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

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?

$$\psi(q;k,\gamma):=-\frac{1}{k}\ln\frac{f^0_{q-1}f^0_{q+1}}{(f^0_{q})^2}+\frac{2}{\gamma}\ln\frac{\gamma+k}{k},$$ 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 $$q$$.