Rigorous results for the Stigler-Luckock model for the evolution of an order book

0490719 - ÚTIA 2019 RIV US eng J - Článek v odborném periodiku
Swart, Jan M.
Rigorous results for the Stigler-Luckock model for the evolution of an order book.
Annals of Applied Probability. Roč. 28, č. 3 (2018), s. 1491-1535. ISSN 1050-5164
Grant CEP: GA ČR(CZ) GA16-15238S; GA ČR GAP201/12/2613
Institucionální podpora: RVO:67985556
Klíčová slova: Continuous double auction * order book * rank-based Markov chain * self-organized criticality * Stigler-Luckock model * market microstructure
Obor OECD: Pure mathematics
Impakt faktor: 1.703, rok: 2018
http://library.utia.cas.cz/separaty/2018/SI/swart-0490719.pdf

In 1964, G. J. Stigler introduced a stochastic model for the evolution of an order book on a stock market. This model was independently rediscovered and generalized by H. Luckock in 2003. In his formulation, traders place buy and sell limit orders of unit size according to independent Poisson processes with possibly different intensities. Newly arriving buy (sell) orders are either immediately matched to the best available matching sell (buy) order or stay in the order book until a matching order arrives. Assuming stationarity, Luckock showed that the distribution functions of the best buy and sell order in the order book solve a differential equation, from which he was able to calculate the position of two prices Jc−<Jc+ such that buy orders below Jc− and sell orders above Jc+ stay in the order book forever while all other orders are eventually matched. We extend Luckock’s model by adding market orders, that is, with a certain rate traders arrive at the market that take the best available buy or sell offer in the order book, if there is one, and do nothing otherwise. We give necessary and sufficient conditions for such an extended model to be positive recurrent and show how these conditions are related to the prices Jc− and Jc+ of Luckock.
Trvalý link: http://hdl.handle.net/11104/0285272