Inventory risk and quote skewing

A market maker earns the spread on every round trip, but between the two halves of a round trip it holds a position, and the price keeps moving. "Market makers and liquidity" showed one such trip; this lesson asks what happens when many trips overlap and positions pile up on one side. It uses the idea from "Random walks and Brownian motion" that price changes behave like a random walk, measures the risk that position creates, and shows the maker's main tool for controlling it: shifting both quotes.

Inventory profit and loss

The maker's inventory $q$ is the number of shares held: positive when long, negative when short. If the mid price changes by $\Delta m$ while the position is unchanged, the profit or loss is

Suppose the mid moves like a random walk whose standard deviation is $\sigma$ per $\sqrt{\text{second}}$. Over $\tau$ seconds $\Delta m$ has mean 0 and standard deviation $\sigma\sqrt{\tau}$, so the position's P&L has mean 0 and

The risk grows in proportion to the size of the position and to the square root of the time it is held. Doubling the position doubles the standard deviation and quadruples the variance. The maker is not paid for this risk in expectation, since the mean is zero, so the sensible aim is to hold only as much of it as the spread income justifies.

How inventory builds

Inventory changes only when a customer trades with the maker. A fill at the bid adds a lot (here 100 shares) and a fill at the ask removes one. Which side fills next is random, so even with quotes that are symmetric around the mid the position drifts. Suppose each side fills at rate $\lambda$ per second. In $T$ seconds the maker sees about $2\lambda T$ fills, each $+1$ or $-1$ lot with equal probability, so the position after $T$ seconds is a random walk with variance $2\lambda T$ lots$^2$. Its typical size grows like $\sqrt{T}$, exactly as a price does.

A quote at distance $\delta$ from the mid fills at the rate $\lambda(\delta) = A\,e^{-k\delta}$: the closer the quote, the more often it is hit. With $A = 0.25$ per second, $k = 30$ per USD and $\delta = 0.02$ USD, each side fills at $0.25\,e^{-0.6} \approx 0.137$ per second, about once every seven seconds. Such a maker will often end a minute holding several lots, none of which it chose.

Skewing the quotes

The maker can influence which side is hit. If it is long, it wants a sell to come before a buy. It achieves that by moving both quotes down: the ask becomes cheaper and is hit more often, the bid becomes less attractive and is hit less often. This is called skewing the quotes. Let $\ell = q/100$ be the inventory in lots and $\theta$ the shift per lot, in USD. The bid and ask are then at distances

from the mid. The quotes' centre is $m - \theta\ell$ and the spread $2\delta$ is unchanged. The expected change in inventory per second is the bid fill rate minus the ask fill rate:

where $\lambda = A e^{-k\delta}$ and the approximation holds for small $k\theta\ell$. The drift is proportional to $-\ell$: long positions are pushed down and short ones up. Inventory is no longer a plain random walk but is pulled back to zero at rate $\kappa = 2\lambda k\theta$. Balancing the random fills against this pull gives a long-run variance of $1/(2k\theta)$ lots$^2$, which does not depend on $A$ and falls as $\theta$ grows.

What skewing costs

Skewing reduces risk but it is not free. When the maker is long, its ask is closer to the mid than $\delta$, so each share it sells earns less than the full half-spread. It also fills less often on the side that would add to its position. Average spread income falls as $\theta$ rises, while the standard deviation of the final position and of the P&L falls too. The figure runs 200 simulated minutes with the same random numbers for every $\theta$, so the changes you see are caused by the skew alone.

Start with the skew at 0. The final inventory has a standard deviation of about 372 shares, the mean P&L is 36.01 USD and the P&L has a standard deviation of 39.62 USD. Drag the marker down, or use the slider, to 0.20 cents per 100 shares: the inventory spread falls to 253 shares and the P&L standard deviation to 34.50 USD, while the mean falls to 33.59 USD, about 7% less. At 0.40 cents the inventory spread is 202 shares and the P&L standard deviation 29.58 USD, but the mean has fallen to 30.39 USD. How much to skew depends on how much the maker dislikes risk, which is the question the next lessons answer precisely.

Hedging instead of skewing

There is a second way to remove inventory: trade it away. The maker can cross the spread, selling at the bid what it holds, and pay the half-spread plus any fee. Or it can trade a correlated instrument, such as a future or an exchange-traded fund, and leave the shares in place; this is the idea of "Delta and hedging", applied to an equity position rather than an option. Crossing the spread is certain and immediate but costs money on every share. Skewing costs nothing up front and works slowly, so it suits small positions and calm markets. Hedging suits large positions, fast markets, or a maker who cannot wait for customers to arrive.

A cost of 6.50 USD for removing a 127.42 USD tail loss is cheap, which is why makers flatten large positions rather than wait. With a position of 50 shares the same trade buys little, and skewing is enough.

Summary

• Inventory $q$ earns $q\,\Delta m$; its P&L standard deviation is $|q|\,\sigma\sqrt{\tau}$, so risk grows with the size of the position and the square root of time.
• With symmetric quotes, random fills make inventory a random walk whose typical size grows like the square root of time.
• Moving both quotes down by $\theta$ per lot when long makes the expected inventory change $-2\lambda\sinh(k\theta\ell)$, which pulls inventory back to zero.
• Skewing lowers the variance of inventory and P&L but gives up some spread income; the figure shows both effects.
• Crossing the spread or trading a correlated instrument removes inventory at a certain cost, which pays off for large positions.

The next lesson, "Optimal quoting: the Avellaneda–Stoikov model", chooses the skew from the maker's risk aversion instead of treating it as a free number.