6.3 Inventory Management & Quote Skewing

Basic Prerequisites: Avellaneda–Stoikov & Optimal Quoting and Probability & Stochastic Control. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

A market maker earns the spread, but he does not keep what he earns unless he controls his inventory. Every time a customer lifts his ask, the maker’s inventory goes down (he sold); every time a customer hits his bid, it goes up (he bought). Buy and sell flow arrive asymmetrically and for reasons the maker does not control, so inventory is a random walk that drifts away from zero and carries real price risk. The spread is compensation for bearing that risk - and if the position moves against you before you can flatten it, the spread is more than given back.

This folder is the inventory control branch of Pillar 6. Its object is one mechanism: quote skewing - shifting both quotes down when you are long (to encourage sells and discourage buys) and up when you are short, so that the natural arrival of noise flow bleeds inventory back to its target. Where the Avellaneda–Stoikov folder is the stochastic-control engine, this folder is the theory of the position it manages - the Ho–Stoll (1981) dealer model that invented the reservation price, the inventory-risk function, and the mean-reverting skew that every production maker implements.

The one-sentence essence. “Hold inventory $I$ for horizon $\tau$ and you bear a certainty-equivalent cost $\frac{1}{2}\gamma {\sigma }^2{I}^2\tau$, so your personal fair value is the reservation price $r(I)=\bar{S}-\gamma {\sigma }^{2}I\tau$ - and you should quote around $r(I)$, not around the mid, shifting both quotes against the position by $\gamma {\sigma }^2\tau$ per share.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). The core formulas are from Ho & Stoll (1981), Optimal dealer pricing under transactions and return uncertainty, JFE 9(1), cross-checked against Avellaneda & Stoikov (2008) and Guéant, Lehalle & Fernandez-Tapia (2013). The numbers in the check column were re-executed and reproduced exactly (see §3).

Notation: $I$ inventory (shares, $+$ long / $-$ short), $\bar{S}$ reference (fair) price, $\tau =T-t$ remaining horizon, ${\sigma }^2$ per-unit-time return variance, $\gamma >0$ absolute risk aversion, $c$ per-share order-processing cost, $k$ fill elasticity, $A$ baseline arrival rate, $\alpha$ skew sensitivity, $Q$ inventory limit.

Critical scaling caveat. The reservation-price skew $\gamma {\sigma }^{2}I\tau$ grows with the horizon $\tau$ and the variance ${\sigma }^2$, not the volatility $\sigma$. Quote skew in Ho–Stoll / A–S is set by ${\sigma }^2\tau$ per share of inventory; a desk quoting off daily volatility mis-skews by an order of magnitude. And the reservation spread $\gamma {\sigma }^2\tau$ is an inventory-risk component distinct from the order-processing half-spread $a^{\ast }=c+1/k$ - the two add, they do not substitute.

3. Computational Implementation - the inventory-controlled quote engine

This runs on NumPy and reproduces every verified number above. It is the same engine the sub-pages use: reservation price (linear inventory skew) + cost-anchored half-markup.

Note the skew: at $I=+4$ the dealer values a share at $98.40$ and quotes both bid and ask below the $100$ reference - he is long and actively trying to sell. The full P&L-versus-inventory-variance comparison of skewing is in 04 · Quote Skewing.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. Inventory limits are binding in the tail. The reservation skew encourages flattening but never forbids a position; an unmanaged maker breaches the cap and is force-liquidated at adverse prices.
2. Forced liquidation converts paper risk into realized loss. Once $|I|$ hits the risk limit, the desk pays the spread to exit - at exactly the wrong time (a trend). Our simulation shows $92\%$ of uncontrolled paths breaching the cap in a trending market.
3. Adverse selection is not inventory risk. Skewing controls the position; it does nothing about informed flow that picks off stale quotes. The two must be combined (Toxic Order Flow & VPIN).

5. References

• Ho, Thomas & Stoll, Hans: Optimal dealer pricing under transactions and return uncertainty, Journal of Financial Economics 9(1), 47–73 (1981). The canonical dealer-inventory model; every inventory-skew rule in production descends from its reservation price.
• Ho, Thomas & Stoll, Hans: The dynamics of dealer markets under competition, Journal of Finance 38(4), 1053–1074 (1983). Competing dealers and how competition compresses the inventory component of the spread.
• Stoll, Hans: The supply of dealer services in securities markets, Journal of Finance 33(4), 1133–1151 (1978). The holding-cost / transaction-cost decomposition of the spread that the markup $a^{\ast }=c+1/k$ formalizes.
• Avellaneda, Marco & Stoikov, Sasha: High-frequency trading in a limit order book, Quantitative Finance 8(3), 217–224 (2008). The continuous-time descendant of Ho–Stoll; the reservation price $s-q\gamma {\sigma }^2(T-t)$ is this folder’s skew.
• Guéant, Lehalle & Fernandez-Tapia: Dealing with the inventory risk, Mathematics and Financial Economics 7(4), 477–507 (2013). *Inventory caps and closed-form asymptotics
• Garman, Mark: Market microstructure, Journal of Financial Economics 3(3), 257–275 (1976). The first inventory-control model, in which an unhedged market maker can go bankrupt.
• Menkveld, Albert: High frequency trading and the new market makers, Journal of Financial Markets 16(4), 712–740 (2013). Empirical: real HFT makers earn the spread, incur inventory costs, and skew their quotes.

6. Connected Graph Bridges

• Foundational base: Stochastic Calculus & Itô · Probability & Stochastic Control
• Sibling topic (in-pillar): Avellaneda–Stoikov & Optimal Quoting (the stochastic-control engine) · Adverse Selection & Glosten–Milgrom · Spread Decomposition & the Roll Model
• Sub-pages (in-folder): 01 From Zero · 02 The Inventory Problem · 03 The Ho–Stoll Model · 04 Quote Skewing · 05 Failure Modes · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05