Inventory: carrying risk, shedding it, and the reservation price

A market maker’s edge comes from the spread, but their losses come from the position. Inventory management is the discipline of staying near flat without paying away the edge to get there — and in an iterative trading game it is the thing that separates a candidate who quotes well from one who merely quotes.

Definition 4.1

The reservation price

Reservation price, $r = \text{theo} - q\,\gamma\,\sigma^2 (T - t)$ — Not what you think the asset is worth, but the price at which *you* — holding $q$ units, with risk aversion $\gamma$ and $T-t$ left to run — would be indifferent to holding it. Long inventory pushes it below theo, short pushes it above, and your quotes are placed symmetrically around it rather than around theo. That single substitution is the whole of inventory skewing.

Proposition 4.2

Which way to skew

Long inventory means you want to sell, so lower *both* sides: your offer becomes more attractive and your bid less so. The width stays roughly the same; the centre moves. Candidates often widen one side instead, which changes what they earn per trade rather than which trade they attract.

Holds when

Skew is about the midpoint of your quote; width is about your uncertainty. They are independent decisions.
Skewing *into* your inventory — bidding higher when already long — is the error the market-making simulator exists to expose.
The right amount of skew grows with position size, with volatility, and with how long you must carry it.

Example 4.3

Your theo is $100$ and you normally quote $99.50$ at $100.50$ . You are now long 50 lots against a limit of 100, and you judge a full position to be worth a quarter of a point of skew. Requote.

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Worked solution

Formula $r = \text{theo} - \text{skew} \times \frac{q}{q_{\max}}$
Substitute $= 100 - 0.25 \times \frac{50}{100}$
Solve $r = 100 - 0.125 = 99.875$
$\text{quote } r \pm 0.50$
Answer $99.375 \text{ at } 100.375$

Sanity check. Both sides moved down by an eighth and the width is unchanged at one point. You are now more likely to be lifted than hit, which is exactly the flow you want while long.

Aspect	Flatten now	Hold and skew
Costs	You cross the spread	You carry overnight risk
Right when	The position is large relative to your limit	The position is modest and flow is two-sided
Also right when	An event is coming that you cannot price	The market is quiet and you expect offsetting flow
The wrong reason	Discomfort with a position that is within limits	Hoping it comes back

Table 4.4 · Flatten or hold?. The two wrong reasons are the ones interviewers probe. Crossing the spread because you feel uneasy is paying real money for a feeling; holding because you hope it recovers has turned a market-making position into a directional bet you never intended.

Why you cannot just stay flat. The obvious answer to inventory risk is to hedge out every trade immediately, and the reason nobody does is that it costs a spread each time. If you earn a tick making a market and pay a tick flattening, you have worked for nothing. So a maker holds a position deliberately, up to the point where the risk of carrying it exceeds the cost of shedding it — and the whole art is in locating that point. Skewing is the cheap intermediate: it sheds inventory through the flow you were going to see anyway, at no cost in spread.

Inventory: carrying risk, shedding it, and the reservation price

After this lesson you should be able to

The reservation price

Which way to skew

The rest of this lesson is in Premium