# 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.

## 1 Answer

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."

• Thanks a lot ! Another small question: I understand that gamma here is used as a discount rate...but can you tell me its significance for a high frequency trader who flattens his position at EoD? – nimbus3000 Jul 9 '19 at 15:58