Market making: the Avellaneda–Stoikov model

A market maker posts a bid and an ask, buys from sellers and sells to buyers, and aims to earn the spread many times a day. Two forces fight that income:

• Inventory risk: after a run of fills on one side, you hold a position exposed to price moves.
• Adverse selection: the counterparties most eager to trade with you are often the ones who know where the price is going.

The Avellaneda–Stoikov model

Avellaneda and Stoikov (2008) derived optimal quotes for a market maker with inventory $q$, risk aversion $\gamma$, mid price $s$ following Brownian motion with volatility $\sigma$, and time $T - t$ remaining in the session.

First, the reservation price — the price at which the market maker is indifferent to holding one more unit:

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

If you are long ($q > 0$), your reservation price sits below the mid, so you quote lower: more eager to sell, less eager to buy. If short, it sits above. This is inventory skew.

Second, the optimal total spread around $r$:

$$\delta^a + \delta^b = \gamma\,\sigma^2\,(T - t) + \frac{2}{\gamma}\ln\!\Big(1 + \frac{\gamma}{k}\Big)$$

where $k$ describes how quickly fill probability falls as you quote further from the mid (order arrival intensity decays like $A e^{-k\delta}$). The bid is placed at $r - \text{spread}/2$ and the ask at $r + \text{spread}/2$.

Reading the formulas

• Higher volatility → wider spread and stronger skew: inventory is riskier.
• Higher risk aversion $\gamma$ → stronger skew, faster inventory mean reversion.
• Larger $k$ (fill probability falls off quickly) → tighter spread, because quoting wide means you rarely trade.
• As $T - t$ shrinks, the inventory terms fade: near the end of the horizon you mainly want to get flat.

Adverse selection and protection

The model assumes fills are random. In reality, fills cluster just before the price moves against you. Practical defences:

• widen or pull quotes when volatility or order-book imbalance spikes,
• skew quotes using short-term signals (micro-price, trade flow),
• cap inventory hard and flatten with a market order if breached,
• cancel quickly; stale quotes are free options for faster traders.