Stochastic Liquidity as a Proxy for Nonlinear Price Impact

Optimal execution and trading algorithms rely on price impact models, such as the propagator model, to quantify trading costs. Empirically, price impact is concave in trade sizes, leading to nonlinear models for which optimization problems are intractable, and even qualitative properties, such as price manipulation, are poorly understood. However, we show that in the diffusion limit of small and frequent orders, the nonlinear model converges to a tractable linear model. In this high-frequency limit, a stochastic liquidity parameter approximates the original impact function’s nonlinearity. We illustrate the approximation’s practical performance using limit order data.