Introducing Dealing with the Inventory Risk: A Solution to the Market Making Problem for after-work quants

Market makers earn the spread—until inventory piles up on the wrong side. How do you keep quoting when risk from stock you already hold starts to dominate?

Dealing with the Inventory Risk: A Solution to the Market Making Problem (Olivier Guéant, Charles-Albert Lehalle, Joaquin Fernandez-Tapia, 2011) takes the famous Avellaneda–Stoikov setup and makes it more workable: they transform the Hamilton–Jacobi–Bellman (HJB) problem into linear ODEs and give closed-form asymptotic quotes under inventory constraints.

Fit for learners: Transforms Avellaneda–Stoikov HJB into linear ODEs and gives closed-form asymptotic quotes under inventory constraints.

This Hub note is a plain-language intro. Not investment advice.

The intuition (without the full PDE)

• Classic continuous-time market-making often ends in a nonlinear HJB that is hard to solve by hand.
• This paper rewrites the problem so the hard part becomes a system of linear ODEs—friendlier for asymptotics and numerics.
• The takeaway quote rules adjust bid/ask as inventory grows: you shade quotes to reduce unwanted risk, not to “predict” the next tick.
• Closed-form asymptotic formulae give after-work readers something concrete to compare against toy simulations later.

Read the PDF for assumptions, proofs, and figures. Do not treat asymptotic quotes as a live trading recipe.

Why after-work learners should care

Inventory is the quiet villain in every market-making notebook. Before you code a reinforcement-learning agent or copy a blog formula, you want a classical solution that says: risk from position size changes how you quote. This open PDF is a standard next step after Avellaneda–Stoikov.

Healthy practice later: simulate a random walk + Poisson fills, track inventory, and plot how a simple inventory penalty changes mid-offset—label it “learning only.”

One-evening reading checklist

1. Skim the abstract until “inventory risk” and “asymptotic” click.
2. Note how they connect to Avellaneda–Stoikov (what stays, what changes).
3. Find where linear ODEs replace the raw nonlinear HJB.
4. Write two sentences on why inventory shading is risk control, not a forecast.

Download the PDF

Download PDF

Landing: arXiv:1105.3115

Important disclaimer

Not investment advice. These models are educational research tools. They are not instructions to quote, hedge, or trade in a live account.

What this is not

• Not CEF / subsidy framing
• Not Quant or Macro course CTA
• Not a broker market-making product pitch

Next step

Single course CTA: SoloQuant Academy’s Introduction to Python Programming—get comfortable with arrays, files, and plots first; continuous-time papers become much less scary afterward.