Measuring Resilience: The Liquidity Half-Life

A market order eats through 4,000 shares resting on the ask, and the visible depth at the top of the book drops from 4,000 shares to 500. The interesting question isn't how thin the book got — it's how long it stays that thin. A book that refills to 3,500 shares within two seconds behaves very differently, for a trader working a large order, than one that takes two minutes to recover. Resiliency is the name microstructure researchers give to that recovery speed, and the liquidity half-life is the standard way to measure it: the time it takes for depth to recover half the distance back to its pre-trade level.

Defining the half-life

Let $D_0$ be the resting depth just before a trade, $D_1$ the depth immediately after, and $D(t)$ the depth $t$ seconds later as market makers replenish their quotes. Depth recovery is usually well approximated by exponential decay toward the old level:

$D(t) = D_0 - (D_0 - D_1) \, e^{-t/\tau}$

Here $\tau$ is a time constant, and the half-life is $t_{1/2} = \tau \ln 2 \approx 0.693\,\tau$ — the time for the depth gap to close by 50%. A smaller half-life means a more resilient book: liquidity providers are quick to notice a gap and step back in with fresh quotes. A larger half-life signals that liquidity providers are cautious, possibly because they suspect the trade that just cleared the book carried information and want to see how the price settles before committing more capital.

A worked example

Before a trade, the ask side shows 4,000 shares. A buy order clears it down to 500 shares. Five seconds later, depth has recovered to 2,250 shares. That's a recovery of $2{,}250 - 500 = 1{,}750$ shares out of a total gap of $4{,}000 - 500 = 3{,}500$ shares — exactly half the gap has closed in five seconds, so the half-life is 5 seconds. Fitting the exponential form gives $\tau = 5 / \ln 2 \approx 7.2$ seconds.

Now compare a second stock with the same starting depth of 4,000 and the same post-trade depth of 500, but where depth only reaches 1,200 shares after five seconds. The gap closed is $1{,}200 - 500 = 700$ out of $3{,}500$, only 20%, so the half-life is much longer — solving $0.5^{t/5}$-style algebra gives roughly 16 seconds. A trader slicing a large order into the first stock can return to that price level in a few seconds with confidence the book has reloaded; in the second stock, hitting the same level again inside ten seconds means trading against a book that's still hollowed out, and paying up for it.

Where this gets used

• Slicing algorithms use estimated half-lives to set the pause between child orders: if a book's half-life at a given price level is 8 seconds, sending the next slice after 2 seconds means trading against a book that hasn't recovered, paying more impact than necessary.
• Comparing venues and names: half-life is a cleaner cross-sectional resiliency metric than raw depth, because a stock can have shallow-but-fast-refilling liquidity (resilient) or deep-but-slow-refilling liquidity (fragile), and depth alone can't tell them apart.
• Stress detection: a sudden lengthening of half-life across many names — the book stops refilling as quickly as it used to — is an early, mechanical signal that liquidity providers are stepping back before spreads or volatility fully reflect it. See Cascading Liquidity Withdrawal In A Selloff for what happens when this compounds.

In interviews

If asked how you'd measure whether a stock's liquidity is "resilient," don't reach for average spread or average depth — those are snapshots. Describe the exponential recovery framework: measure depth right after a trade, measure it again some seconds later, and report the time to close half the gap. That answer signals you understand liquidity as a dynamic process, which is the distinction this topic is testing for, and it connects directly to Permanent Versus Temporary Impact — a market that heals fast is one where temporary impact decays quickly.

It's also worth naming the two failure modes an interviewer might probe: a short half-life with shallow post-trade depth (fast to refill, but never refills to much) versus a long half-life with deep post-trade depth (slow to refill, but eventually generous). The two describe genuinely different execution problems — the first rewards patience between clips, the second rewards spacing clips far enough apart that you aren't repeatedly trading into the same unhealed gap — and a strong answer distinguishes them rather than treating "resilient" as a single scalar good or bad.