Dealing with the Inventory Risk (Lehalle, Gueant, Tapia): Delta T parameter and actual order duration

Reference paper: Dealing with the Inventory Risk (Lehalle, Gueant, Tapia)

∆T is the time horizon over which you compute the intensities of aggressive orders (you get the k and A parameters)

I have two questions:

1. The choice of ∆T is not discussed in detail. How can we determine it empirically ?

What I have investigated: Tapia, in his thesis Modeling, optimization and estimation for the on-line control of trading algorithms in limit-order markets (TAPIA) mentions that in practice the value of the parameter ∆T should be calibrated in a way that the relationship between A and σ is as weak as possible, i.e. volatility measures price risk and it is not contaminated with local oscillations due to trading intensity.

What I am planning to do: do the calibration exercise using a different set of ∆T (example: 1, 2, ... 20 seconds). And for each set, compute the correlation between A and σ. The choice of ∆T will be the one that exhibits the smallest correlation. What do you think about this approach ? Anyone could maybe share some examples of used value ?

1. In practice, this market making algorithm will quote around a Fair Value with some tick sensitivity that can trigger cancellation of the orders. The smaller the tick sensitivity, the shorter will be the duration of the orders. It can of course be totally different with the ∆T from the model. My question is: How do we reconcile the model ∆T from the practical orders life time ? Or maybe, the right approach with this kind of model is to force a certain order duration for the orders ? In other terms, what is the approach that makes the most sense when it comes to the reality of quoting?

you are right, the choice of $$\Delta T$$ is subtle, and as suggested by Joaquin, it is important to preserve the (relative) independence between
1. the price formation process, that corresponds to the "liquidity game" market participants are involved into when trying to get liquidity at the best possible price (or for market makers to earn the bid-ask spread). In this model $$A$$ and $$k$$ are meant to capture this.
In the paper, for the sake of simplicity, the two are assumed to be independent but of course in reality it is not really the case. Typically when exogenous information hit the book, $$A$$ and $$k$$ may change and can become asymmetric (ie not being the same on the bid and ask sides of the book)... It means that this model is efficient during the liquidity game, and you probably should switch to another model if you can detect something very asymmetric is happening, to protect your inventory from adverse selection.