Modeling market impact and timing risk in volume time

Abstract

Intraday volatility and market impact models in volume time are proposed. We build an intraday volatility profile to capture non-stationarity of intraday price returns and utilize a fractional Brownian motion process to measure deviations from square root scaling rule of volatility. We propose a generalized, scalable market impact model that encompasses two mainstream approaches: an aggregated impact of a series of trades on a sufficiently long trading horizon and a transient impact of individual trades. We give an intuitive interpretation of the model parameters and provide a generalized formulation of the optimal trading horizon and efficient trading frontier. The self-similarity feature of an aggregated model allows for its application to smaller trading horizons and modeling of transient impact of sliced orders. We formulate conditions when the impact of sliced orders can be consistently aggregated to the total impact of the original order and deduce relationships between parameters of macro and micro level models to enforce such consistency. We demonstrate that the parameters of aggregated and transient impact models are intimately related to the auto-covariance function of trade signs. We give an explicit formulation of such a relationship when the stated auto-covariance function has a power law form.

An Optimal Execution Problem with S-shaped Market Impact Functions

“…Gatheral and Schied [8] pointed out that we should regard the time parameter, t, not as physical time but as volume time. Volume time implies a stochastic clock, which is measured by a market trading volume process [3,9,20,25]. If we consider the model on a volume time line, we may find the optimality of the VWAP strategy in a similar way to Theorem 7.…”