[Paper Review] Permanent market impact can be nonlinear

Olivier Guéant|arXiv (Cornell University)|May 2, 2013

Financial Markets and Investment StrategiesEconomics, Econometrics and Finance19 references21 citations

TL;DR

This paper proposes a novel framework that reconciles nonlinear permanent market impact with the absence of dynamic arbitrage by modeling it as a function of cumulative traded volume. It generalizes Almgren et al.'s [10] estimation methodology, enabling unbiased statistical estimation of both permanent and instantaneous market impact using slippage, pre-trade, and post-trade prices.

ABSTRACT

There are two schools of thought regarding market impact modeling. On the one hand, seminal papers by Almgren and Chriss introduced a decomposition between a permanent market impact and a temporary (or instantaneous) market impact. This decomposition is used by most practitioners in execution models. On the other hand, recent research advocates for the use of a new modeling framework that goes down to the resilient dynamics of order books: transient market impact. One of the main criticisms against permanent market impact is that it has to be linear to avoid dynamic arbitrage. This important discovery made by Huberman and Stanzl and Gatheral favors the transient market impact framework, as linear permanent market impact is at odds with reality. In this paper, we reconsider the point made by Gatheral using a simple model for market impact and show that permanent market impact can be nonlinear. Also, and this is the most important part from a practical point of view, we propose different statistics to estimate permanent market impact and execution costs that generalize the ones proposed in Almgren at al. (2005).

Motivation & Objective

• To resolve the theoretical conflict between nonlinear permanent market impact and the absence of dynamic arbitrage, which has long favored transient impact models.
• To demonstrate that nonlinear permanent impact is compatible with no-arbitrage when the impact depends on cumulative volume, not just current trade size.
• To provide a statistically robust method for estimating permanent and instantaneous market impact components from observable market data.
• To generalize Almgren et al.'s [10] estimation formula to accommodate nonlinear permanent impact, particularly the square root form commonly observed empirically.
• To correct bias in instantaneous market impact estimation that arises when nonlinear permanent impact is ignored in standard models.

Proposed method

• Introduces a modified market impact model where permanent impact depends on the cumulative volume executed so far, not just the current trade size.
• Uses a power-law functional form for permanent impact: $ k \cdot \text{sgn}(q_0) \cdot |q_0|^\alpha $, with $ \alpha \neq 1 $, to model concave impact.
• Derives a stochastic model of price dynamics incorporating both permanent and temporary impact, with price impact driven by a Brownian motion.
• Establishes a key identity linking slippage, price return, and cumulated instantaneous impact, showing that both components can be estimated from observable data.
• Applies Itô's lemma and stochastic calculus to derive the joint distribution of estimation errors in the generalized framework.
• Proposes Theorem 2 as a generalization of Almgren et al. [10], providing explicit formulas for estimating permanent and instantaneous impact using pre-trade price, post-trade price, and slippage.

Experimental results

Research questions

• RQ1 Can nonlinear permanent market impact coexist with the absence of dynamic arbitrage in optimal execution models?
• RQ2 Is the standard linear permanent impact assumption in Almgren-Chriss models empirically justifiable, given the observed square root form in practice?
• RQ3 How can permanent and instantaneous market impact be consistently estimated when permanent impact is nonlinear?
• RQ4 What is the impact of nonlinear permanent impact on the statistical estimation of the instantaneous impact function?
• RQ5 Can the classical Almgren et al. [10] estimation framework be generalized to accommodate nonlinear permanent impact without introducing bias?

Key findings

• Nonlinear permanent market impact is compatible with the absence of dynamic arbitrage when it depends on cumulative traded volume rather than instantaneous trade size.
• The proposed framework allows for a concave permanent impact function, such as the empirically observed square root form, without violating no-arbitrage conditions.
• Theorem 2 provides a generalized estimation formula that allows consistent recovery of both permanent and instantaneous market impact from slippage, pre-trade price, and post-trade price.
• When permanent impact is nonlinear (e.g., $ \alpha \neq 1 $), using the linear [10] formula leads to biased estimation of the instantaneous impact function.
• The estimation error terms $ \epsilon_1 $ and $ \epsilon_2 $ are jointly normally distributed with known variance-covariance structure, enabling statistical inference.
• The framework preserves the structure of classical optimal execution models, as the permanent impact component does not affect optimal execution strategies, only estimation accuracy.