Before revision:  We propose in this paper closed-form approximations for the bid and ask quotes a market maker should set for optimally managing the risk of his/her multi-asset market making book. Our work is an extension of the paper "Optimal market making" by Gu\'eant. We also consider generalizations with asymmetric arrival rate intensities. Moreover, we consider the case where the reference price is modeled with a drift and adverse selection .
After revision:  We propose in this paper closed-form approximations for the bid and ask quotes a market maker should set for optimally managing the risk of his/her multi-asset extensions of the Avellaneda-Stoikov model. These approximations or proxies can be used (i) as heuristic evaluation functions, (ii) as initial value functions in reinforcement learning algorithms, and/or (iii) directly to design quoting strategies through a greedy approach. Regarding the latter, our results lead to new and easily interpretable closed-form approximations for the optimal quotes, both in the finite-horizon case and in the asymptotic (ergodic) regime. Furthermore, we propose a perturbative approach to improve our closed-form approximations through Monte-Carlo simulations .