How to propertly change time horizon in Avellaneda-Stoikov model?

Question

I'm working in the Avellaneda-Stoikov implementation using Python. My implementation reproduces the authors' results, but I don't know how to properly adapt the algorithm in order to consider a larger time horizon. From the equation

r = s - q * gamma * sigma**2 * (T-t)

if we use larger T, the indifference price computed could become too big compared with the mid price s, or even a negative value, when q is positive.

Is there an a-dimensional implementation? How to independize from T? Is there a possibility to independize also from the election of s?

Answer 1

In Dealing with the inventory risk: a solution to the market making problem (preprint available at arxiv) we extend the approximation proposed by Marco and Sasha to a rigorous mathematical framework and provide exact steady-state approximation. The bid-ask spread that should be quoted is

$$\psi(q;k,\gamma):=-\frac{1}{k}\ln\frac{f^0_{q-1}f^0_{q+1}}{(f^0_{q})^2}+\frac{2}{\gamma}\ln\frac{\gamma+k}{k},$$ where $k$ is a characteristic of the liquidity consuming flow, $\gamma$ is the MM risk aversion, and the $f^0_q$ are solution of the steady-state version of the ODE driving the value of accepting an inventory of $q$.

The paper explain all this and has been followed by some others. I especially recommend the ones driven by the two other authors:

• Joaquin Fernandez-Tapia went towards ergodic versions of the problem during his PhD thesis (identifying connections with reinforcement learning), and later to versions for online bets (especially for internet ad banners)
• Olivier Guéant went to more formal versions of the problem, and later to reinforcement learning to numerically solve multi-dimensional versions