Fredholm Approach to Nonlinear Propagator Models

TL;DR

This work studies optimal execution under a nonlinear, non-Markovian price-impact model driven by a general propagator $G$ and a concave impact function $h$, incorporating predictive alpha signals. A variational approach yields a nonlinear stochastic Fredholm equation that characterizes the optimal trading strategy; under a monotone $\mathbf A$, existence and uniqueness follow with $\gamma$-strong concavity, and an iterative scheme with provable convergence solves the problem numerically. The authors extend existence beyond monotonicity to countable probability spaces and provide a rigorous convergence rate for the scheme, along with stability results with respect to kernel and signal perturbations. Numerical illustrations demonstrate convergence, stability, and the effects of power-law decay and concavity on optimal strategies, including approximations of fractional kernels by sums of exponentials. Altogether, the paper delivers a mathematically rigorous framework and practical algorithm for optimal liquidation under realistic, nonlocal, nonlinear price-impact dynamics with non-Markovian features.

Abstract

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.

Figures (9)

• Figure 1: Impact function specification \ref{['eq:impact_function_specification']} compared to the linear $x \mapsto x$ and the square-root impact functions $x \mapsto \xi \text{sign}(x)\sqrt{|x|}, \; \xi = 0.6$, and its derivative \ref{['eq:impact_function_specification_derivative']} for various values of $c$. Note that we don't display the derivative of the square-root impact function $x \in \mathbb{R}^{*} \mapsto \frac{\xi}{\sqrt{2|x|}}$ which explodes when approaching the origin.
• Figure 2: Arrow-Pratt relative measure of risk ratio for various values of $c$, see Proposition \ref{['P:arrow_pratt_ratio_for_f_x_0_c']}.
• Figure 3: Illustration of convergence at exponential rate of the numerical scheme \ref{['scheme_initialization']}--\ref{['scheme_update']} for various kernels in the deterministic case, with $\gamma = 1$, impact function $h_{x_{0}, c}$ specified in \ref{['eq:impact_function_specification']} and deterministic signal given by \ref{['eq:drift_signal_specification']}--\ref{['eq:signal_specification']} with $\tilde{\theta}=-40$, $\tilde{\kappa}=1$, $\tilde{\xi} = 0$, $\tilde{I}_0=20$.
• Figure 4: Convergence of the numerical solution obtained by the scheme \ref{['scheme_initialization']}-\ref{['scheme_update']} in the stochastic case to the explicit benchmark \ref{['eq:explicit_benchmark_smooth']}-\ref{['eq:explicit_benchmark_bulk_trades']} when decreasing $\gamma$ in the concave case, i.e., $\left(x_{0}, c\right) = \left(0.5,0.8\right)$ in the impact function specification \ref{['eq:impact_function_specification']}. The stochastic signal is parametrized as the integral of an Ornstein-Uhlenbeck as in \ref{['eq:drift_signal_specification']} with volatility $\tilde{\xi} = 0.5$, mean level $\tilde{\theta} = -4$, mean-reversion speed $\tilde{\kappa}=1$ and initial value $\tilde{I}_0 = 2$. We run $n=30$ iterations for each case, with $N = 200$ time-steps and $M=10000$ sample trajectories. Both the scaling parameter and the mean-reversion rate $\tau$ of the exponential decay are fixed to $1$. The regression basis is set as $\left( u^{[n-1]}, \int_{0}^{\cdot} u^{[n-1]}_{s} \mathrm{d} s, \int_{0}^{\cdot} e^{-\tilde{\kappa}(\cdot-s)}u^{[n-1]}_{s} \mathrm{d} s \right)$ at each iteration and Laguerre polynomials up to degree $d=4$ are selected for the expansion basis. The respective numerical error metrics $E^{N,M}$ from \ref{['eq:empirical_error_metric']} are $6e-5$, $1e-3$, $5e-2$ and $1e-1$ for $\gamma = 1, \; 0.1, \; 0.01, \; 0.002$.
• Figure 5: Power-law decay approximation by multiple exponential decays: without shift ($\epsilon=0$) on the left, and with shift ($\epsilon>0$) on the right.

Theorems & Definitions (38)

• Definition 2.1
• Remark 2.2: Flexibility of the model specification
• Definition 2.3
• Remark 2.4: Assumptions on the impact function $h$
• Remark 2.5: Linear transient impact
• Theorem 2.6
• Remark 2.7
• Proposition 2.8
• Remark 2.9
• Theorem 2.10