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Market makers have to continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization problem in which their return, based on the bid-ask spread they quote and the frequency they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, the authors provide optimal bid and ask quotes and closed-form approximations are derived using spectral arguments. The optimization of the intra-day trading process on electronic markets was born with the need to split large trades to make the balance between trading too fast (and possibly degrade the obtained price via "Market impact") and trading too slow (and suffer from a too long exposure to "Market risk").
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