Market Making and Liquidity Provision › Inventory Risk

The central challenge in market making is inventory risk, the risk that accumulated positions will lose value before they can be unwound. If you buy more than you sell, you accumulate a long position. If prices then fall, you suffer losses that may exceed the spread revenue earned. This risk is unavoidable because you cannot control the timing or direction of incoming orders. You must quote continuously, but you cannot determine which of your quotes will be executed.

Consider a scenario where you start the day flat (no position). Over the course of trading, order arrivals are random. Sometimes more buyers arrive, causing you to sell and become short. Other times more sellers arrive, causing you to buy and become long. Your inventory thus evolves as a consequence of the random sequence of order arrivals, not as a result of any deliberate trading decision.

We can model inventory evolution as a random process. This mathematical framework helps us understand why inventory accumulation is such a persistent concern. Let $q_t$ denote your inventory at time $t$. Each trade changes inventory by $\pm 1$ (assuming unit trade sizes):

$$q_{t+1} = q_t + \Delta q_t$$

where:

• $q_t$: current inventory level
• $\Delta q_t$: change in inventory (+1 for buy, -1 for sell)
• $q_{t+1}$: inventory level after the trade

If a market sell order arrives, you buy ($\Delta q_t = +1$). If a market buy order arrives, you sell ($\Delta q_t = -1$). Notice that you take the opposite side of each incoming order, which is what it means to "make a market." The inventory change is thus determined by the arriving order, not by your preference.

The simulation shows that even with perfectly balanced order flow, where buys and sells arrive with equal probability, inventory follows a random walk and can drift far from zero. This is a mathematical consequence of the random walk properties: the expected position is zero, but the variance of the position grows with the number of trades. This drift is dangerous because it exposes you to directional price risk, exactly the risk you don't want to take. You aim to profit from the spread, not from predicting price direction. Yet without active management, you inevitably accumulate positions that bet on price direction.

To understand how to manage inventory risk, we must first quantify it precisely. The P&L impact of inventory comes from price changes. If you hold inventory $q$ and the price changes by $\Delta P$, the inventory P&L is:

$$\text{Inventory P\&L} = q \times \Delta P$$

where:

• $q$: current inventory position
• $\Delta P$: change in asset price

This formula captures the essence of the risk: your profit or loss depends on how much inventory you hold and how much the price moves. A long position profits when prices rise and loses when prices fall; a short position has the opposite exposure.

The variance of inventory P&L over a short time interval $\Delta t$ can be derived from the price variance. This derivation connects our understanding of price dynamics to the specific risk you face. Assuming the asset price $P$ follows a random walk with percentage volatility $\sigma$:

$$\begin{aligned} \text{Var}[\text{Inventory P\&L}] &= \text{Var}[q \Delta P] \\ &= q^2 \text{Var}[\Delta P] && \text{(properties of variance)} \\ &= q^2 (P \sigma)^2 \Delta t && \text{(substitute price variance)} \end{aligned}$$

where:

• $q$: inventory size
• $\Delta P$: change in asset price
• $P$: current asset price
• $\sigma$: annualized volatility (percentage)
• $\Delta t$: time interval

This result reveals a critical insight: inventory risk grows quadratically with position size. Holding twice the inventory creates four times the variance, not twice the variance. This non-linear relationship means that large positions are disproportionately risky. If you hold 100 shares, you face four times the risk variance of holding 50 shares, even though you hold only twice as many shares. This quadratic scaling provides strong incentive to keep positions small.