Regularity of Market Impact Models with Stochastic Price Impact

We introduce a market impact model for stochastic linear transient impact, extending the model of Gatheral (2010) with the possibility of randomly fluctuating liquidity. We discuss regularity conditions for market impact models, i.e. properties of optimal liquidation strategies in these models. By many examples, we illustrate how regularity might fail and what consequences arise. In particular, there can be arbitrage opportunities although the unaffected price process is a martingale. For our stochastic market impact model, we give a necessary condition, and for exponentially decaying impact a sufficient condition for the regularity of the model. In a numerical example we show that regularity can strongly depend on the liquidation time horizon. Furthermore, we show that even if the liquidity parameter is a martingale, deterministic strategies can be suboptimal.