Market Microstructure Foundation Dealer Inventory Control Ho & Stoll (1981)

Ho-Stoll (1981) Market Making & Dealer Pricing Lab

Explore Thomas Ho and Hans Stoll's (1981) foundational dealer inventory control model. Simulate optimal bid-ask quote shading, reservation price skewing under CARA utility, and inventory mean-reversion versus naive unconstrained quoting.

Theoretical Foundations: Optimal Dealer Pricing & Inventory Risk
1. Reservation Price Indifference Valuation

Under Constant Absolute Risk Aversion (CARA) utility $U(W) = -\exp(-\gamma W)$ and mid-price volatility $\sigma$, a market maker holding inventory position $q$ values the asset at an endogenous reservation price $R(S, q, t)$:

$$R(S, q, t) = S - q \gamma \sigma^2 (T - t)$$

When $q > 0$ (long inventory), the dealer incurs inventory risk variance and values the asset strictly below mid-price ($R < S$), creating a compelling economic incentive to lower quotes.

2. Asymmetric Bid-Ask Quote Shading

The dealer sets optimal ask $P_a = S + \delta_a^*(q)$ and bid $P_b = S - \delta_b^*(q)$ with asymmetric inventory adjustments around base spread $s_0$:

$$\delta_a^*(q) = \frac{1}{2}s_0 + \frac{1}{2}(1 - 2q)\gamma \sigma^2 (T - t), \quad \delta_b^*(q) = \frac{1}{2}s_0 + \frac{1}{2}(1 + 2q)\gamma \sigma^2 (T - t)$$

Total spread $S^*(q) = s_0 + \gamma \sigma^2 (T - t)$ remains invariant to $q$, but the quotes shift down when $q > 0$ and up when $q < 0$, naturally stabilizing inventory.

Institutional Market Regimes & Presets

NYSE Designated Market Maker (DMM): Balanced equity specialist, moderate risk aversion (γ = 0.10), high liquidity intensity, and active inventory mean-reversion.

Ho-Stoll Model Parameters
Base Spread ($s_0$)
$1.33
Elasticity Limit
Ho-Stoll PnL
$34.20
Skew-Managed
Naive Fixed PnL
-$12.40
Uncontrolled
PnL Alpha Premium
+$46.60
Ho-Stoll Edge
Dealer Inventory Risk Control: Ho-Stoll vs Naive Comparison
Strategy Max Peak Position $|q|_{\max}$ Inventory Variance $\text{Var}(q)$ Inventory Control Method
Ho-Stoll Optimal Dealer 3 units 2.14 Asymmetric Quote Shading ($\delta_a^*, \delta_b^*$)
Naive Fixed-Spread Dealer 11 units 18.62 Symmetric Quotes ($S \pm s_0/2$), Zero Shading
Sensitivity 1: Inventory ($q$) vs Risk Aversion ($\gamma$) on Reservation Skew ($R - S$)
Sensitivity 2: Volatility ($\sigma$) vs Horizon ($T-t$) on Inventory Spread Premium ($\gamma\sigma^2\tau$)
Ho-Stoll (1981) Market Making Mastery Quiz
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Frequently Asked Questions: Ho-Stoll (1981) Market Making

The Ho and Stoll (1981) model, published in the Journal of Financial Economics, is the seminal economic framework for optimal dealer pricing under transactions and return uncertainty. It models a risk-averse market maker facing stochastic inventory fluctuations and shows how the dealer sets asymmetric bid and ask prices around an endogenous reservation price to mean-revert inventory.

The reservation price $R(S, q, t)$ represents the dealer's subjective indifference valuation of the asset. Under CARA utility and Brownian motion, $R(S, q, t) = S - q \cdot \gamma \sigma^2 (T - t)$. When inventory is positive ($q > 0$, long position), the reservation price drops below the midprice ($R < S$), compelling the dealer to lower both bid and ask quotes to shed inventory. When $q < 0$ (short position), $R$ rises above $S$, raising quotes to attract sellers.

The optimal half-spreads are asymmetric functions of inventory: $\delta_a^*(q) = \frac{1}{2} s_0 + \frac{1}{2}(1 - 2q)\gamma \sigma^2 (T - t)$ and $\delta_b^*(q) = \frac{1}{2} s_0 + \frac{1}{2}(1 + 2q)\gamma \sigma^2 (T - t)$, where $s_0$ is the base spread determined by order flow elasticity and risk aversion. When long ($q > 0$), $\delta_a$ shrinks and $\delta_b$ widens, making selling to the market maker less attractive.

Ho and Stoll (1981) founded discrete/multi-period dealer pricing using dynamic programming and CARA utility over terminal wealth. Avellaneda and Stoikov (2008) modernized and extended Ho-Stoll into continuous-time stochastic optimal control using the Hamilton-Jacobi-Bellman (HJB) PDE with explicit exponential Poisson order arrival intensities, yielding closed-form asymptotic quoting schedules.

A naive dealer who quotes a constant, symmetric spread around the midprice ignores inventory accumulation. Because order arrivals are stochastic, a naive dealer's inventory undergoes an unconstrained random walk, accumulating massive directional exposure that causes catastrophic mark-to-market drawdowns during trending markets.