Ho-Stoll Inventory Model

Ho-Stoll (1981): The Inventory Risk Model

The trading-fundamentals article introduced the market maker’s dilemma: earn the spread, but bear the risk of inventory moving against you. Ho and Stoll (1981) turned this intuition into a precise result using expected utility theory.

This is the first of the three canonical microstructure models. It answers: how should a risk-averse market maker set quotes given their current inventory?

Setup

A monopolist market maker quotes a bid $b$ and ask $a$ for a single asset. The true (efficient) mid-price is $\mu$. Between quote updates, the asset price moves with variance ${\sigma }^2$. The maker currently holds inventory $q$ units.

The maker has CARA (constant absolute risk aversion) utility:

$U(W)=-e^{-\gamma W}$

where $\gamma >0$ is the coefficient of absolute risk aversion and $W$ is terminal wealth.

Why CARA?

CARA gives a closed-form solution because it separates the effect of wealth level from risk attitude. Under CARA, the certainty equivalent of a normal gamble $X\sim \mathcal{N}({\mu }_X,{\sigma }_X^2)$ is:

$CE={\mu }_X-\frac{\gamma }{2}{\sigma }_X^2$

This mean-variance tradeoff is exact (not approximate) for CARA + normal returns — a convenient analytical property that makes the derivation tractable.

Derivation

Step 1: Wealth Under Each Scenario

If a buy order arrives (the maker sells one unit at the ask $a$):

$W_{\text{sell}}=W_0+a+(q-1)\cdot \tilde{P}$

The maker receives $a$, but now holds $q-1$ units exposed to the uncertain future price $\tilde{P}\sim \mathcal{N}(\mu ,{\sigma }^2)$.

If a sell order arrives (the maker buys one unit at the bid $b$):

$W_{\text{buy}}=W_0-b+(q+1)\cdot \tilde{P}$

If no trade occurs:

$W_0+q\cdot \tilde{P}$

Step 2: Indifference Condition

The maker sets $a$ and $b$ so that they are indifferent between trading and not trading. Using the CARA certainty equivalent:

Ask (indifference between selling and no trade):

$\underset{\text{CE if sell}}{\underbrace{a+(q-1)\mu -\frac{\gamma }{2}(q-1{)}^2{\sigma }^2}}=\underset{\text{CE if no trade}}{\underbrace{q\mu -\frac{\gamma }{2}{q}^2{\sigma }^2}}$

Solving for $a$:

$a=\mu +\frac{\gamma {\sigma }^2}{2}(2q-1)$

Bid (indifference between buying and no trade):

$\underset{\text{CE if buy}}{\underbrace{-b+(q+1)\mu -\frac{\gamma }{2}(q+1{)}^2{\sigma }^2}}=\underset{\text{CE if no trade}}{\underbrace{q\mu -\frac{\gamma }{2}{q}^2{\sigma }^2}}$

Solving for $b$:

$b=\mu -\frac{\gamma {\sigma }^2}{2}(2q+1)$

Step 3: The Key Results

The adjusted mid-price (midpoint of the maker’s quotes):

$\text{adjusted mid}=\frac{a+b}{2}=\mu -\gamma {\sigma }^2\cdot q$

This is the central result:

$\boxed{\text{adjusted\_mid}=\text{true\_mid}-\gamma {\sigma }^2\cdot q}$

The maker shifts their entire quote schedule away from inventory:

• Long inventory ($q>0$): lower the mid to attract sellers and repel buyers, reducing the position
• Short inventory ($q<0$): raise the mid to attract buyers

The magnitude of the shift scales with risk aversion $\gamma$, price variance ${\sigma }^2$, and inventory level $q$. Each multiplier is intuitive: more risk-averse makers adjust more; more volatile assets require larger adjustments; larger positions demand more aggressive rebalancing.

The spread:

$s=a-b=\gamma {\sigma }^2$

The spread is independent of inventory. Inventory affects where the quotes are centered, not how wide they are. Width is determined solely by risk aversion and volatility.

Interpretation

The Ho-Stoll model separates two distinct effects:

1. Spread width ($\gamma {\sigma }^2$): compensation for bearing one unit of price risk across one trading interval
2. Quote skew ($-\gamma {\sigma }^{2}q$): inventory management through asymmetric pricing

A market maker with zero inventory quotes symmetrically around the true mid. As inventory accumulates, quotes slide — the maker uses price to manage risk, not just to earn fees.

Connection to AMM Impermanent Loss

Constant-product AMMs (Uniswap v2) are market makers with $\gamma =0$ — they never adjust their mid in response to inventory. The bonding curve $x\cdot y=k$ mechanically sets prices based on reserves, but it cannot skew quotes to shed unwanted inventory.

This is precisely why impermanent loss exists: the AMM keeps quoting symmetrically even as informed flow pushes its inventory in one direction. A Ho-Stoll market maker would widen or skew; the AMM cannot. The LP absorbs the inventory risk that a rational market maker would price.

Uniswap v3’s concentrated liquidity gives LPs partial control — they can choose a range, effectively setting a conditional skew. But it is still reactive (withdraw and reposition) rather than continuous (adjust quotes in real time).

Limitations

• The model assumes a monopolist market maker. In competitive markets, spreads compress below $\gamma {\sigma }^2$ as makers compete for flow.
• No adverse selection: all traders are equally uninformed. This is the gap that glosten-milgrom-model fills.
• Single period: the maker optimizes over one interval. Multi-period extensions (Avellaneda & Stoikov 2008) add optimal control.
• Normal returns: CARA + normal is tractable but ignores fat tails.

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Companion notebook: notebooks/market-microstructure/03-ho-stoll.py — simulate a Ho-Stoll market maker, visualize quote skew as inventory changes, compare P&L against a symmetric quoter.

Questions to sit with:

1. If $\gamma {\sigma }^2$ sets the spread, what happens in a market where volatility suddenly doubles? How quickly must the maker react?
2. The AMM has no $\gamma$ parameter — it always trades. Is there a DeFi mechanism that could introduce inventory-dependent skew on-chain?
3. The spread is independent of inventory in this model. Is that realistic? Under what conditions would you expect spread to widen with inventory?