Stochastic Liquidity as a Proxy for Nonlinear Price Impact

Abstract

For the “In This Issue” column: Trading costs play a central role in designing and implementing quantitative trading strategies. To quantify trading costs, optimal execution and trading algorithms rely on price impact models, such as the propagator model. 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. This paper shows 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. This allows us to derive simple formulas for optimal trading strategies and sharp conditions on market volumes to rule out price manipulation. A detailed empirical study using high-frequency limit-order data illustrates the practical performance of the theoretical results.