The Obizhaeva-Wang Transient Impact Model

Trade a large order and the price moves against you — that's Market Impact. But impact doesn't just appear and sit there forever. Eat through the offers, and a minute later new sellers show up at prices close to where you started, because the book heals. Simple impact models — push the price by an amount proportional to your trading rate, full stop — can't tell the difference between one order that eats a lot fast and pauses, versus one that eats a little steadily. Obizhaeva-Wang (2013) can, because it models the healing directly.

The picture: a block of liquidity you dig into

Think of the order book not as a list of prices but as a reservoir of resting volume sitting at each price level, with density $q$ shares per cent of price movement. A market order of size $x$ digs a hole of depth $x/q$ into that reservoir — the price moves by $x/q$ to absorb it. Left alone, the hole doesn't stay: liquidity providers refill it, and the price relaxes back toward where it was, at some decay rate $\rho$. Trade again before it's healed, and you're digging into a shallower reservoir, so the same size order moves the price further.

Formally, the price impact from a single trade of size $x$ at time $0$ decays as

$D(t) = \frac{x}{q} e^{-\rho t}$

In words: the price displacement starts at $x/q$ (same as before) and decays exponentially with half-life $\ln 2 / \rho$. Trade continuously at rate $\dot{x}_s$ and the impacts from every past instant sum up, each decaying at its own age — the total displacement at time $t$ is the accumulated, still-live residue of every trade so far.

Worked example: two trades, same total size

Suppose $q = 10{,}000$ shares per $1 of price, and $\rho$ gives a 5-minute half-life ($\rho \approx 0.139$ per minute). You need to buy 4,000 shares total.

Strategy A — one block, then wait. Buy all 4,000 shares in one clip. Impact: $4000/10000 =$ $0.40. Five minutes later, half has decayed: displacement is down to $0.20.

Strategy B — split into two, five minutes apart. Buy 2,000 now: impact $0.20. Wait five minutes — that impact decays to $0.10. Buy the remaining 2,000: adds another $0.20 on top of the $0.10 still outstanding, for $0.30 total, versus $0.40 for the block. Splitting and waiting for partial healing bought you a cheaper second clip, because you weren't trading against a book you'd already dug into as deeply.

Where it changes the answer

Almgren-Chriss (the workhorse cost-vs-risk optimal execution model) typically assumes impact is either fully permanent or fully instantaneous and temporary — it has no notion of partial healing over time. Obizhaeva-Wang shows that with resilience, the optimal trajectory front-loads trading with an initial block, then trades smoothly, then closes with another block — a "U-shaped" schedule — rather than the smooth, symmetric curve simple models suggest. The initial and final blocks exploit the fact that a single sharp trade against a fresh reservoir is efficient, while ongoing continuous trading avoids re-digging a hole that hasn't healed yet.

In interviews

Be able to state the mechanism in one sentence — "impact decays exponentially as the book refills, so recent trades matter more than old ones" — before touching the formula. If pushed on why a U-shaped schedule beats a flat one, the honest answer is that resilience makes concentrated trading against a fresh book cheaper per share than trading continuously into a book you're actively depleting, up to the point where impact starts compounding. Compare against the smoother schedules implied by The Square-Root Impact Law and plain Optimal Execution.