Optimal execution with deterministically time varying liquidity: well posedness and price manipulation

We investigate the well-posedness in the Hadamard sense and the absence of price manipulation in the optimal execution problem within the Almgren-Chriss framework, where the temporary and permanent impact parameters vary deterministically over time. We present sufficient conditions for the existence of a unique solution and provide second-order conditions for the problem, with a particular focus on scenarios where impact parameters change monotonically over time. Additionally, we establish conditions to prevent transaction-triggered price manipulation in the optimal solution, i.e. the occurence of buying and selling in the same trading program. Our findings are supported by numerical analyses that explore various regimes in simple parametric settings for the dynamics of impact parameters.

💡 Research Summary

The paper investigates the well‑posedness and the absence of price manipulation in optimal execution problems when both temporary and permanent market‑impact coefficients are allowed to vary deterministically over time. Building on the classic Almgren‑Chriss framework, the authors replace the constant impact parameters (θ, η) with deterministic functions θ(t) and η(t), reflecting the empirically observed intraday dynamics of liquidity. The objective is to minimize the expected implementation shortfall, which in the discrete‑time setting can be written as a quadratic form C(ξ)=½ ξᵀAξ, where A encodes the time‑varying impact through a structured matrix containing the temporary impacts on the diagonal and cumulative permanent impacts on the off‑diagonal entries.

A central contribution is the identification of precise mathematical conditions under which the optimization problem is Hadamard‑well‑posed (i.e., a unique, stable solution exists) and financially “strongly well‑posed” (the optimal liquidation trajectory is monotone, precluding transaction‑triggered price manipulation). In discrete time, the authors show that if the impact matrix A is a B‑matrix in the sense of Peña (2001)—that is, it has positive diagonal entries, non‑negative off‑diagonal entries, and satisfies diagonal dominance—then A is automatically symmetric positive definite (SPD). Consequently, the quadratic program has a unique minimizer ξ* = –Q (1ᵀA⁻¹1)⁻¹ A⁻¹1, which is static (F₀‑measurable) and monotone, guaranteeing the absence of manipulation. This B‑matrix condition translates into a simple, verifiable inequality involving the sampled values of θₖ and ηₖ: the maximal decay rate of permanent impact must not exceed the minimal temporary impact over the trading horizon (up to a time‑scale factor).

In the continuous‑time limit (N→∞), the cost functional becomes
∫₀ᵀ