Skip to content
All library documents

How the HJB Equation Models Inventory and Quote Decisions

Article Quant Q&A · Author: frank
Read the original

Summary

The document gives an introductory explanation of the Hamilton–Jacobi–Bellman equation in a market-making model. The value function depends on time and the trader’s inventory, and the goal is to choose bid and ask spreads that optimize it. The explanation connects this setup to Bellman’s principle: compare the value after a short time step with the gain or loss from trading during that step.

The account describes a Taylor expansion for the value function and says that buy and sell orders change cash and inventory while generating spread revenue. Their contributions are weighted by their arrival intensities, and inventory limits constrain the problem. This is a conceptual sketch rather than a derivation: it does not state the full equation, define its terms precisely, or show how to solve for optimal quotes. It is useful as orientation, but readers need the paper or a fuller optimal-control treatment to work through the mathematics.

Key ideas

  • The market-making value function depends on time and current inventory.
  • The HJB equation expresses how a trader can optimize value over a short time step.
  • Buy and sell arrivals affect cash and inventory while producing spread gains.
  • Order arrival intensities and inventory limits enter the optimization problem.

Tags

Market makingStatistics
Full text
# Where this HJB equation comes from?


# Where this HJB equation comes from?












I am reading the paper: "Dealing with the Inventory Risk. A solution to the market making problem", by Olivier Guéant, C.-A. L and Joaquin Fernandez Tapia.

On top of page 6, there is a HJB equation I don't know where it is from and how to understand it. I am new to this topic so I searched many books regarding to optimal control and HJB but still couldn't figure it out. Could you help please?

## Answer by numerairX (score 2, accepted)

https://quant.stackexchange.com/a/49411

without knowing what you're trying to ask exactly:

Overall HJB is used in continuous time optimal control problems. In this paper and this equation particularly, the objective is to solve for optimal bid/offer spread given current time and your current inventory. To do this we must obtain the value function using these two parameters.

By bellman's principal of optimality, at each time step we want to minimize (or maximize when we have utility gain instead of cost function) value function update + utility loss (or gain). Value function update from time $t$ to $t+dt$ is first two items of taylor series expansion of value function;

For gain, think of at each time $t$, if your inventory didn't hit upper and lower limit $Q$, you can either sell or buy from market participants, and earn that spread accordingly. By doing each, your cash inventory, would decrease or increase by $s$, and increase by the bid/offer spread you earned. Since incoming orders on both sides have different arrival intensity (probability of occurrence), you "discount" your utility gain by the probability.

With the system set up and constraints, you can solve for optimal value function. Hope this helps.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.