Bid ask spread in a competitive market with institutions and order size

Abstract

Large orders, particularly from institutions, are quite common these days and hence there is interest to know if institutional trading has any bearing on the price effect associated with large trades. Recent empirical studies contradict earlier evidence of negative price effect on selling large blocks and find no price effect associated with large trades. Existing theoretical framework suggests a monotonic and increasing adverse price effect for large trades, where the motivation for a large trade is private information. We model a trading system where pure information, information-liquidity, and pure liquidity traders trade small and large sizes. The pure information traders strategically choose an order size. Institutions trade only large sizes because of their low execution costs for large trades; they are information-liquidity traders whose ability to use an information signal to determine their trades is subject to a binding liquidity constraint. We show that in such a market a separating equilibrium where trade size is informative does not exist and hence there is no price effect for large trades. Trade size may be revealing only if there is a buy sell asymmetry (large buy size is not equal to large sell size) or the corresponding price effect is asymmetric (price effect due to a large buy is not equal to that of a large sell). Further for a pooling equilibrium to exist, where trade size is not informative, the width of the market denoted by the ratio of order size (large size/small size) needs to be small, while the shallowness (inverse depth) of the market denoted by the ratio between pure information and institutional trades and the information signal needs to be stronger (higher). Our results on bid and ask prices and spread confirm recent empirical evidence on price effect of large and institutional trades found in the literature.

Appendix

The conditional probabilities are computed as follows:

$$ \delta ({S_1 })=\delta \frac{\varepsilon \alpha \theta +\varepsilon \pi ({1-\tau })+({1-\varepsilon } )({1-\tau })}{\delta \varepsilon \alpha \theta +\varepsilon \pi ({1-\tau })+({1-\varepsilon } )({1-\tau })} $$

$$ \delta ({S_2 })=\delta \frac{\varepsilon \alpha ( {1-\theta })+\varepsilon ({1-\alpha -\pi } )\gamma _H }{\delta \varepsilon \alpha ({1-\theta } )+\delta \varepsilon ({1-\alpha -\pi })\gamma _H +\varepsilon ({1-\delta })({1-\alpha -\pi } )\gamma _L } $$

$$ \delta ({B_1 })=\delta \frac{\varepsilon \pi \tau +({1-\varepsilon })\tau }{({1-\delta } )\varepsilon \alpha \theta +\varepsilon \pi \tau +( {1-\varepsilon })\tau } $$

$$ \delta ({B_2 })=\delta \frac{\varepsilon ( {1-\alpha -\pi })\gamma _L }{({1-\delta } )\varepsilon \alpha ({1-\theta })+\varepsilon \delta ({1-\alpha -\pi })\gamma _L +\varepsilon ( {1-\delta })({1-\alpha -\pi })\gamma _H } $$

Proof of Lemma 1

Note that separating conditions imply θ = 0. Hence $\delta ({S_1 })=\delta \left({\varepsilon \pi ({1-\tau } )+({1-\varepsilon })({1-\tau } )}/{\varepsilon \pi ({1-\tau })+( {1-\varepsilon })({1-\tau })}\right)$ which implies δ(S ₁) = δ.

Also note that $\delta ({S_2 })=\delta \left({\alpha +({1-\alpha -\pi })\gamma _H }/{\delta \alpha +\delta ({1-\alpha -\pi })\gamma _H +({1-\delta } )({1-\alpha -\pi })\gamma _L }\right)$. δ(S ₂) > δ if and only if α + (1−α −π )γ _H > δα + δ(1−α −π )γ _H + (1−δ)(1−α −π)γ _L , which implies $({1-\delta })\left[{\alpha +({1-\alpha -\pi })({\gamma _H -\gamma _L })} \right] \ge 0$. If all priors are positive, δ(S ₂) ≥ δ. Similarly, it is shown that δ(B ₁) = δ and δ(B ₂) ≤ δ.

Proof of Lemma 2

When 0 < θ < 1 informed traders trade both small and large quantities and hence

$$ \delta ({S_1 })=\delta \frac{\varepsilon \alpha \theta +\varepsilon \alpha ({1-\tau })+({1-\varepsilon })({1-\tau })}{\delta \varepsilon \alpha \theta +\varepsilon \alpha ({1-\tau })+({1-\varepsilon })({1-\tau })}. $$

