Optimal Execution with Nonlinear Transient Price Impact

Summary

This document formulates an optimal trading problem that combines alpha signals with nonlinear transient price impact. It uses a general propagator model, including power-law impact decay, and derives a nonlinear stochastic Fredholm equation for the optimal strategy through a variational approach. The equation has forward and backward coefficients. The authors establish existence and uniqueness under a monotonicity condition tied to the impact nonlinearity, and give an additional existence result for countable probability spaces when that condition does not hold.

The work also introduces an iterative method and proves convergence to the optimal strategy. A numerical implementation illustrates convergence and stability, as well as how concavity affects execution under exponential and power-law decay. The description provides theoretical guarantees and numerical illustrations, but no specific market dataset, trading-performance figures, or practical calibration guidance. The conclusions therefore concern the stated model and conditions rather than demonstrated live-market outcomes.

Key ideas

• Alpha signals are incorporated into an optimal execution problem with nonlinear transient price impact.
• The optimal strategy satisfies a stochastic Fredholm equation with forward and backward coefficients.
• Existence and uniqueness are established under a monotonicity condition on the impact nonlinearity.
• A convergent iterative scheme is proposed to compute the optimal strategy.
• Numerical examples examine stability and the effect of concavity under exponential and power-law decay.

Tags

# Fredholm Approach to Nonlinear Propagator Models


# Fredholm Approach to Nonlinear Propagator Models









We formulate and solve an optimal trading problem with alpha signals, where transactions induce a nonlinear transient price impact described by a general propagator model, including power-law decay. Using a variational approach, we demonstrate that the optimal trading strategy satisfies a nonlinear stochastic Fredholm equation with both forward and backward coefficients. We prove the existence and uniqueness of the solution under a monotonicity condition reflecting the nonlinearity of the price impact. Moreover, we derive an existence result for the optimal strategy beyond this condition when the underlying probability space is countable. In addition, we introduce a novel iterative scheme and establish its convergence to the optimal trading strategy. Finally, we provide a numerical implementation of the scheme that illustrates its convergence, stability, and the effects of concavity on optimal execution strategies under exponential and power-law decay.

Shown in full with attribution under the source's licence. Licence: abstract CC0

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