A Semi-Markovian Modeling of Limit Order Markets

Abstract

R. Cont and A. de Larrard [5] introduced a tractable stochastic model for the dynamics of a limit order book, computing various quantities of interest such as the probability of a price increase or the diffusion limit of the price process. As suggested by empirical observations, we extend their framework to 1) arbitrary distributions for book events inter-arrival times (possibly non-exponential) and 2) both the nature of a new book event and its corresponding inter-arrival time depend on the nature of the previous book event. We do so by resorting to Markov renewal processes to model the dynamics of the bid and ask queues. We keep analytical tractability via explicit expressions for the Laplace transforms of various quantities of interest. We justify and illustrate our approach by calibrating our model to the five stocks Amazon, Apple, Google, Intel and Microsoft on June 21 st 2012. As in [5], the bid-ask spread remains constant equal to one tick, only the bid and ask queues are modeled (they are independent from each other and get reinitialized after a price change), and all orders have the same size.

General Compound Hawkes Processes in Limit Order Books

“…N (t) is the renewal process, then the model (12) were used in [42] for a semi-Markovian modelling of limit order markets.…”

“…Following the martingale method from [42], we have the following weak convergence in the Skorokhod topology (see [40]):…”

“…Regime-switching models for pricing of European and American options were considered in Buffington et al (2000) [2] and [3], respectively. Semi-Markov process was applied to limit order books in [42] to model the mid-price. We note, that a level-1 limit order books with time dependent arrival rates λ(t) were studied in [13], including the asymptotic distribution of the price process.…”

Quantitative and Comparative Analyses of Limit Order Books with General Compound Hawkes Processes

“…As suggested by empirical observations, Reference Swishchuk and Vadori (2017) extended their framework to (1) arbitrary distributions for book events inter-arrival times (possibly non-exponential) and (2) both the nature of a new book event and its corresponding inter-arrival time depend on the nature of the previous book event. They did so by resorting to Markov renewal processes to model the dynamics of the bid and ask queues.…”

General Semi-Markov Model for Limit Order Books: Theory, Implementation and Numerics

“…Also available at SRRN (see Swishchuk & Vadori (2015b)) and arXiv (see Swishchuk & Vadori (2016a)). The paper was submitted to SIAM Journal of Financial Mathematics in April 2015 and an extended version was resubmitted in March 2016.…”