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Trading of financial instruments has largely moved away from floor trading and onto electronic exchanges. Orders to buy and sell are queued at these exchanges in a {em limit-order book}. While a full analysis of the dynamics of a limit-order book requires an understanding of strategic play among multiple agents, and is thus extremely complex, so-called {em zero-intelligence Poisson models} have been shown to capture many of the statistical features of limit-order book evolution. These models can be addressed by traditional queueing theory techniques, including Laplace transform analysis. In this article, we demonstrate in a simple setting that another queueing theory technique, approximating the Poisson model by a diffusion model identified as the limit of a sequence of scaled Poisson models, can also be implemented. We identify the diffusion limit, find an embedded semi-Markov model in the limit, and determine the statistics of the embedded semi-Markov model. Along the way, we introduce and study a new type of process, a generalization of skew Brownian motion that we call {em two-speed Brownian motion}.
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We propose a parametric model for the simulation of limit order books. We assume that limit orders, market orders and cancellations are submitted according to point processes with state-dependent intensities. We propose new functional forms for these intensities, as well as new models for the placement of limit orders and cancellations. For cancellations, we introduce the concept of priority index to describe the selection of orders to be cancelled in the order book. Parameters of the model are estimated using likelihood maximization. We illustrate the performance of the model by providing extensive simulation results, with a comparison to empirical data and a standard Poisson reference.
An inhomogeneous first--order integer--valued autoregressive (INAR(1)) process is investigated, where the autoregressive type coefficient slowly converges to one. It is shown that the process converges weakly to a Poisson or a compound Poisson distribution.
The fractional non-homogeneous Poisson process was introduced by a time-change of the non-homogeneous Poisson process with the inverse $alpha$-stable subordinator. We propose a similar definition for the (non-homogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional non-homogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombes theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.
Machine learning (especially reinforcement learning) methods for trading are increasingly reliant on simulation for agent training and testing. Furthermore, simulation is important for validation of hand-coded trading strategies and for testing hypotheses about market structure. A challenge, however, concerns the robustness of policies validated in simulation because the simulations lack fidelity. In fact, researchers have shown that many market simulation approaches fail to reproduce statistics and stylized facts seen in real markets. As a step towards addressing this we surveyed the literature to collect a set of reference metrics and applied them to real market data and simulation output. Our paper provides a comprehensive catalog of these metrics including mathematical formulations where appropriate. Our results show that there are still significant discrepancies between simulated markets and real ones. However, this work serves as a benchmark against which we can measure future improvement.
We examine the dynamics of the bid and ask queues of a limit order book and their relationship with the intensity of trade arrivals. In particular, we study the probability of price movements and trade arrivals as a function of the quote imbalance at the top of the limit order book. We propose a stochastic model in an attempt to capture the joint dynamics of the top of the book queues and the trading process, and describe a semi-analytic approach to calculate the relative probability of market events. We calibrate the model using historical market data and discuss the quality of fit and practical applications of the results.
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