6.3.2 The Inventory Problem

Basic Prerequisites: 01 · From Zero and Probability.

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1. Intuition & Practical Objective

Before choosing quotes, you must understand what inventory does to you. This page quantifies the two costs of holding a position and derives the optimal inventory position - the target the skew drives toward. The practical objective: three numbers you should be able to state for any position - the P&L standard deviation $|I|\sigma \sqrt{\tau }$, the certainty-equivalent cost $\frac{1}{2}\gamma {\sigma }^2{I}^2\tau$, and the resulting reservation-price shift $\gamma {\sigma }^{2}I\tau$ per share.

The economics, in one line each:

1. Inventory risk grows linearly in position size, in volatility, and in the square root of holding time. Hold 40 shares of a $\sigma =2$ asset for a year and the P&L swings $\pm 80$ (std $=|I|\sigma \sqrt{\tau }=40\cdot 2\cdot 1=80$). The risk is proportional to $|I|$, not $I^2$ - but the cost of bearing it is quadratic.
2. The cost of a position is quadratic in size. A risk-averse dealer with risk aversion $\gamma$ pays a certainty-equivalent penalty $\frac{1}{2}\gamma {\sigma }^2{I}^2\tau$ for holding $I$. The marginal share (moving from $I$ to $I+1$) costs $\gamma {\sigma }^2(I+\frac{1}{2})\tau$ - so the next share you buy costs more than the last. This increasing marginal cost is exactly why there is an optimal inventory.
3. The optimal inventory is zero (in a zero-drift market). With no forecast, every dollar of inventory is pure unremunerated risk; the expected P&L is zero and the variance positive, so $I^{\ast }=0$. All deviation from zero is an accident of order flow that the skew must correct.

The core identity. “A position $I$ held for horizon $\tau$ has P&L standard deviation $|I|\sigma \sqrt{\tau }$ and certainty-equivalent cost $\frac{1}{2}\gamma {\sigma }^2{I}^2\tau$; the marginal cost $\gamma {\sigma }^{2}I\tau$ is the reservation-price skew - so the quote shift that makes a dealer indifferent to one more share is exactly the inventory risk that share adds.”

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2. Mathematical Ground Truth & Derivations

Inventory risk is variance. If the reference price follows $dS=\sigma dW$ over horizon $\tau$, a fixed position of $I$ shares has terminal wealth $I\cdot S_T$ with variance

$$\mathrm{Var}(I{S}_T)=I^2{\sigma }^2\tau ,\text{std}=|I|\sigma \sqrt{\tau }.$$

This is the directional risk you take by being a dealer instead of a pure spread collector: linear in $|I|$, linear in $\sigma$, square-root in time.

The certainty-equivalent cost. Under CARA utility $u(W)=-e^{-\gamma W}$, holding a position with P&L variance $V$ and zero mean is worth $-e^{-\gamma \cdot (\text{CE})}$ where the certainty equivalent satisfies

$$\mathbb{E}[-e^{-\gamma I{S}_T}]=-e^{-\gamma (0-\frac{\gamma }{2}{I}^2{\sigma }^2\tau )},$$

so the dealer’s personal valuation of a share at inventory $I$ is the reservation price

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

and the total cost of the position, in certainty-equivalent dollars, is $\frac{1}{2}\gamma {\sigma }^2{I}^2\tau$. Differentiating in $I$:

$$\frac{\partial }{\partial I}\left(\frac{1}{2}\gamma {\sigma }^2{I}^2\tau \right)=\gamma {\sigma }^{2}I\tau$$

• the marginal cost of the $I$-th share is the skew. This is the linear reservation price of 03 · The Ho–Stoll Model.

The optimal inventory position. The dealer maximizes expected CARA utility of terminal wealth. Over a horizon with no drift the objective reduces to

$$\mathbb{E}[W]-\frac{\gamma }{2}\mathrm{Var}(W)=\text{spread revenue}-\frac{1}{2}\gamma {\sigma }^2\mathbb{E}[I^2]\tau .$$

Because spread revenue does not require holding a position (you earn it round-trip), the variance penalty is minimized at $\mathbb{E}[I^2]=0$, i.e. $I^{\ast }=0$. Any forecast $\mu$ pulls the target away from zero: with drift, the optimal position shifts to $I^{\ast }=\frac{\mu }{\gamma {\sigma }^2}$ (risk-adjusted trade of return vs. variance).

Inventory mean-reversion. An unmanaged position is a random walk: $q_{t+1}=q_t+{\epsilon }_t$ with $\mathrm{Var}(q_T)\propto T$. Quote skewing injects a restoring drift: the probability of the inventory-reducing trade rises with $|q|$, so the position satisfies an Ornstein–Uhlenbeck-type equation

$$d{q}_t=-\kappa q_{t}dt+\text{flow noise},$$

where the reversion speed $\kappa$ is set by the skew strength $\gamma {\sigma }^2\tau$ and the fill elasticity $k$. The optimal liquidation of a large position is the controlled version of this mean reversion (06 · Advanced Extensions).

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3. Computational Implementation - the cost of a position, verified

We hold a fixed position $I$ to time $T$ (no rebalancing, pure price risk) and measure the P&L distribution across 600 000 paths, checking the two identities above.

The simulated P&L std matches $|I|\sigma \sqrt{\tau }$ to three decimals ($20.02,40.04,80.09$) and the certainty-equivalent cost matches $\frac{1}{2}\gamma {\sigma }^2{I}^2\tau$ ($20,80,320$) exactly. The mean P&L is ~0 (the position earns nothing on average - pure risk). Doubling the position quadruples the cost (quadratic in $I$); the optimal target is $I=0$, and every share above it is exactly what the skew of 04 must bleed off.

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4. Failure Modes & First-Principles Breakdowns

1. Treating inventory risk as linear. The P&L std is linear in $|I|$, but the cost is quadratic - a 2x position is 4x as costly. Position limits set on the linear scale systematically understate the pain of the tail.
2. “I’ll hold it, it’ll come back” - no mean reversion. Without skewing, inventory is a driftless random walk; there is no force returning it to zero, and holding longer only adds $\sqrt{\tau }$ more variance.
3. Zero drift is an assumption, not a fact. The target $I^{\ast }=0$ assumes no forecast. A real signal $(\mu \ne 0)$ moves the optimal position to $\frac{\mu }{\gamma {\sigma }^2}$; a maker who ignores a real drift is knowingly short or long the market. But a maker who believes in a spurious drift becomes a directional bet - the calibration failure mode of 06.
4. Risk aversion is a choice. $\gamma$ sets how fast the skew fights inventory. Too small, and positions run to the limit (05); too large, and the desk quotes itself out of the market to avoid $I\ne 0$.

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5. References

• Ho & Stoll (1981), Optimal dealer pricing under transactions and return uncertainty, JFE 9(1)
• Stoll (1978), The supply of dealer services in securities markets, JFE 3(2), 113–124
• Avellaneda & Stoikov (2008), High-frequency trading in a limit order book, Quantitative Finance 8(3)
• Garman (1976), Market microstructure, JFE 3(3)

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6. Connected Graph Bridges

• Back: 01 · From Zero
• Forward: 03 · The Ho–Stoll Model · Index Hub
• Sibling: A–S: The Market-Maker’s Problem · VaR & Expected Shortfall