Why Exponential Utility Makes Optimal Limit-Order Quotes Wealth-Independent

Summary

The document asks why a limit-order-book model’s function for inventory-related value depends on the terminal number of shares held but not on cash accumulated during trading. The answer links this feature to the exponential utility used in the model and to the terminal condition in its Hamilton-Jacobi-Bellman equation.

With exponential utility, the value function can be written as an exponential factor for cash multiplied by another function of the market state, inventory, and time. Substituting this form into the equation factors out the cash term, so the optimal quote choices do not depend on current wealth. This result is specific to the chosen utility structure and model setup; it is not a general claim that wealth never affects quoting decisions. The discussion gives a conceptual derivation through a value-function ansatz, but does not provide the full equation or explore how alternative utility functions would change the result.

Key ideas

• The model uses exponential utility to represent the agent’s preference over wealth.
• An exponential factorization separates cash from the remaining state variables in the value function.
• Substitution into the Hamilton-Jacobi-Bellman equation removes current cash from the quote optimization.
• Optimal quotes are independent of current wealth under this utility specification and model setup.
• The wealth-independence conclusion need not hold for other utility functions.

Tags

# High Frequency Trading in LoB - Sasha Stoikov and Marco Avellaneda


# High Frequency Trading in LoB - Sasha Stoikov and Marco Avellaneda












I am reading the paper High Frequecy Trading in a Limit Order Book by Sasha Stoikov and Marco Avellaneda. There is a point that I am having trouble understanding.

The authors give a definition of the of the optimization problem that they want to solve.

My question is, why is $\theta$ not a function of cash that I have generated while trading and only depends on the terminal condition in terms of the numbers of stocks held?

I have only little understanding of Hamilton-Jacobi-Bellman equation. Not sure if some of the results follow from there.

## Answer by Freelunch (score 3, accepted)

https://quant.stackexchange.com/a/46515

The terminal condition for the HJB equation implies that you can factor the value function into \begin{equation*} u(s, x, q, t) = \exp(-\gamma x)\exp(-\gamma \theta(s,q,t)): \end{equation*} and a direct substitution using this ansatz allows you to factor out $\exp(-\gamma x)$ from the equation. This is because of the exponential utility function used, which means that the optimal quotes will be independent of the current wealth. They state this in the beginning of the article: "This choice of convex risk measure is particularly convenient, since it will allow us to define reservation (or indifference) prices which are independent of the agent’s wealth."

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.