Skewing Quotes To Manage Inventory

A market maker who is flat quotes symmetrically around fair value: buy at 99.99, sell at 100.01, one cent off the mid on each side. Now suppose a wave of buying has left them long 5,000 shares. Quoting symmetrically again just invites more of the same — the market clearly wants to buy, so the maker's offer at 100.01 keeps getting hit and the bid at 99.99 sits untouched, and the position grows. The fix is skewing: move both quotes down. Bid at 99.97, offer at 99.99. The bid is now less attractive to sellers (fewer people want to sell to you at a worse price), while the offer is now more attractive to buyers (they can buy from you cheaper than the old 100.01) — so the maker is more likely to sell off the unwanted long and less likely to add to it.

Why skew instead of just cancelling one side

Pulling the offer entirely stops the bleeding but earns nothing and signals to the market that you're stuck long. Skewing keeps both quotes live — still earning spread on whichever side fills — while tilting the odds of which side fills next. It's the difference between shutting the shop and just changing the prices.

The mechanics: reservation price and inventory penalty

The Avellaneda-Stoikov framework (see The Avellaneda-Stoikov Model) formalises this with a reservation price $r$ that sits away from fair value $s$ in proportion to current inventory $q$:

$r = s - q \gamma \sigma^2 (T - t)$

In words: fair value $s$ gets adjusted down by an amount that grows with how long you're inventoried ($q$), how averse you are to risk ($\gamma$), how volatile the asset is ($\sigma^2$), and how much time remains before you must be flat ($T-t$). A market maker then centres both quotes on $r$, not on $s$, and posts a spread around it. Being long ($q>0$) pulls $r$ below $s$, which is exactly the "quote lower to sell more, buy less" move from the example above.

Worked example

Fair value $s$ = $50.00. Maker is long $q = 2{,}000$ shares. Risk aversion $\gamma = 0.0005$, volatility $\sigma^2 = 0.04$ (annualised), and 0.5 trading days remain until the desk wants to be flat ($T - t = 0.5/252 \approx 0.00198$ years).

$r = 50.00 - 2000 \times 0.0005 \times 0.04 \times 0.00198 \approx 50.00 - 0.000079$

With this small inventory and short horizon the skew is tiny — a fraction of a tenth of a cent — because $\gamma$, $\sigma^2$, and the remaining time are all small numbers multiplying each other. Push the same $q$ out to 40,000 shares and a full trading day remaining ($T-t = 1/252 \approx 0.00397$):

$r = 50.00 - 40000 \times 0.0005 \times 0.04 \times 0.00397 \approx 50.00 - 0.0032$

Now the reservation price sits about a third of a cent below fair value — small per share, but it's the direction and consistency that matters: every quote update leans the same way until the position comes down, compounding into a meaningful nudge over thousands of quotes.

In interviews

Be ready to explain skew without the formula first — "shift both quotes in the direction that makes the market more likely to take your unwanted position off you" — then bring in $r = s - q\gamma\sigma^2(T-t)$ to show why each variable belongs: bigger position, more volatility, or more time pressure all mean skew harder.