How do we relate time-horizon with reducing inventory risk in the Avellaneda-Stoikov model?

Question

On the Avellaneda Stoikov Model we have the following definition of reserve price:

This means that when q > 0 and the market maker has a long position, the reservation price decreases. So there is a greater chance that the market maker sells inventory, because he will place asks at a lower price. That seems quite logical.

However, when we get to close to the terminal price T, that factor of the equation tends to zero. What seems confusing to me is that if q > 0 when we are approaching the terminal time, asks will be higher and not lower as mentioned before. When the market maker is approaching terminal time he intends to reduce inventory risk, thus he has to sell his long position right?

I'm sure there is something I am missing here. But I am not understanding correctly what the formula intends to describe as the market maker behavior approaching to the terminal time.

Answer 1

Be aware that the Avellaneda-Stoikov model is a Taylor extension of the full model when $q$ is small. We developed (following their seminal paper) the full model in Guéant, Olivier, L, and Joaquin Fernandez-Tapia. "Dealing with the inventory risk: a solution to the market making problem." Mathematics and financial economics 7.4 (2013): 477-507.

Your qualitative reasoning is targeting "large remaining quantity close to the end of the period": this formula does not apply. What you see in the formula is that for small $q$, being close to the terminal time implies that there is less uncertainty in holding a position, hence you do not "fear" it that much. It the full model, the market maker fears the terminal time when $q$ is not small.