High-frequency trading model for a complex trading hierarchy

Abstract

Financial markets exhibit a complex hierarchy among different processes, e.g. a trading time marks the initiation of a trade, and a trade triggers a price change. High-frequency trading data arrive at random times. By combining stochastic and agent-based approaches, we develop a model for trading time, trading volume, and price changes. We generate intertrade time (time between successive trades) Δt _i , and the number of shares traded q(Δt _i ) as two independent but power-law autocorrelated processes, where Δt _i is subordinated to q(Δt _i ), and Δt _i is more strongly correlated than q(Δt _i ). These two power-law autocorrelated processes are responsible for the emergence of strong power-law correlations in (a) the total number of shares traded N(ΔT) and (b) the share volume Q _ΔT calculated as the sum of the number of shares q _i traded in a fixed time interval ΔT. We find that even though q(Δt _i ) is weakly power-law correlated, due to strong power-law correlations in Δt _i , the (integrated) share volume exhibits strong long-range power-law correlations. We propose that intertrade times and bid–ask price changes share the same volatility mechanism, yielding the power-law autocorrelations in absolute values of price change and power-law tails in the distribution of price changes. The model generates the log-linear functional relationship between the average bid–ask spread ⟨S⟩_ΔT and the number of trade occurrences N _ΔT , and between ⟨S⟩_ΔT and Q _ΔT . We find that both results agree with empirical findings.

Keywords:

• Subordinated processes
• High-frequency data
• Fractionally integrated processes

Acknowledgements

We thank the Ministry of Science of Croatia, the Keck Foundation Future Initiatives Program, and NSF grants PHY-0855453, CMMI-1125290, CHE-0911389, and CHE-0908218 for financial support.