Mathematical Models to Study and Control the Price Formation Process.

Summary The price formation process is at the heart of all financial mathematics models. First approximated by a Brownian motion (cf. [Karatzas & Shreve, 1998]), then by integrating jumps (see [Shiryaev, 1999]) as long as long time scales were involved, the taking into account of the volumes exchanged came later: - In an econometric framework (see for example [Tauchen & Pitts, 1983]) to try to better explain the dynamics of prices - in the framework of theoretical studies of general equilibria where the "auction game" is modelled (as in [Kyle, 1985], [Ho & Stoll, 1983] or [Glosten & Milgrom, 1985]) As regulation pushes more and more exchanges towards electronic markets, very large databases are now available. They contain not only the actual transactions, but also the declarations of interest of all participants. This has opened the door to empirical studies (such as [Lillo et al. , 2003]) that provide interesting avenues for new families of models (see for example [Bacry & Muzy, 2013]). Properly modeling the price formation process helps guide regulators, who try to encourage trading in electronic markets (as they are more easily trac ̧able). It also allows for the development of optimal trading techniques, which, when used by investors and financial intermediaries, will minimize the disruption of prices due to trading intensity.