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this pathological situation, because it forces traders to improve by non-infinitesimal amounts.
Thus, intuitively, imposing a minimum price variation can be a way to reduce the expected
spread, despite the rounding effect, because it makes the market more resilient. We demonstrate
this claim by providing a numerical example. The values of the parameters are as in Example
3 except that r =0.97 (i.e. ¸ =0.49, and the market is weakly resilient), so that the condition
r > rc is satisfied.24 Table 3 gives all the monetary spreads on the equilibrium path for two
different values of the tick size: (1) " =0 and (2) " =0.0625. The two last lines of the table give
the expected spread and the resiliency obtained for each regime. First, observe the  rounding
effect - the thirteen smallest spreads are lower when " =0, than in the case of " =0.0625.
Second, observe the  spread improvement effect - the spread reduction is quicker for every spread
level if a minimum price variation is enforced. This explains why market resiliency is smaller when
there is no minimum price variation. For this reason, the expected spread turns out to be larger
in this case ($1.58 instead of $1.48).
24
Given the values of the parameters rc H" 0.92.
27
Table 3 - Rounding and Spread Improvement Effects
(Parameter Values: » =1,Km =2.5, ´1 =0.1, ´2 =0.25, r =0.97)
hnm(" =0) nm(" =0.0625)
h h
1$0.1$0.125
2$0.294 $0.375
3$0.482 $0.625
4$0.665 $0.813
5$0.842 $1
6$1.014 $1.188
7$1.181 $1.375
8$1.343 $1.563
9$1.5$1.75
10 $1.652 $1.938
11 $1.799 $2.125
12 $1.942 $2.313
13 $2.081 $2.5
14 $2.216 NA
15 $2.347 NA
16 $2.474 NA
17 $2.5 NA
Expected Spread $1.58 $1.48
Resiliency 1.1 × 10-5 1.9 × 10-4
So far we have compared a situation with and without a mandatory minimum price varia-
tion. More generally, the  spread improvement effect implies that the expected spread does not
necessarily decrease when the tick size is reduced. In order to see this point, consider Table 4. It
1 1 1
demonstrates which of the following tick sizes, {100, , }, minimizes the expected spread for dif-
16 8
1
ferent values of r. Consistent with the above argument " = does not minimize the expected
100
spread for low values of r. However as r increases, inducing traders to make large improvements
by imposing a large minimum price variation becomes less effective, since they already submit
aggressive orders. For this reason, the  spread improvement effect becomes of second order
compared to the  rounding effect . In fact Table 4 shows that the tick size which minimizes
28
the expected spread decreases with r and that once r e" 1 the expected spread is minimized at
1
" = .
100
Table 4 - The Tick Size Minimizing the Expected Spread
1 1 1
(Parameter Values: » =1, Km =2.5, ´1 =0.1, ´2 =0.25, " " {100, , })
16 8
r 0.7 0.8 0.9 0.93 0.97 1 1.1 1.2 1.3
"" 1 1 1 1 1 1 1 1 1
8 8 8 16 16 100 100 100 100
Finally we briefly discuss the case in which r
spreads byaninfinitesimal amount. Thus the quotes are always set arbitrarily close to the largest
possible ask price, A, or the smallest possible bid price, B.25 Thus market resiliency is zero, as
when r goes to rc. Imposing a minimum price variation is a way to restore market resiliency
since spread improvements are non-infinitesimal as soon as " > 0 (Proposition 5).
To sum up, reducing or even eliminating the tick size may or may not reduce the average
spread. The impact depends on the proportion of patient traders in the market, r. Many
empirical papers have found a decline in the average quoted spreads following a reduction in tick
size. These papers, however, do not control for the ratio of patient to impatient traders. One
difficulty of course is that this ratio cannot be directly observed. In Section 5, we argue that
the proportion of patient traders is likely to decrease over the trading day. In this case, the
impact of a decrease in the tick size on the quoted spread should vary throughout the trading
day. Specifically, a decrease in the tick size may increase the average spread at the end of the
trading day. To the best of our knowledge, there exists no test of this hypothesis.
4.2 Fast vs. Slow Markets
In this section, we analyze the effect of orders arrival rate (») on the dynamics of the spread
and the expected spread. We compare two markets, F and S, which differ only with respect
to orders arrival rate, ». Specifically, »F > »S, which implies that the average waiting time
between orders in market F is smaller than in market S. Thus, other things being equal, events
(orders and trades) happen faster in clock time in market F. For this reason, we refer to market
25
This would also be the case if patient traders waiting cost were equal to zero (´1 =0). When r
the equilibrium (when there is no minimum price variation) is difficult to describe formally since traders improve
upon prevailing quotes by an infinitesimal, but strictly positive, amount.
29
F as a fast market and market S as a slow market. Proposition 5 and Corollary 1 immediately
yield the next result.
Corollary 5 : Consider two markets with differing orders arrival rates: »F >»S. Then:
1. The spreads on the equilibrium path in markets F and S are such that: (1) nh(»F ) d" nh(»S),
for h
spreads in the fast market is shifted to the left compared to the support of possible spreads
in the slow market.
2. The slow market is more resilient than the fast market.
The economic intuition of these results is as follows. On the one hand, the waiting time of
a trader with a given priority level in the queue of limit orders is smaller in the fast market
(see Proposition 4), thus patient traders require a smaller compensation for waiting. This effect
explains the first part of the proposition. On the other hand, spread improvements are larger and
the spread narrows more quickly in the slow market (see the discussion following Proposition 5).
Hence the slow market is more resilient.
These two effects have an opposite impact on the average spread. Unfortunately it is not [ Pobierz całość w formacie PDF ]

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