Trading with market resistance and concave price impact

We consider an optimal trading problem under a market impact model with endogenous market resistance generated by a sophisticated trader who (partially) detects metaorders and trades against them to exploit price overreactions induced by the order flow. The model features a concave transient impact driven by a power-law propagator with a resistance term responding to the trader’s rate via a fixed-point equation involving a general resistance function. We derive a (non)linear stochastic Fredholm equation as the first-order optimality condition satisfied by optimal trading strategies. Existence and uniqueness of the optimal control are established when the resistance function is linear, and an existence result is obtained when it is strictly convex using coercivity and weak lower semicontinuity of the associated profit-and-loss functional. We also propose an iterative scheme to solve the nonlinear stochastic Fredholm equation and prove an exponential convergence rate. Numerical experiments confirm this behavior and illustrate optimal round-trip strategies under “buy” signals with various decay profiles and different market resistance specifications.

💡 Research Summary

The paper develops a novel optimal trading framework that incorporates endogenous market resistance generated by sophisticated traders who detect and trade against meta‑orders. Building on the linear propagator model, the authors specify the impact kernel as a sum of a permanent component κ and a power‑law transient component G_{λ,ν}(t)=λ t^{‑ν} (ν∈(0,1)), which captures the long‑memory decay observed empirically. The key innovation is the introduction of a resistance flow r_u(t) that satisfies a fixed‑point equation r_u(t)=U\bigl(∫_0^t G(t‑s)(u(s)‑r_u(s))ds\bigr), where U is a monotone resistance function. A natural choice U(x)=c x² reproduces the square‑root scaling of market impact with participation rate.

The trader’s objective is to maximize expected profit‑and‑loss, balancing a stochastic signal α(t) against the total impact I_u(t)=∫_0^t G(t‑s)(u(s)‑r_u(s))ds. By applying variational calculus, the first‑order optimality condition is derived as a (non)linear stochastic Fredholm equation: u(t)=B^{-1}