Inventory models (Ho–Stoll, Amihud–Mendelson)
M1 — Market microstructure
Glosten–Milgrom explained spreads with information. Now switch information off completely — nobody is informed, every trader is noise, no order tells the maker anything about value.
Spreads still exist. Something else is going on, and M0.4 already named it: inventory risk. This lesson is that component, done properly.
The reason it deserves its own model is that it produces price behaviour that looks identical to information on a chart and is economically its opposite. Telling them apart is a live problem, and M1.5 is entirely about it.
The maker’s real problem
A maker who buys at the bid is not sitting on a profit. It is sitting on a position it did not want, exposed to the price moving before it can get flat. Do that a few hundred times in one direction and the accumulated position, not the spread, dominates the P&L.
So the maker is not neutral about which side it trades. It is actively steering. Ho and Stoll (1981) formalise this by making the maker risk-averse and asking what quotes it should post given the position it already has.
Deriving the skew
Take a maker with constant absolute risk aversion, U(x) = −exp(−γx), currently holding q units of an asset whose price is a random walk with variance rate σ², over a horizon τ until it expects to be flat.
Under CARA with normal outcomes, the certainty equivalent is mean minus half gamma times variance:
holding q units to the horizon:
E[terminal wealth] = q·S
Var[terminal wealth] = q²σ²τ
CE(q) = q·S − ½ γ q² σ² τ
─────────────
the risk penalty — QUADRATIC in position
The maker’s reservation price for one more unit is the price at which it is indifferent between holding q and holding q + 1 — the most it would pay, or the least it would accept:
r = CE(q+1) − CE(q)
= S − ½γσ²τ·[(q+1)² − q²]
= S − ½γσ²τ·(2q + 1)
≈ S − γ q σ² τ (for one unit against a large q)
────────────────────────────────────────────────────────────────
r = S − γ q σ² τ
That single term γqσ²τ is the whole lesson. Read it slowly:
- Linear in inventory q, and signed. Long → reservation price below mid → the maker shades its quotes down so its ask gets hit and it sheds risk. Short → shades up. Flat → quotes symmetric about the mid.
- Linear in variance σ², so quadratic in volatility. Double the volatility and the skew quadruples. This is why makers pull in hard when things get noisy.
- Linear in horizon τ. Risk you must carry longer costs more, which is exactly why makers flatten into the close — as τ → 0 the penalty vanishes, and a position held overnight has a much larger τ.
- γ is the maker’s own risk aversion, not a property of the market. Two makers facing identical books will quote differently, which is why there is competition to model.
What it feels like in numbers
Stock at £100, daily volatility 2% so σ = £2 and σ² = 4, horizon τ = 1 day, risk aversion γ = 10⁻⁵.
inventory q σ (daily) skew = γqσ²τ maker quotes centred on
──────────────────────────────────────────────────────────────────────
0 £2 0 100.00 (symmetric)
+5,000 long £2 −£0.20 99.80 (leaning to SELL)
+10,000 long £2 −£0.40 99.60
−5,000 short £2 +£0.20 100.20 (leaning to BUY)
+5,000 long £4 −£0.80 99.20 ◀ vol doubled,
skew ×4
The mid is still 100.00 in every row. The market’s belief about value has not moved at all. What moved is one maker’s willingness to hold more of it.
The distinction that matters most in M1
ADVERSE SELECTION (M1.2) INVENTORY (M1.3)
─────────────────────────────────────────────────────────────────
the maker LEARNED something the maker learned NOTHING
the MID moves the mid stays; QUOTES skew
around it
driven by the market’s belief driven by one participant’s
about value position and risk appetite
the move is PERMANENT the move MEAN-REVERTS once
the maker is flat again
compensation for being picked compensation for warehousing
off by the informed risk nobody else wanted
That third pair of rows is the one to keep. Information changes the price. Inventory changes who wants to trade at it. On a chart both look like “price went down”; only one of them stays down.
This is also the honest reason a real spread can’t be read off either model. M0.4’s three components are: order-processing (small, boring), inventory (this lesson), and adverse selection (M1.2). A live quote carries all three at once, and decomposing an observed spread into them is an empirical problem, not an algebraic one.
Amihud–Mendelson: illiquidity has a price
Ho–Stoll is about one maker on one day. Amihud and Mendelson (1986) ask the cross-sectional question: if trading costs money, what does that do to asset prices?
The argument is short. If you must pay half the spread to get in and half to get out, your net return is the gross return minus the round trip. Rational investors will only hold a wide-spread asset if its gross return is high enough to compensate. So:
Illiquidity is priced. In equilibrium, assets with wider spreads must offer higher expected gross returns — an illiquidity premium.
With a genuinely useful corollary, the clientele effect: the cost is paid per round trip, so it is amortised over your holding period. A trader turning over daily cannot afford a wide spread; a pension fund holding for a decade barely notices it. Illiquid assets therefore migrate to long-horizon holders, and each asset ends up with the clientele whose horizon matches its spread.
This is where microstructure stops being a story about plumbing and starts affecting what things are worth — a wider spread doesn’t just cost you on the way in, it permanently lowers the price everyone will pay.
Source: Ho & Stoll (1981) for the original; the modern, implementable statement is Avellaneda & Stoikov (2008), “High-frequency trading in a limit order book”, whose reservation price is exactly the r above and which is short and unusually clear. Amihud & Mendelson (1986), “Asset pricing and the bid-ask spread”, for the illiquidity premium. O’Hara ch.2 covers the inventory literature.