The random walk of high frequency trading

This paper builds a model of high-frequency equity returns by separately modeling the dynamics of trade-time returns and trade arrivals. Our main contributions are threefold. First, we characterize the distributional behavior of high-frequency asset returns both in ordinary clock time and in trade time. We show that when controlling for pre-scheduled market news events, trade-time returns of the near-month E-mini S&P 500 futures contract are well characterized by a Gaussian distribution at very fine time scales. Second, we develop a structured and parsimonious model of clock-time returns using a time-changed Brownian motion composed with a general, non-Lévy directing process. Particular cases of this model allow for leptokurtosis and volatility clustering in clock-time returns, even when trade- time returns are Gaussian. Finally, we highlight conditions on the directing process which are required in order to generate proper volatility dynamics while simultaneously matching the unconditional distribution of returns. In-sample fitting and out-of-sample realized volatility forecasting demonstrate the strength of our model relative to leading candidates.