Stochastic Modeling of Limit Order Books: Convergence of the Price Process, Simulation and Applications

In the past two decades, electronic limit order books (LOBs) have become the most important mechanism through which securities are traded. A LOB contains the current supply and demand of a security at different prices and it can be modeled as a random, state-dependent, and high-dimensional system since typically a great number of orders are placed at many different prices at a millisecond time scale. These features lead to an inherent mathematical complexity which is extremely hard to describe in a tractable manner. Thus, depending on the purpose, different models have been proposed to capture specific properties of the underlying trading mechanism, making LOB modeling a trending topic in the quantitative and investment finance literature for the past few years. Some of the most important objectives for which a LOB model is designed are to provide algorithmic trading strategies, bottom-up estimates for a variety of parameters, better understanding of asset price formation. In the present work, two continuous-time models for the level I of a LOB are proposed. As with many articles in the literature, arrivals of limit orders, market orders, and cancellations are assumed to be mutually independent, memoryless, and stationary, but, unlike the earlier approaches, the proposed models also account for some of the “sparsity” and “memory” exhibited by real LOB dynamics. Specifically, the first proposed model allows for variable price shifts after each price change in order to account for some of the larger than-usual “gaps” between levels (sparsity property) that has been pointed out in some empirical studies. A more realistic approach is pursued in a second model by keeping the information about the standing orders at the opposite side of the book after each price change (memory property), and also incorporating arrivals of new orders within the spread, which in turn leads to a variable spread. To illustrate the applicability of the latter model, analytical expressions for some important quantities of interest, such as the distribution of the time span between price changes and the probability of consecutive price increments conditioned on the current state of the book, are derived. In spite of the inherent model complexity, the long-run asymptotic behavior of the resultant mid-price process is fully characterized for both models and, hence, our analysis shed further light on the relation between the macro price dynamics and some more detailed LOB features than those considered in earlier works. The asymptotic results are illustrated with a numerical Monte Carlo study for which an efficient simulation scheme is also developed. The interplay between the micro-structure of the market and the macro-price dynamics is further investigated in a natural extension of the above models, in which multiple levels are considered. Arrivals of limit orders, market orders, and cancellations, again, follow independent Poisson processes at each level, and conditions under which the LOB becomes a positive recurrent Markov process are being investigated. After such conditions are found, a similar approach to the one followed in the second model described above can be applied to analyze the convergence of the price process, in the long run. Finally, a separate important problem of interest is considered. Concretely, a natural question is to find appealing conditions on the LOB dynamics, under which the resulting (mid-)price process, converges to a non-homogeneous diffusion in order to account for the inherent stochastic volatility of the price process for the given time scales. This leads to propose a model under which the limiting (mid-)price process would follow a time-changed Wiener process