6.3.3 The Ho–Stoll Dealer Model

Basic Prerequisites: 02 · The Inventory Problem and Stochastic Calculus & Itô’s Lemma.

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1. Intuition & Practical Objective

Ho & Stoll (1981) is the ancestor of every inventory-skew model, including Avellaneda–Stoikov. It is the first dealer model that asks the right question: a dealer chooses bid and ask to control his inventory over a horizon, balancing the spread he earns against the return uncertainty of the position he holds. Its answer is the two objects this folder is built on - the reservation price (linear in inventory) and the cost-anchored spread (interior, not degenerate).

The practical objective: understand where the skew rule comes from. The A–S reservation price of the A–S folder is the continuous-time descendant of Ho–Stoll’s discrete recursion; every production quote engine is a variant of the two-step recipe below:

1. Compute the reservation price $r(I)=\bar{S}-\gamma {\sigma }^{2}I\tau$ - the dealer’s personal fair value at inventory $I$.
2. Add a cost-anchored half-spread $a^{\ast }=c+\frac{1}{k}$ on each side - set by the order-processing cost $c$ and the fill elasticity $k$.

The two-step recipe is not a mathematical accident: the reservation price comes from inventory (return) uncertainty, the half-spread from transaction costs. The model keeps them separate, which is exactly why you can measure them separately (the econometrics lives in Spread Decomposition & the Roll Model).

The one-sentence result. “A CARA dealer with inventory $I$, horizon $\tau$, risk aversion $\gamma$ and per-share transaction cost $c$ quotes $\text{bid}=r(I)-a^{\ast }$ and $\text{ask}=r(I)+a^{\ast }$ with reservation price $r(I)=\bar{S}-\gamma {\sigma }^{2}I\tau$ and half-markup $a^{\ast }=c+\frac{1}{k}$ - an interior spread plus a linear inventory skew.”

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2. Mathematical Ground Truth & Derivations

2.1 The setup and the inventory recursion

Time is discrete with $N$ periods of length $\Delta$; the dealer’s horizon is $T$, $\tau =T-t$ remaining. State is inventory $I\in \{-Q,\dots ,Q\}$. In each period the dealer posts a bid $B$ and ask $A$. Over the period, exactly one of three things happens:

• a customer buys at the ask (dealer sells): $I\to I-1$, cash $+A$, probability ${\pi }_a(A)$;
• a customer sells at the bid (dealer buys): $I\to I+1$, cash $-B$, probability ${\pi }_b(B)$;
• no trade: $I$ unchanged, probability $1-{\pi }_a-{\pi }_b$.

The dealer maximizes the expected CARA utility of terminal wealth $W_T=$ cash $+I{S}_T$, where the reference price follows $S_{t+1}=S_t+\epsilon$, $\epsilon \sim \mathcal{N}(0,{\sigma }^2\Delta )$. The value function solves the Bellman recursion - the Ho–Stoll inventory recursion:

$$V_t(I)=\underset{A,B}{\max }[{\pi }_a(A)\mathbb{E}[V_{t+1}(I-1)]+{\pi }_b(B)\mathbb{E}[V_{t+1}(I+1)]+(1-{\pi }_a-{\pi }_b)\mathbb{E}[V_{t+1}(I)]].(\text{HS }1)$$

Substituting the exponential form $V_t(I)=-e^{-\gamma \text{cash}}{u}_t(I)$ and folding the Gaussian price jump through its moment generating function (the factor $c_J=\exp (\frac{1}{2}{\gamma }^2{J}^2{\sigma }^2\Delta )$ for holding $J$ over one step), the recursion on $u$ becomes

$$u_t(I)=\underset{A,B}{\min }[{\pi }_a{e}^{-\gamma A}{c}_{I-1}{u}_{t+1}(I-1)+{\pi }_b{e}^{+\gamma B}{c}_{I+1}{u}_{t+1}(I+1)+(1-{\pi }_a-{\pi }_b)c_I{u}_{t+1}(I)],(\text{HS }2)$$

with terminal condition $u_T(I)=1$. (The sign flips to a min because the exponential utility is negative; this is the same bookkeeping that makes the A–S HJB a minimization of $-\theta$.)

2.2 The reservation price

If the dealer is forced to hold $I$ shares to $T$ with no trading, terminal wealth is $I{S}_T$, whose certainty equivalent (computed in 02) is $\bar{S}I-\frac{1}{2}\gamma {\sigma }^2{I}^2\tau$. The reservation price - the price at which the dealer is indifferent between trading one more share and not - is the marginal value of a share:

$$\boxed{r(I)=\bar{S}-\gamma {\sigma }^{2}I\tau }$$

Linear in $I$ with slope $-\gamma {\sigma }^2\tau$ per share. This is Ho–Stoll’s central object and the A–S reservation price $s-q\gamma {\sigma }^2(T-t)$ is exactly this. In Ho–Stoll’s two-period solution the same structure appears through the reservation bid and ask,

$$r^a(I)=\bar{S}+(1-2I)\frac{\gamma {\sigma }^2\tau }{2},r^b(I)=\bar{S}+(-1-2I)\frac{\gamma {\sigma }^2\tau }{2},$$

whose difference is the reservation spread $r^a-r^b=\gamma {\sigma }^2\tau$ - the inventory-risk component of the quoted spread, which widens with return uncertainty ${\sigma }^2$ and with the horizon $\tau$.

