Abstract
The high-frequency dynamics of prices, supply and demand in electronic markets maybe described in terms of a limit order book, whose dynamics has a natural description as a point process or queueing system [1,3]. We present a model for the arrival and execution of limit and market orders which allows for correlated durations between orders, random heterogeneous and correlated order sizes and temporal dependence in the order flow. Our models includes Poisson, self-exciting and ACD models as special cases. We derive a functional central limit theorem for the limit order book, which shows that, in liquid markets where orders arrive with high frequency, the dynamics of buy and sell queues may be approximated by a Markovian jump-diffusion process whose parameters are readily estimated from the observed order flow [4]. This approximation provides an analytically tractable description of the dynamics of the order book and the market price and yields a quantitative link between statistical properties of the price process and properties of the order flow, without recourse to ad-hoc assumptions on price impact [2,5]. In particular, we obtain an expression for the (low-frequency) volatility of the price in terms of the first and second moments of the order flow variables at high frequency. This relation is found to be in good agreement with empirical data for liquid US stocks.
Keywords: limit order book, high frequency trading, functional central limit theorem, queueing systems, point processes, volatility.
References:
[1] Rama Cont, Sasha Stoikov and Rishi Talreja (2010) A stochastic model for order book dynamics, Operations Research, Volume 58, No. 3, 549-563.
[2] Rama Cont and Adrien de Larrard (2010) Price dynamics in a Markovian limit order market, http://ssrn.com/abstract=1735338
[3] Rama Cont (2011) Statistical Modeling of High Frequency Financial Data: Facts, Models and Challenges, IEEE Signal Processing.
http://ssrn.com/abstract=1748022
[4] Rama Cont and Adrien de Larrard (2011) Order book dynamics in liquid markets: heavy traffic limits and diffusion approximations, Working Paper.
[5] Rama Cont and Adrien de Larrard (2011) Price dynamics in limit order markets: linking volatility and order flow, Working Paper.