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Trading with market resistance and concave price impact

Nathan De Carvalho, Youssef Ouazzani Chahdi, Grégoire Szymanski

TL;DR

This paper studies optimal trading under a market impact model that includes endogenous market resistance generated by sophisticated traders who detect metaorders and trade against their transient effects. It derives a first-order condition in the form of a nonlinear stochastic Fredholm equation, proves well-posedness and existence results (uniqueness in the linear case; existence under convex resistance), and develops a Nyström-based numerical scheme with an exponential convergence rate to compute optimal trading strategies. The framework combines a power-law transient propagator with a fixed-point resistance mechanism, yielding a rich nonlinear optimization problem that captures square-root-like impact and resistance dynamics. Numerical experiments demonstrate optimal round-trip strategies under convex resistance, power-law decay, and stochastic buy signals, highlighting the practical viability of the approach for real-time execution with endogenously adjusted impact.

Abstract

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.

Trading with market resistance and concave price impact

TL;DR

This paper studies optimal trading under a market impact model that includes endogenous market resistance generated by sophisticated traders who detect metaorders and trade against their transient effects. It derives a first-order condition in the form of a nonlinear stochastic Fredholm equation, proves well-posedness and existence results (uniqueness in the linear case; existence under convex resistance), and develops a Nyström-based numerical scheme with an exponential convergence rate to compute optimal trading strategies. The framework combines a power-law transient propagator with a fixed-point resistance mechanism, yielding a rich nonlinear optimization problem that captures square-root-like impact and resistance dynamics. Numerical experiments demonstrate optimal round-trip strategies under convex resistance, power-law decay, and stochastic buy signals, highlighting the practical viability of the approach for real-time execution with endogenously adjusted impact.

Abstract

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.
Paper Structure (46 sections, 20 theorems, 207 equations, 8 figures)

This paper contains 46 sections, 20 theorems, 207 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Inception: a sophisticated trader as a market resistance
  3. A two-player toy model
  4. A general market resistance model
  5. Properties of the market resistance
  6. Optimal trading with market resistance
  7. Problem formulation
  8. Reformulation with operators on $\mathcal{L}^2$.
  9. The linear case
  10. The general case
  11. Numerical scheme and application to optimal round-trips
  12. Approximation of operators
  13. Nyström approximation of Volterra integral operators.
  14. Numerical approximation of the nonlinear operator $\mathbf{A}$ from \ref{['eq:A_nonlinear_operator']}.
  15. Iterative numerical scheme
  16. ...and 31 more sections

Key Result

Lemma 3.2

Let $u \in \mathcal{L}^2$. Suppose that Assumptions assumption:G and assumption:H hold and that $\mathcal{U}$ is Lipschitz continuous, then there exists a unique solution $r^{u} \in \mathcal{L}^2$ to eq:r_u. Moreover, the mapping $u \mapsto r^u$ is Lipschitz continuous on $\mathcal{L}^2$.

Figures (8)

  • Figure 1: Market impact and trading rates for $\nu = 0.5$, $\lambda = 1$, $u(t) = 0.3\,\mathbf{1}_{0 \leq t \leq 1}$, and $\mathcal{U}(x)=x^2$.
  • Figure 2: Market impact as a function of $\gamma$ for $\nu=0.5$, $\lambda=1$, and $\mathcal{U}(x)=x^2$. A power-law fit of the form $MI = 1.172 \cdot \gamma^{0.6086}$ is shown.
  • Figure 3: Resistance function specification \ref{['eq:resistance_specification']} and its derivative \ref{['eq:derivative_resistance_specification']} for various values of $c$, while fixing $\delta = 1.2$.
  • Figure 4: Optimal round-trips for three stochastic "buy" signals with different signal decays $\kappa$ from \ref{['eq:drift_signal_specification']}. For each quantity, the shaded regions represent their normal $95\%$ confidence intervals estimated from the optimal $2000$ sample trajectories, the empty dot markers denote the corresponding sample means, and we also display $5$ sample trajectories.
  • Figure 5: Convergence of the numerical scheme \ref{['eq:approximate_FOC_scheme']}--\ref{['eq:update_resistance_scheme']} for three stochastic "buy" signals with different signal decays $\kappa$ from \ref{['eq:drift_signal_specification']}: the left-hand plot displays the error $E_{N,M}^{1}$ of the FOC defined in \ref{['eq:max_error_foc']} while the right-hand plot shows the numerical error $E_{N,M}^{bf}$ of the backward scheme \ref{['eq:numerical_solution_f']} defined in \ref{['eq:max_error_backward_scheme']} as a function of the iteration step of the scheme. Note that the numerical error $E_{N,M}^{2}$, defined in from \ref{['eq:max_error_resistance']}, and due to Picard iterations when computing the resistance function is set to be lower than $1e-16$ at each iteration.
  • ...and 3 more figures

Theorems & Definitions (41)

  • Definition 3.1
  • Remark 1
  • Lemma 3.2
  • proof
  • Remark 2
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • Theorem 3.6
  • ...and 31 more