Inventory Risk Near Expiry in the Avellaneda–Stoikov Model
Summary
The discussion explains why the reservation-price adjustment for inventory in the Avellaneda–Stoikov market-making model can seem to conflict with the goal of reducing inventory near the end of a trading period. For a long position, the small-inventory approximation lowers the reservation price, encouraging asks that can help sell inventory. As the horizon approaches its endpoint, that adjustment shrinks, which may appear to weaken the incentive to sell.
The answer resolves the apparent conflict by stressing the approximation’s scope: the familiar formula is a Taylor expansion intended for small inventory. Near expiry, a small position carries less uncertainty, so the approximate model assigns it less inventory penalty. A trader holding a large quantity may still face terminal inventory risk, but that behavior requires the full model rather than the small-inventory expression. The post points to a full-model treatment, but does not derive it or provide numerical evidence; the explanation is qualitative and depends on the model assumptions.
Key ideas
- The quoted reservation-price formula is a small-inventory approximation to the full market-making model.
- For a small long position, the inventory adjustment lowers the reservation price and can encourage selling.
- As expiry approaches, a small position entails less uncertainty about being held, so its penalty can decline.
- The formula does not describe how the model treats large remaining inventory near the terminal time.
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# How do we relate time-horizon with reducing inventory risk in the Avellaneda-Stoikov model? # How do we relate time-horizon with reducing inventory risk in the Avellaneda-Stoikov model? 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 by lehalle (score 1) https://quant.stackexchange.com/a/71834 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.
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