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Liquidity supply and adverse selection in a pure limit order book market 107
Assuming the linear specification in Eq. (1) for g (m, X), and dividing by the
unconditional probability, P (m ≥ Q), Eq. (16) simplifies to
R Emm ½ j QŠ
X ¼0: (17)
Eq. (17) highlights that the expected profit of a limit order trader depends on the
upper tail expectation of the market order distribution.
Assuming exponentially distributed market order sizes as in Eq. (2) we have
Using R=p − γ this yields
Emm ½ j QŠ
¼ Q þ (18)
p X
Q ¼
; (19)
which is a generalized form of Eq. (3). Without the distributional assumption, the
equivalent of Eq. (19) is
p X
Emm ½ j QŠ
¼
Replacing E[m∣m ≥ Q] by the conditional sample mean bE½mm j QŠ,
i.e. the
observed upper tail market order distribution in the sample, one can construct
update and break even moment conditions for GMM estimation which do not
require a parametric assumption of market order sizes.
So far, the results are valid for an order book with a continuous price grid. We
now focus on a specific offer side quote with price p +k and corresponding limit
order volume q +k. Abstracting from the discreteness of limit order size shares and
assuming that the execution probabilities for all units at the quote tick p +k are
identical, we calculate the expected profit of all limit orders with identical limit
price p +k by integrating the left hand side of equation, Eq. (17), viz 17
Z Qþk
ðpþkEmm ½ j QŠ
X ÞdQ Pm ð Qþk 1Þ:
(21)
Qþk 1
Assuming exponentially distributed order sizes and subtracting quote specific
fixed execution costs ξ yields the total expected profit of the limit order volume at
price p+k. Dividing by the volume at quote q+k, yields the average expected profit
per share at the +kth quote,
pþk X
qþk
Qþk þ
qþk
2
(20)
Pm ð Qþk 1Þ:
(22)
17 The same result can be derived using the precise probabilities and a first-order Taylor
approximation for the emerging exponential terms.