δ(S ₁) ≥ δ if and only if ɛ α θ (1−δ) ≥ 0. If all priors are positive δ(S ₁) ≥ δ. Similarly, δ(S ₂) ≥ δ while δ(B ₁) ≤ δ and δ(B ₂) ≤ δ. Further for θ = (1−θ ) = 0.5, δ(S ₂) ≥ δ(S ₁) implies:

$$ \frac{\alpha \theta +({1-\alpha -\pi })\gamma _H }{\delta \alpha \theta +\delta ({1-\alpha -\pi })\gamma _H +({1-\delta })({1-\alpha -\pi })\gamma _L } \ge \frac{\varepsilon \alpha \theta +\varepsilon \alpha ( {1-\tau })+({1-\varepsilon })({1-\tau } )}{\delta \varepsilon \alpha \theta +\varepsilon \alpha ( {1-\tau })+({1-\varepsilon })({1-\tau } )}. $$

Clearly for δ > 0,

$$ \frac{\varepsilon \alpha \theta +\varepsilon \alpha ({1-\tau } )+({1-\varepsilon })({1-\tau } )}{\delta \varepsilon \alpha \theta +\varepsilon \alpha ( {1-\tau })+({1-\varepsilon })({1-\tau } )} > 1 $$

and

$$ \frac{\alpha \theta +({1-\alpha -\pi })\gamma _H }{\delta \alpha \theta +\delta ({1-\alpha -\pi })\gamma _H +({1-\delta })({1-\alpha -\pi })\gamma _L } > 1. $$

Therefore, θ = (1−θ) does not lead to conditional probabilities of small and large order sizes to be equal. Nevertheless, there exists a set of other priors for which δ(S ₂) ≥ δ(S ₁).

Proof of Proposition 1

A separating equilibrium exists if in case of a low (high) signal, the following two conditions are satisfied:

• (1) Pure information motivated traders uniformly prefer large size to small size, S ₂ over S ₁ (B ₂ over B ₁). Information motivated traders would do so if S ₂ (b ₂ − v _L ) >S ₁(b ₁ − v _L ) which implies $\frac{S_2 }{S_1 } > \frac{({b_1 -v_L })}{({b_2 -v_L })} > 1$.

• (2) Further information-liquidity traders trade large sell (buy) over large buy (sell) implying $\frac{S_2 }{B_2 } > \frac{({v_H -a_2 })}{({b_2 -v_L })}$.

Computation shows

$$ \frac{b_1 -v_L }{b_2 -v_L }=1+\delta \left[ {\frac{\gamma _H }{\gamma _L }-1+\frac{\alpha }{({1-\alpha -\pi })\gamma _L }} \right]. $$

Therefore a separating equilibrium exists on the sell side if

$$ \hbox{(I)}\; \; \frac{S_2 }{S_1 } > 1+\delta \left[{\frac{\gamma_H }{\gamma_L }-1+\frac{\alpha }{({1-\alpha -\pi })\gamma_L }}\right ] \quad \hbox{and}\quad \hbox{(II)} \; \; \frac{S_2 }{B_2 } > \frac{({v_H -a_2 } )}{({b_2 -v_L })}.$$

The buy side is symmetric and a separating equilibrium exists if

$$ \hbox{(I)}\; \; \frac{B_2 }{B_1 } > 1+({1-\delta }) \left[ {\frac{\gamma _H }{\gamma _L }-1+\frac{\alpha }{({1-\alpha -\pi })\gamma _L }} \right] \quad \hbox{and} \quad \hbox{(II)}\; \; \frac{B_2 }{S_2 } > \frac{({b_2 -v_L })}{({v_H -a_2 })}.$$

Proof of Proposition 2

A pooling equilibrium is feasible on the sell side if there exists a fraction of informed traders, θ > 0 who prefer a small size to a large size implying S ₁(b ₁ − v _L )  = S ₂(b ₂ − v _L ). Note that for θ > 0 b ₂ and b ₁ are as follows:

$$ \begin{aligned} b_1&=&\delta ({S_1 })v_L +({1-\delta ({S_1 })})v_H \hbox{, } \\ b_2&=&\delta ({S_2 })v_L +({1-\delta ({S_2 })})v_H \\ \end{aligned} $$

where

$$ \delta ({S_1 })=\delta \frac{\varepsilon \alpha \theta +\varepsilon \pi ({1-\tau })+({1-\varepsilon })({1-\tau })}{\delta \varepsilon \alpha \theta +\varepsilon \pi ({1-\tau })+({1-\varepsilon })({1-\tau })} $$

and

$$ \delta ({S_2 })=\delta \frac{\varepsilon \alpha ({1-\theta })+\varepsilon ({1-\alpha -\pi })\gamma _H }{\delta \varepsilon \alpha ({1-\theta })+\delta \varepsilon ({1-\alpha -\pi })\gamma _H +\varepsilon ({1-\delta })({1-\alpha -\pi })\gamma _L }. $$

