Measuring the Concentration of Hidden Liquidity

In this section, we put the above observations to a robust test. To this end, we introduce a range of measures to estimate the degree ofdispersionorlocalisationand assess the difference between hidden and displayed liquidity. We analyse dispersion along both dimensions, price and time.

6This reasoning would be in line with prior empirical findings that suggest that hidden orders are mainly used by large investors, see Bessembinder et al. (2009) for instance.

We introduce some notation. Denotex^h_ij (x^d_ij) the hidden (displayed) depth at timet_i (i= 1, ..., n) at price quotepj (0≤j ≤m) for some stock. The hidden and displayed liquidity dis-tribution at timetiis then given byx^h_i = (x^h_i1, x^h_i2, x^h_i3, ..., x^h_im)andx^h_i = (x^h_i1, x^h_i2, x^h_i3, ..., x^h_im) respectively. We assume that the list of prices is finite andmdenotes the maximum number of price quotes. Moreover, we denote byy^h_i (y_i^d) the total hidden (displayed) depth at timeti, i.e.

y^h_i = ^Pm_j=1x^h_ij (y_i^d = ^Pm_j=1x^h_ij). Their time evolution is denoted by y^h (y^d), that is to say y^h = (y^h₁, y^h₂, y^h₃, ..., y_m^h)andy^d= (y^d₁, y^d₂, y₃^d, ..., y^d_m)hold.

Concentration and Dispersion Measures

The first measure will be the coefficient-of-variation (C) as in table 1.1. In line with our de-sire to capture dispersion along both, the time and the price domain, we define two measures accordingly

C^time:= σ(y^q)

µ(y^q), C^book(i) := σ(x^q_i)

µ(x^q_i), (1.4.1) whereσdenotes the respective sample’s standard-deviation andµits the standard mean.

The second measure is motivated by the concept of entropy.⁷ The temporalφ_time and the spatialororder-book entropyφbook read

φ^time:=− 1 log(n)

Xn i=1

g^q_i log g_i^q, (1.4.2)

φ^book(i) :=− 1 log(m)

Xm j=1

h^q_ijlog h^q_ij, (1.4.3) g_i^q and h^q_ij denote the corresponding empirical density distributions, i.e. g^q_i := y_i^q/(^Pm_i=1y_i^q) and h^q_ij := x^q_ij/(^Pm_j=1x^q_ij) withq = d, h and i = 1, ..., n. The choice of the normalisation factors, 1/log(n) and 1/log(m), where n and m denote the respective sample (state) sizes, ensures that entropy is normalised and values range between0and1. This eases cross-sectional comparison and comparisons across different sample sizes.⁸ It is well-known that the entropy measure is non-negative and that it takes on its maximum value for equi-distributed weights (i.e.

state of highest dispersion) and its minimum when all but one weight is non-zero (i.e. state of highest of localisation).

Although both measures capture sample dispersion, the obtained numbers hardly allow for an illustrative understanding, particularly in the case of entropy. To account for this deficit, we

7In thermodynamics, entropy is understood to represent the degree of dispersion (or disorder) in the thermodynamical system’s micro state-space. To be more precise, according to the famous Gibbs formula the EntropyS is defined according toS = −kBP

ipilogpi, wherepi represents the probability of a finite system to reside in the statei, wherekBdenotes the Boltzmann-constant. Nowadays, entropy finds more and more use in social and economic sciences. For instance, see Hart (1971)

8Observe that without the chosen normalisation, entropy would increase with the sample size and thus the notion ofconcentrationwould consequently depend on the sample sizen.

provide an additional measure:the minimum fraction of elements of a vector that account for at least a fractionsof the total sum of the vector. More formally, letxe^q_i andye^qdenote the vectors y^qandx^q_i in descending order. We consider their partial sumsy^q_j =^Pj_i=1y_l^q

The smaller this number, the fewer states occupy spercent of the overall depth. To check that this construction does well behave, consider the casey^q = (0,0,0,0, Q) withQ >0. We have y_j =Qfor allj≥1. Hence,

Indeed 20 percent of the statej = 5occupy more than the fractionsof the total depth.

