Limit order books, diffusion approximations and reflected SPDEs: from microscopic to macroscopic models

Motivated by a zero-intelligence approach, the aim of this paper is to connect the microscopic
(discrete price and volume), mesoscopic (discrete price and continuous volume) and macroscopic
(continuous price and volume) frameworks for the modelling of limit order books, with a view to
providing a natural probabilistic description of their behaviour in a high to ultra high-frequency
setting. Starting with a microscopic framework, we first examine the limiting behaviour of
the order book process when order arrival and cancellation rates are sent to infinity and when
volumes are considered to be of infinitesimal size. We then consider the transition between
this mesoscopic model and a macroscopic model for the limit order book, obtained by letting
the tick size tend to zero. The macroscopic limit can then be described using reflected SPDEs
which typically arise in stochastic interface models. We then use financial data to discuss a
possible calibration procedure for the model and illustrate numerically how it can reproduce
observed behaviour of prices. This could then be used as a market simulator for short-term
price prediction or for testing optimal execution strategies.