according to the views of market participants. This creates needless frictions and sloppiness in the price (strong mean reversion at the high frequency level), and also favors speed (race to the top of the book). Moreover, market takers pay a large extra cost in order to obtain liquidity.
If the tick value is not satisfying, exchanges often have the possibility to change it. Such a modification implies changes in various market quantities (number of trades, spread, liquidity, etc). The first thing the platform designer needs to do is to define the desired effects of this change of tick value, which is already a difficult question. Even in the case where market designers have a clear idea of the situation they want to reach, they still face the problem of the way to reach it. Indeed, it is commonly acknowledged that tick values have to be determined by trial and error and that the success of a change in the tick value can only be assessed ex post, on the basis of the obtained effects. Thus, only few predictive models have been designed in the literature and the consequences of a change in the tick value have been essentially studied from an empirical point of view. We offer in [1] a methodology that we believe will help exchanges choose the correct tick value. Starting from a large tick asset, we provide a closed form formula for the optimal tick value.
The forecasting formula
In the case of large tick assets, our approach enables us to forecast ex ante the consequences of a change in the tick value on some market quantities, in particular the crucial parameter η which quantifies the intensity of microstructure effects. Let us start from a situation where the tick value is α0, the microstructure parameter is equal to η0 and the daily number of trades is M0. Assuming the long term volatility and the daily turnover do not depend on the tick value, if we change the tick value from α0 to α , we get the following prediction formula for the new value of η:
η ≈ η0 (α0 / α) ^ (1-β / 2),
with β a parameter between 1/2 and 1 depending on the shape of the implicit supply and demand curves. This formula has been successfully tested on the Bobl contract, which changed tick value on June 15, 2009, see Figure 2.
4) Optimal tick value
Optimal tick value formula
Defining an optimal tick value is a very complicated issue. Indeed, different types of market participants can have opposite views on what is a good tick value. Thanks to our framework, we can suggest a reasonable notion of optimal tick value. Of course the optimality notion we are about to define is arguable and we do not take into account some elements, for example the fact that a given asset can be traded on different platforms, with possibly different tick values (however, note that according to recent regulatory proposals, it could be required for the tick value of a given asset to be the same on all trading platforms). Nevertheless, we still think it is a first quantitative step towards solving the tick value question. We consider that a tick value is optimal if:
- The (average) ex post cost of a limit order is equal to the (average) ex post cost of a market order, both of them equal to zero.
- The spread is stable and close to one tick.
Such a situation can be seen as reasonable for both market makers and market takers. Indeed, it removes any implicit costs or gains due to the microstructure. Moreover, having a stable spread close to one tick prevents sparse order books which can drive liquidity away.
It is easy to see that getting an optimal tick value is equivalent to have η = 1/2 together with a spread which is still equal to one tick. Thus, we refer to this last situation as the optimal tick size case. Note that in term of the microstructure parameter η, the optimal tick size is the same for any asset η = 1/2, whereas the optimal tick value depends on the features of the asset. Remark that in the optimal situation, we can show that the following properties follow for the microstructure:
- The last traded price can be seen as a sampled Brownian motion.
- Consequently, the signature plot is flat.
Starting from a large tick asset, our approach enables us to reach the optimal tick size situation. Indeed, it is possible to obtain η = 1/2 and a spread close to one tick by changing the tick value only assuming that η increases continuously when the tick value decreases. Then, when modifying the tick value, the spread remains equal to one tick as long as α / 2-ηα ≥ 0. Indeed, if α * denotes the largest tick value such that η = 1/2 then for all α> α * market makers make positive profits with a spread of one tick and consequently maintain this spread Then, we obtain the following formula for the optimal tick value leading to η = 1/2:
α ≈ α0 (2η0) ^ (1 / (1-β / 2)).
Of course we do not pretend that in practice, applying such rule will exactly lead to an optimal tick value (in our sense). However, we do believe that this simple formula gives the right order of magnitude for the relevant tick value of a given asset.
Optimal tick value for small tick assets
A crucial point in our approach is that when changing the tick value of a large tick asset, the spread remains equal to one tick as long as market makers make profit with such a spread. So the spread (in tick unit) is invariant when the tick value is modified. For a small tick asset, when enlarging the tick value, both the spread and the number of trades adjust so that the spread and the volatility per trade have of the same order of magnitude. The way these two variables are jointly modified is intricate and this is why our method cannot, a priori, be used for small tick assets. However, let us stress the fact that it is still possible for the exchange to use a two steps procedure in the case of a small tick asset:
– Step 1: Enlarge sufficiently the tick value so that the asset becomes a large tick asset.
– Step 2: Use our methodology for large tick assets.
To contact the authors:
khalildayri@gmail.com
mathieu.rosenbaum@upmc.fr
Bibliography
- Dayri, K., and M. Rosenbaum, 2013, Large tick assets: implicit spread and optimal tick size, Preprint.
- Madhavan, A., Richardson, M., and M. Roomans, 1997, Why do security prices change? A transaction-level analysis of NYSE stocks, Review of Financial Studies 10, 1035-1064.
- Robert, CY, and M. Rosenbaum, 2011, A new approach for the dynamics of ultra-high-frequency data: The model with uncertainty zones, Journal of Financial Econometrics 9, 344-366.
- Wyart, M., Bouchaud, JP, Kockelkoren, J., Potters, M., and M. Vettorazzo, 2008, Relation between bid-ask spread, impact and volatility in double auction markets, Quantitative Finance 8, 41 – 57.