High-frequency trading model for a complex trading hierarchy

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) Δ t i , and the number of shares traded q (Δ t i ) as two independent but power-law autocorrelated processes, where Δ t i is subordinated to q (Δ t i ), and Δ t i is more strongly correlated than q (Δ t 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 (Δ T ) and (b) the share volume Q Δ T calculated as the sum of the number of shares q i traded in a fixed time interval Δ T . We find that even though q (Δ t i ) is weakly power-law correlated, due to strong power-law correlations in Δ t 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 ⟩ Δ T and the number of trade occurrences N Δ T , and between ⟨ S ⟩ Δ T and Q Δ T . We find that both results agree with empirical findings.