  Market makers provide liquidity to other market participants: they propose
prices at which they stand ready to buy and sell a wide variety of assets. They
face a complex optimization problem with both static and dynamic components.
They need indeed to propose bid and offer/ask prices in an optimal way for
making money out of the difference between these two prices (their bid-ask
spread). Since they seldom buy and sell simultaneously, and therefore hold long
and/or short inventories, they also need to mitigate the risk associated with
price changes, and subsequently skew their quotes dynamically. In this paper,
(i) we propose a general modeling framework which generalizes (and reconciles)
the various modeling approaches proposed in the literature since the
publication of the seminal paper "High-frequency trading in a limit order book"
by Avellaneda and Stoikov, (ii) we prove new general results on the existence
and the characterization of optimal market making strategies, (iii) we obtain
new closed-form approximations for the optimal quotes, (iv) we extend the
modeling framework to the case of multi-asset market making and we obtain
general closed-form approximations for the optimal quotes of a multi-asset
market maker, and (v) we show how the model can be used in practice in the
specific (and original) case of two credit indices.
