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.
