# Infinite horizon agent in Avellaneda-Stoikov model

I am trying to understand the Avellaneda-Stoikov model for high frequency trading, in particular the optimizing agent with infinite horizon.

The reservation ask/bid prices for such an agent are defined in the paper as: and It is said in the paper that $\omega$ serves as an upper bound on the inventory position.

I took the argument of the natural logarithm from the reservation bid price and wrote the inequality satisfying the logarithm:

Looking closer at the formula for $\omega$, we see that it depends on some $q_{max}$ parameter.

What I can't answer for myself is the question whether $q_{max}$ is heuristically chosen or is it estimated by some established method.

## 1 Answer

I'm doing this from memory, but as I recall $$q_{\text{max}}$$ is the maximum inventory on any side that you wish to take (otherwise you might build up a huge position if you are adversely selected).

Later papers such as this one https://arxiv.org/pdf/1105.3115.pdf helped my understanding.

As it actually happens, I implemented these algorithms and had a go doing HFT style MM on Bitmex. Although they do work in certain situations, I'll paraphrase something Sinclair wrote in his book, "Making money market making is trivial. Keeping it is a lot harder." These algos are an interesting starting point, but remember they are based on an idealised distribution of incoming orders, perfect connectivity to the exchange, and always being front-of-the-queue.