Order flow adverse selection and price impact

Definition Each trade $k$ gets a sign $\varepsilon_k = +1$ if it is buyer-initiated and $\varepsilon_k = -1$ if it is seller-initiated, where the initiator is the side whose marketable order arrived and executed against resting liquidity. With trade sizes $v_k > 0$ in shares, the signed order flow over an interval is the net buyer-initiated volume

in shares. The sum of signs $\sum_k \varepsilon_k$ is a different statistic, which counts trades instead of shares. When the data do not record the initiator, the sign must be inferred by a stated classification rule, for example by comparing the trade price with the prevailing midpoint, and under that rule a trade exactly at the midpoint stays ambiguous. The value of the statistic depends on the interval, the size measure and the sign rule, so all three are fixed before it is interpreted. A positive signed flow says that more volume was initiated by buyers. It does not say that the price must rise.

Definition A price response statistic relates later price changes to signed flow over a chosen lag. For trades indexed by $k$, with sign $\varepsilon_k$ and midpoint $m_k$ just before trade $k$, and for a stationary sequence of trades, a common form is the response function

where the lag $\ell$ counts trades. It is estimated by the sample average of $\varepsilon_k (m_{k+\ell} - m_k)$ over the trades. Another common form is the slope of a regression of interval midpoint changes on the signed flow of the interval. Defined this way, a response is an association. Information, the mechanical depletion of liquidity, dealer inventory, autocorrelated order flow (for example a large order split into many trades of the same sign), a price trend that began before the trades, and common news all contribute to it. A positive response does not identify which of these moved the price. Reading a response as the causal effect of trading needs a research design that separates these sources, for example trades whose timing is randomized independently of any information.

Definition A liquidity provider faces adverse selection when the trades it receives are more likely to come from counterparties whose information makes the trade unfavorable to it. Let $V$ be the value of the share in currency units, with mean $\mathbb{E}[V]$. A dealer who sells at the ask $a$ to a buyer gains $a - V$ per share and a dealer who buys at the bid $b$ from a seller gains $V - b$. Adverse selection means $\mathbb{E}[V \mid \text{buy}] > \mathbb{E}[V]$ and $\mathbb{E}[V \mid \text{sell}] < \mathbb{E}[V]$. A risk-neutral dealer whose competitors drive expected profit to zero sets each quote equal to the conditional expected value given that a trade arrives on that side,

so that $\mathbb{E}[a - V \mid \text{buy}] = 0$ and $\mathbb{E}[V - b \mid \text{sell}] = 0$. The spread then balances expected losses to informed traders against expected gains from uninformed ones. It can be positive with no inventory risk, no costs and no market power.

Definition The following finite model is constructed for this lesson in the spirit of the Glosten-Milgrom model. One trader arrives and trades one share. The value per share is $V \in \{V_L, V_H\}$ with $V_L < V_H$ and prior $\mathbb{P}(H) = \tfrac12$, where $H$ is the event $V = V_H$ and $L$ the event $V = V_L$. Independently of $V$, the trader is informed with probability $\theta \in [0, 1]$. An informed trader buys under $H$ and sells under $L$, and an uninformed trader buys or sells with probability $\tfrac12$ each, independently of $V$. A risk-neutral competitive dealer sets $a = \mathbb{E}[V \mid \text{buy}]$ and $b = \mathbb{E}[V \mid \text{sell}]$, with no fees, no tick size and no inventory cost. Under $H$ an informed trader buys for sure and an uninformed one with probability $\tfrac12$, so $\mathbb{P}(\text{buy} \mid H) = \theta + \tfrac12 (1 - \theta)$, and under $L$ only uninformed traders buy. Hence

so $\mathbb{P}(\text{buy}) = \tfrac12$ and Bayes’ rule gives

With the prior mean $\bar V = \tfrac12 (V_L + V_H)$, the ask is

and the same computation after a sell, where $\mathbb{P}(H \mid \text{sell}) = \tfrac12 (1 - \theta)$, gives $b = \bar V - \tfrac{\theta}{2} (V_H - V_L)$. The informational spread is $a - b = \theta\, (V_H - V_L)$. It increases with $\theta$. It vanishes at $\theta = 0$, because the order then has the same likelihood under both values and reveals nothing, and it equals $V_H - V_L$ at $\theta = 1$, when every trade reveals the value. The spread in this model comes from information alone. Fees, the tick size, inventory costs and order processing costs can create spreads that the model leaves out.

Definition In a stipulated impact model a trade of signed size $q$ shares, positive for a buy and negative for a sell, moves the midpoint from $m$ to

and the shift persists over the horizon of the model. The coefficient $\gamma \ge 0$ is in currency per share squared, so $\gamma q$ is in currency per share. Calling impact permanent is a modeling assumption that needs a stated horizon and a stated mechanism, and the word supplies neither. It does not say whether the shift comes from information or from mechanics, and the model takes $\gamma$ as given rather than estimating it. A dynamic impact model should also be checked on round trips. If buying and then selling the same quantity could earn a profit from the modeled price displacement alone, the model allows price manipulation. Whether a model avoids this depends on its impact functions, its timing convention, its costs and the admissible trading paths taken together. A positive fitted coefficient does not settle it, and this lesson derives no general no-manipulation theorem.

Definition Temporary impact is a displacement of the execution price that applies only to the trade that causes it and has dissipated by the next observation of the midpoint. In the linear stipulated model it is $\eta q$, with $\eta \ge 0$ in currency per share squared. The execution price also needs a timing convention. Here the trade is assumed to execute at the fully shifted midpoint plus the temporary displacement,

after which the midpoint is $m + \gamma q$. The cost of the trade relative to the pre-trade midpoint is then

which is nonnegative for buys and sells alike. Other conventions give other execution prices, for example a trade that sweeps through the book while the permanent shift builds up. The convention is part of the model, in the same way that the conditioning rule is part of the informational model.