Dealing with Inventory Risk

Question

I am reading a paper$^\color{magenta}{\dagger}$ on market-making and having trouble understanding a point. Towards the end of section 2, the authors stated that:

$$\sup_{(\delta_t^a)_t, (\delta_t^b)_t \in \mathcal A} {\Bbb E} \left[ - \exp \left( - \gamma \left( X_T + q_T S_T \right) \right) \right]$$

where $\mathcal A$ is the set of predictable process bounded from below. Can someone please explain what this means - in the mathematical context and in the trading context for a trader?

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_{$\color{magenta}{\dagger}$ Olivier Guéant, Charles-Albert Lehalle, Joaquin Fernandez Tapia, Dealing with the Inventory Risk. A solution to the market making problem, 2011.}

Answer 1

It means that the quotes $\delta^a,\delta^b$ of the market maker

1. cannot use future information (it is what predictable means)
2. has to be finite (this is for bounded from below).

The second assumption is technical (to obtain the theorems). If the first one seems natural, do not forget that in some market making academic papers (like Kyle 87), some traders have future information. Typically the asset managers are meant to be able to understand news and other data about the companies, and it is because of that that they take position.

But in the context of this paper, that I recommend by the way ;{)}, the market maker only observes flows (that drives the intensity of being hit at the bid or ask), that reflects in its inventory, to take decisions.