2.3 The cost-anchored spread

Why does the dealer quote wide at all, and why is the spread interior rather than degenerate? Purely from inventory risk, with symmetric arrivals, the maximization is bang-bang (quote at the boundary). The interior spread comes from the transaction (order-processing) cost $c$ per trade and the elasticity of fill arrivals $\pi (\delta )=\Lambda e^{-k\delta }$ (prefactor $\Lambda$, markup $\delta$). The dealer’s expected revenue per sell at ask markup $\delta$ is $\Lambda e^{-k\delta }(\delta -c)$; the FOC $-k(\delta -c)+1=0$ gives

$${\delta }^{\ast }=c+\frac{1}{k}.(\text{HS }3)$$

So the half-spread ${\delta }^{\ast }=c+\frac{1}{k}$ balances the fixed cost $c$ (quote tight enough to still be profitable) against the fill elasticity $k$ (quote far enough to slow the bleed). It is interior, independent of inventory, and additive with the reservation spread: the full quote is

$$\text{ask}=r(I)+{\delta }^{\ast },\text{bid}=r(I)-{\delta }^{\ast }.$$

Reading the structure. The reservation price carries all the inventory dependence (the skew $-\gamma {\sigma }^{2}I\tau$); the half-spread carries the transaction-cost component. This clean separation - inventory risk in the centre, transaction cost in the width - is the Ho–Stoll legacy and the reason the spread can be decomposed empirically.

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3. Computational Implementation - reservation price + interior spread

We reproduce the two closed-form results: (1) the reservation price is linear in $I$ with slope $-\gamma {\sigma }^{2}T$; (2) the cost-anchored half-markup is interior at $a^{\ast }=c+1/k$; (3) the reservation spread widens with $\sigma$.

The reservation price is exactly linear in $I$ - slope $-0.4000$ per share, matching $-\gamma {\sigma }^{2}T=-0.1\cdot 4\cdot 1$. The revenue-optimizing half-markup is interior at $0.7167=c+1/k$. At $I=+4$ both quotes sit below the $100$ reference (centre $98.40$): the dealer is long and his own quotes pull him back to flat. And the reservation spread is quadratic in volatility ($0.10,0.40,1.60$ for $\sigma =1,2,4$) - the return-uncertainty component of the Ho–Stoll spread.

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4. Failure Modes & First-Principles Breakdowns

1. Pure-inventory quotes are bang-bang, not interior. If you drop the transaction cost $c$, the recursion’s optimum degenerates to corner quotes (quote at the boundary). The interior spread needs the cost-anchoring of $a^{\ast }=c+1/k$ - a spread with no processing-cost term is a modelling artefact, not a real strategy.
2. The reservation-price separation hides in discrete time. The clean split - inventory risk in the centre, transaction cost in the width - holds in the standard two-period/linearized solution. In full generality the two interact (holding a position changes the marginal value of the next trade); don’t over-claim the separation far from the solution’s assumptions.
3. The skew assumes $c$ and $k$ are known. The half-spread is set by the fill elasticity $k$, which must be estimated from the book and is non-stationary - it collapses in news exactly when quotes matter most. Mis-set $k$, and the interior optimum moves to the wrong place.
4. No adverse selection. Ho–Stoll prices pure inventory and transaction risk over symmetric noise flow. Informed flow that picks off the stale quote is outside the model - the reason the model must be combined with Glosten–Milgrom.

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5. References

• Ho & Stoll (1981), Optimal dealer pricing under transactions and return uncertainty, JFE 9(1), 47–73
• Ho & Stoll (1983), The dynamics of dealer markets under competition, Journal of Finance 38(4), 1053–1074
• Stoll (1978), The supply of dealer services in securities markets, JFE 3(2)
• Avellaneda & Stoikov (2008), High-frequency trading in a limit order book, Quantitative Finance 8(3)
• Guéant, Lehalle & Fernandez-Tapia (2013), Dealing with the inventory risk, Mathematics and Financial Economics 7(4)

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6. Connected Graph Bridges

• Back: 02 · The Inventory Problem
• Forward: 04 · Quote Skewing · Index Hub
• Sibling: A–S: Reservation Price & Optimal Spread · Spread Decomposition & the Roll Model