Cross-sectional averages for the time dispersion measures are obtained in the usual way. For instance, consider the estimates for the time entropy of stockk, i.e. φ^time_k as per (1.4.3). We construct the cross-sectional average as follows

φ^time :=

P_n

k=1σ⁻²(φ^time_k )φ^time_k Pn

k=1σ⁻²(φ^time_k ) . (1.4.6)

Similarly, to obtain cross-sectional averages for the price dispersion measures, we first con-struct the measures for each time and stock and average first over time and afterwards over the cross-section. For instance, consider the price entropies for each stock and time, i.e. φ^book_k =

1 n

Pn

i=1φ^book_k (i). And denote σ²(φ^book_k )the respective standard variance. Then we define the cross-sectional average according to

The same procedure is applied to the other measures, i.e.C^time,C^bookandL^timemL^book. Estimation Results

Results are shown in the tables 1.2 and 1.3 and grouped into liquidity quintiles. Estimates for the entropy (φ) are grouped in the first column, the coefficient of variation (C) in the second and the localisation measure (L) fors= 0.50,0.80are shown in the third and forth column of each table. We show estimates for hidden, displayed as well as total posted liquidity. Table 1.2 reports estimation of the averageprice-dispersionof liquidity, while table 1.3 reports estimates for thetime-dispersionof liquidity.

The results of table 1.2 and 1.3 can be broadly summarised in three points. First, the es-timates confirm the intuition that hidden liquidity is concentrated around few price quotes and few points in time. For instance, according to theL^book measure in table 1.2, on average 26% of price quotes already contain more than 80% of the hidden volume. The same degree of displayed liquidity is distributed across80%of the price quotes on average.

Second, the difference in the degree of dispersion between hidden and displayed liquidity seems is larger in the time domain than in the price domain.

Third, we observe that hidden liquidity varies significantly more than displayed liquidity.

Our unconditional coefficient of variation reports2.72for hidden and0.73for displayed liquidity in the price domain and2.88and1.23in the time domain.

Table 1.2: Estimates of localisation of hidden, displayed and total depth in the order book.

Estimates are shown for four measures,Φ^book,C^book,L^book_0.50 andL^book_0.80 Liquidity

Quintiles

Φ^book C^book L^book_0.50 L^book_0.80

hidden displayed total hidden displayed total hidden displayed total hidden displayed total

q₁

(least) 0.88 0.97 0.97 2.45 0.82 0.92 0.09 0.24 0.24 0.30 0.54 0.54 q₂ 0.87 0.97 0.97 2.61 0.77 0.87 0.09 0.26 0.25 0.30 0.55 0.56 q₃ 0.86 0.97 0.97 2.69 0.72 0.84 0.08 0.26 0.26 0.26 0.56 0.56 q₄ 0.84 0.98 0.97 3.04 0.69 0.88 0.06 0.28 0.26 0.23 0.57 0.57 q₅

(most) 0.85 0.98 0.98 2.81 0.66 0.78 0.06 0.29 0.27 0.22 0.59 0.59 all 0.86 0.97 0.97 2.72 0.73 0.86 0.08 0.27 0.26 0.26 0.56 0.56 Table 1.3:Estimates of localisation of hidden, displayed and total depth in time. Estimates are shown for four measures,φ^time,C^time,L^time_0.50 andL^time_0.80

Liquidity Quintiles

Φ^time C^time L^time_0.50 L^time_0.80

hidden displayed total hidden displayed total hidden displayed total hidden displayed total

q₁

(least) 0.20 0.62 0.58 3.11 1.46 1.87 0.10 0.20 0.15 0.13 0.37 0.29 q₂ 0.21 0.66 0.64 2.87 1.35 1.62 0.10 0.21 0.17 0.14 0.4 0.34 q₃ 0.20 0.72 0.69 2.81 1.22 1.41 0.11 0.23 0.20 0.14 0.44 0.39 q₄ 0.17 0.75 0.73 2.82 1.15 1.31 0.10 0.24 0.21 0.14 0.47 0.42 q₅

(most) 0.16 0.82 0.78 2.80 0.95 1.10 0.11 0.28 0.25 0.14 0.53 0.49 all 0.19 0.71 0.68 2.88 1.23 1.46 0.10 0.23 0.20 0.14 0.44 0.38