A Mathematical Approach to Order Book Modelling

By Frédéric Abergel, A. Jedidi. 2010

121 citations recorded by the source corpus.

We present a mathematical study of the order book as a multidimensional continuous- time Markov chain where the order flow is modeled by independent Poisson processes. Our aim is to bridge the gap between the microscopic description of price formation (agent- based modeling), and the Stochastic Differential Equations approach used classically to describe price evolution at macroscopic time scales. To do this, we rely on the theory of infinitesimal generators and Foster-Lyapunov stability criteria for Markov chains. We motivate our approach using an elementary example where the spread is kept constant ("perfect market making"). Then we compute the infinitesimal generator associated with the order book in a general setting, and link the price dynamics to the instantaneous state of the order book. In the last section, we prove that the order book is ergodic - in particular it has a stationary distribution - that it converges to its stationary state exponentially fast, and that the large-scale limit of the price process is a Brownian motion.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/52e085c8ae33686eba3cdb9aee6a265b2d8f7206

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Ban Zheng
2. Franccois Roueff
3. Nakahiro Yoshida
4. Frédéric Abergel
5. M. A. Porter

In Discrete Mathematics and Combinatorics

1. Ban Zheng
2. Franccois Roueff
3. Frédéric Abergel
4. Sam D. Howison

Citations in the corpus

• Limit order books (2010, background): “ and interacting strategies, and nding a set of agent rules that produces a specic behaviour from an ABM provides no guarantee that such a set of rules is the only one to do so (Preis et al., 2007). Abergel and Jedidi (2011) attempted to address these issues by studying systems of stochastic dierent”
• A one-level limit order book model with memory and variable spread (2017, background): “Abergel and Jedidi [1] also obtain a diffusive approximation for the mid-price process; however, our model cannot be framed within their approach since the spread in their model remains bounded, while, in our model, it can take values on Z + .”
• Ergodicity and scaling limit of a constrained multivariate Hawkes process (2013, methodology): “More recently point processes have been introduced to financial application.”
• Statistical inference for ergodic point processes and application to Limit Order Book (2015, background): “As an example, the former is often taken as the best ask price or the best bid price, see [1].”
• Deep learning for limit order books (2016, unknown): “Cont et al. (2010), Cont and De Larrard (2013), Cont and De Larrard (2012), Abergel and Jedidi (2013), Donier et al. (2015), Moallemi and Saglam (2013), Maglaras et al. (2014), Carmona and Webster (2013), Carmona and Webster (2012a), Carmona and Webster (2012b), Avellaneda and Stoikov (2008),…”