Clearly for some 0 < θ < 1 the above conditions are true. Thus a pooling equilibrium is feasible on the sell and buy sides of the market respectively if the following conditions hold:

$$(\hbox{I)}\; \; \frac{S_2 }{S_1 }\le 1+\delta \left[ {\frac{\gamma _H }{\gamma _L }-1+\frac{\alpha }{({1-\alpha -\pi})\gamma _L }} \right ] \quad \hbox{and}\quad (\hbox{II)}\; \; \frac{S_2}{B_2}=\frac{({v_H -a_2 })}{({b_2 -v_L })}$$

$$(\hbox{I)}\; \; \frac{B_2 }{B_1 }\le 1+({1-\delta }) \left[ {\frac{\gamma _H }{\gamma _L }-1+\frac{\alpha }{({1-\alpha -\pi })\gamma _L }} \right ]\quad \hbox{and} \quad (\hbox{II)}\; \; \frac{B_2}{S_2}=\frac{({b_2 -v_L })}{({v_H -a_2 })}$$

Proof of Proposition 3

Spreads for small and large trades under two separating and pooling equilibrium conditions are as follows:

$$ sp_1^s =a_1^s -b_1^s =0 $$

$$ sp_2^s =\Upomega \frac{[ {\alpha +({1-\alpha -\pi } )({\gamma _H -\gamma _L })} ][ {\alpha +({1-\alpha -\pi })} ]}{[ {\alpha \delta +\delta ({1-\alpha -\pi })\gamma _H +({1-\delta } )({1-\alpha -\pi })\gamma _L } ][ {\alpha ({1-\delta })+\delta ({1-\alpha -\pi } )\gamma _L +({1-\delta })({1-\alpha -\pi } )\gamma _H } ]} $$

$$ sp_1^p =\Upomega \frac{\varepsilon \alpha \theta [ {\varepsilon \alpha \theta +\varepsilon \pi +({1-\varepsilon })} ]}{[ {\varepsilon \alpha \theta ({1-\delta } )+\varepsilon \pi \tau +({1-\varepsilon })\tau } ][ {\varepsilon \alpha \theta \delta +\varepsilon \pi ({1-\tau })+({1-\varepsilon })( {1-\tau })} ]} $$

$$ sp_2^p =\Upomega \frac{[ {\alpha ({1-\theta } )+({1-\alpha -\pi })({\gamma _H -\gamma _L })} ][ {\alpha ({1-\theta })+( {1-\alpha -\pi })} ]}{[ {\alpha \delta ( {1-\theta })+\delta ({1-\alpha -\pi })\gamma _H +({1-\delta })({1-\alpha -\pi })\gamma _L } ][ {\alpha ({1-\delta })({1-\theta } )+\delta ({1-\alpha -\pi })\gamma _L +( {1-\delta })({1-\alpha -\pi })\gamma _H } ]} $$

where $\Upomega=\frac{\sigma _v^2 }{({v_H -v_L })^{2}}$ .

We show below the derivations for pooling equilibrium spread for δ = .5. For a separating equilibrium, θ = 0.

$$ {sp_2^p}_{|\delta =0.5}=\frac{\sigma _v^2 }{({v_H -v_L} )^{2}}\frac{[ {\alpha ({1-\theta })+( {1-\alpha -\pi })({\gamma _H -\gamma _L })} ]}{.25({1-\alpha \theta -\pi })} $$

Therefore,

$$ \frac{\partial sp_2^p }{\partial \alpha }=\frac{\sigma _v^2 }{( {v_H -v_L })^{2}}\frac{[ {({1-\theta } )-({\gamma _H -\gamma _L })} ][ {1-\alpha \theta -\pi } ]+[ {\alpha ({1-\theta } )+({1-\alpha -\theta })({\gamma _H -\gamma _L })} ]\theta }{.25({1-\alpha \theta -\pi } )^{2}} > 0 $$

Similarly,

$$ \frac{\partial sp_2^p }{\partial \gamma _H }=\frac{\sigma _v^2 }{({v_H -v_L })^{2}}\frac{[ {1-\alpha -\pi } ]}{.25[ {\alpha ({1-\theta })+( {1-\alpha -\pi })} ]^{2}} > 0 $$

but

$$ \frac{\partial sp_2^p }{\partial \gamma _L }=-\frac{\sigma _v^2 }{({v_H -v_L })^{2}}\frac{[ {1-\alpha -\pi } ]}{.25[ {\alpha ({1-\theta })+( {1-\alpha -\pi })} ]^{2}} < 0 $$