Econophysics & Microstructure Transient Market Impact Bouchaud et al. (2004)

Bouchaud Propagator Market Impact Lab

Simulate Jean-Philippe Bouchaud's (2004) market impact propagator model. Resolve the long-memory order flow paradox, analyze power-law bare kernels $G(l)$, and examine the critical balance condition $\beta = \frac{1 - \gamma}{2}$ ensuring price efficiency.

Theoretical Foundations: Long-Memory Order Flow, Bare Propagators & Critical Diffusivity
1. The Long-Memory Efficiency Paradox & Propagator Kernel

Empirical trade signs $\epsilon_t \in \{-1, +1\}$ exhibit long-range autocorrelation $C(l) \sim l^{-\gamma}$ with $\gamma \in (0, 1)$. If market impact were permanent, prices would be super-diffusive ($\text{Var} \sim t^{2-\gamma}$). Bouchaud et al. proved impact is transient and decays via a bare power-law propagator kernel:

$$P_t = P_0 + \sum_{s < t} G(t - s)\epsilon_s + \sum_{s=1}^t \eta_s, \quad G(l) = \frac{\Gamma_0}{(1 + l)^\beta}$$

Each trade shifts the price by bare impact $\Gamma_0$, which gradually relaxes as limit orders replenish the book at decay exponent $\beta > 0$.

2. Critical Balance Condition for Martingale Efficiency

For price returns to be purely diffusive (i.e. random walk / zero serial correlation $\mathbb{E}[\Delta P_t \Delta P_{t+l}] = 0$), the kernel decay $\beta$ must exactly match the flow persistence $\gamma$:

$$\beta = \frac{1 - \gamma}{2} \iff \alpha = 1 - \frac{\gamma + \beta}{2} = 0.50 \quad \text{(Brownian Diffusion)}$$

When $\beta < \frac{1-\gamma}{2}$, prices are super-diffusive ($\alpha > 0.50$, momentum); when $\beta > \frac{1-\gamma}{2}$, prices are sub-diffusive ($\alpha < 0.50$, mean-reverting).

Model Parameters

Empirical Equity Flow: γ = 0.60, β = 0.20. Satisfies Bouchaud's critical balance condition β ≈ (1 - γ)/2 = 0.20, yielding purely diffusive, efficient prices.

Theoretical Hurst α 0.50 Critical Diffusive
Critical Balance Gap 0.000 β - (1-γ)/2
Bare Impact $G(10)$ 3.10¢ Decayed Bare
Apparent Plateau $R_\infty$ 5.82¢ Permanent Level
Net Signed Flow +14 trades
Realized Price Shift +$0.824
Return Autocorr $\rho_1$ -0.018
Initial Impact $\Gamma_0$ 5.0¢/tr
Dual 5×5 Microstructure Sensitivity Grids
Sensitivity Grid 1: Decay β vs Memory γ → Diffusive Exponent $\alpha = 1 - \frac{\gamma + \beta}{2}$

Identifies the critical knife-edge $\alpha = 0.50$ (martingale diffusion) versus super-diffusive momentum ($\alpha > 0.50$) and sub-diffusive mean reversion ($\alpha < 0.50$):

Sensitivity Grid 2: Propagator Scale $\Gamma_0$ vs Flow Imbalance → Realized Price Displacement ($)

Calculates expected cumulative price displacement across market impact scales and signed order flow imbalance:

Market Microstructure Mastery: Bouchaud Propagator & Transient Impact
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Frequently Asked Questions: Bouchaud Propagator & Long-Memory Order Flow

The Propagator model, introduced by Jean-Philippe Bouchaud, Yonatan Gefen, Marc Potters, and Matthieu Wyart (2004), is a landmark quantitative finance framework that represents market impact as a linear convolution of past signed trade flow with a decaying power-law propagator kernel $G(l) = \frac{\Gamma_0}{(1 + l)^\beta}$.

Empirical order flow displays extreme long memory with autocorrelation $C(l) \sim l^{-\gamma}$ where $\gamma < 1$. If individual trades had fixed permanent impact, cumulative autocorrelated order flow would cause price variance to explode super-diffusively ($\text{Var} \sim t^{2 - \gamma}$), making price returns wildly predictable and violating weak-form market efficiency.

Bouchaud et al. proved that price returns remain strictly diffusive (uncorrelated martingale increments with Hurst exponent $\alpha = 0.50$) if and only if the propagator decay exponent $\beta$ and the order flow correlation exponent $\gamma$ satisfy the exact critical balance: $\beta = \frac{1 - \gamma}{2}$.

The bare impact $G(l)$ is the isolated physical impulse response of a single trade, which decays toward zero as the limit order book replenishes. In contrast, the apparent impact $R(l)$ is what econometricians measure empirically: because order flow is autocorrelated, subsequent trades continually replenish price displacement, causing $R(l)$ to plateau into a pseudo-permanent impact.

If $\beta < \frac{1 - \gamma}{2}$, liquidity recovery is too slow (under-damped), leading to super-diffusive momentum ($\alpha > 0.50$) and trending liquidity bubbles. If $\beta > \frac{1 - \gamma}{2}$, replenishment is over-damped, producing sub-diffusive mean-reversion ($\alpha < 0.50$). Financial markets operate naturally close to the critical knife-edge.