recent developments in high frequency financial ... - Index of

Liquidity supply and adverse selection in a pure limit order book market 95
the liquidity supply equations which are implied by the zero expected profit
condition can be written as
0
1
B
qþk;t ¼ 2B
@
0
B
q k;t ¼ 2B
@
pþk;t Xt
Xt p k;t
qþk;t
q k;t
þ
C
A Qþk 1;t k ¼ 1; 2; ::: ðask sideÞ
1
C
A Q kþ1;t k ¼ 1; 2; ::: ðbid sideÞ:
ξ denotes the fixed cost component which is assumed to be identical for each price
tick in the order book. To derive the equations in Eq. (10), we have retained the
parametric assumption about the distribution of trade sizes. Considering a nonparametric
alternative along the lines described in the previous subsection is also feasible.
Proceeding as above, i.e. by eliminating the unobserved fundamental asset value Xt by
adding the bid and ask side equations for quote +k and −k yields the following
unconditional moment restrictions which we refer to as average break even conditions,
E p k;t 2
qþk;t q k;t
1
2 Qþk;t þ 2 þ 1
2 Q k;t
(10)
¼ 0 k ¼ 1; 2;:::;
where Δp ± k,t = p +k,t − p − k,t. Subtracting deviations from the implied depths at the kth
quote at time t +1andt and taking expectation yields the following equations which
we refer to as average update conditions,
E pþk;tþ1 qþk;t
(11)
þ
qþk;t 1 2 Qþk;tþ1 ð Qþk;tÞ
mt ¼0 k ¼1; 2; :::
E p k;tþ1þ
qþk;t qþk;t
þ ðQk;tþ1 1 2
Q k;tÞ
mt ¼0 k ¼1; 2;:::;
(12)
where Δp j,t +1 = p j, t +1− p j ,t. The average break even and update conditions replace
the marginal break even and update conditions of Eqs. (4) and (5).
4 Empirical results
4.1 Performance comparisons
Using the DAX30 order book data we follow Såndas (2001) and estimate the
model parameters exploiting the marginal break even conditions (4) and the
marginal updating conditions (5) along with Eq. (6). To construct the moment
conditions we use the respective first four best quotes, i.e. k=1, . . . ,4 on the bid and
the ask side of the visible order book. This yields thirteen moment conditions: four