SMC-ABC methods for the estimation of stochastic si- mulation models of the limit order book
Résumé
In this chapter we consider classes of models that have been recently developed for quantitative finance that involve modelling a highly complex multivariate, multi-attribute stochastic process known as the Limit Order Book (LOB). The LOB is the primary data structure recorded each day intra-daily for the majority of assets on electronic exchanges around the world in which trading takes place. As such, it represents one of the most important fundamental structures to study from a stochastic process perspective if one wishes to characterize features of stochastic dynamics for price, volume, liquidity and other important attributes for a traded asset. In this paper we aim to adopt the model structure recently proposed by Panayi and Peters [2015], which develops a stochastic model framework for the LOB of a given asset and to explain how to perform calibration of this stochastic model to real observed LOB data for a range of different assets. One can consider this class of problems as truly a setting in which both the likelihood is intractable to evaluate pointwise, but trivial to simulate, and in addition the amount of data is massive. This is a true example of big-data application as for each day and for each asset one can have anywhere between 100,000-500,000 data vectors for the calibration of the models.
The class of calibration techniques we will consider here involves a Bayesian Approximate Bayesian Computation (ABC) reformulation of the indirect inference framework developed un- der the multi-objective optimization formulation proposed recently by Panayi and Peters [2015]. To facilitate an equivalent comparison for the two frameworks, we also adopt a reformulation of the class of genetic stochastic search algorithms utilised by Panayi and Peters [2015], known as NGSA-II [Deb et al., 2002]. We adapt this widely utilised stochastic genetic search algorithm from the multi-objective optimization algorithm literature to allow it to be utilised as a mutation kernel in a class of Sequential Monte Carlo Samplers (SMC Sampler) algorithms in the ABC context. We begin with the problem and model formulation, then we discuss the estimation frameworks and finish with some real data simulation results for equity data from a highly utilised pan-European secondary exchange formerly known as Chi-X, before it was recently aquired by BATS to form BATS Chi-X Europe in 2014 (https://www.batstrading.co.uk/).