The Limit Order Book and the Break-Even ... - Marriott School

THE LIMIT ORDER BOOK AND THE BREAK-EVEN CONDITIONS REVISITED 7
the order size, v(δ) = E[ṽ|˜δ = δ], and v + (δ) = E[ṽ|˜δ > δ]. The expected profit of a
marginal offer is
u(x, y) = E(x − ṽ)I { δ>y} e = x(1 − F (y)) −
= (x − v + (y))(1 − F (y))
∫ ∞
y
v(u)dF (u)
The relevant region is D = {(x, y) : F (y) < 1}, and the probability of execution is
ɛ(x, y) = (1 − F (y)).
2. The Break Even Conditions
Following Rock (1990) and Glosten (1994), it is common to posit that when there
are infinitely many liquidity suppliers and it is costless to post offers then the supply
function satisfies the break even conditions:
[
]
x ≤ E ṽ|˜δ(y(·)) > y(x)
with equality whenever dy(x) > 0. The right hand side is commonly called the “upper
tail expectation.” We may have a strict inequality because there are prices at which
even competitive liquidity suppliers do not offer shares (e.g., prices that are below
the ask price).
An economic environment is said to be regular if: (i) There is a unique supply function,
which we denote by y ∞ (·), that satisfies the break even condition, and (ii) In the
relevant region D, the sign of u(·, ·) is negative above the graph of y ∞ (x) and positive
below the graph. 8 We denote by ask ∞ the ask price that corresponds to the supply
function y ∞ (·).
Though our analysis is general, we are only interested in regular economic environments
where the argument for the break even conditions sounds appealing. In a
regular economic environment, the following procedure can be used to construct the
supply function y ∞ (·): ask ∞ is the smallest positive root of x → u(x, 0). For all
x < ask ∞ , we set y ∞ (x) = 0. For all x > ask ∞ , we set y ∞ (x) to be the value where
the function y → u(x, y) changes signs.
8 The graph of y ∞ (·) is a subset of the positive quadrant: it is the collection of all the pairs (x, y ∞ (x)).