Liquidation stops copying a fixed schedule: a neural policy times trades from fractional-Brownian memory and beats Almgren-Chriss away from the martingale case
Synopsis
In liquidation with linear market impact and a price driven by fractional Brownian motion, the deterministic Almgren-Chriss schedule is no longer optimal. A policy-gradient-trained neural network takes past price increments as input and maximizes CARA utility directly; the learned policy beats the Almgren-Chriss benchmark at every tested Hurst exponent, recovers it in the martingale case, improves more as the exponent moves away from 0.5, and gains more in the anti-persistent regime.
Interpretation
The liquidation strategy changes from a fixed deterministic schedule to a dynamic policy conditioned on past price increments. In the original Almgren-Chriss setting the price follows ordinary Brownian motion and the optimal solution is deterministic; when the price is instead driven by fractional Brownian motion, that deterministic solution is no longer optimal. At every tested Hurst exponent the learned policy outperforms the Almgren-Chriss benchmark, and in the martingale case it recovers that benchmark's value.
The policy compresses the increment correlation of fractional Brownian motion into a single scalar forecast and sets the selling pace from it. Policy gradient is used to estimate optimal dynamic controls; here a neural network parameterizes the policy and the Almgren-Chriss strategy serves as a baseline that reduces gradient variance. Training and evaluation use parameters set by the Almgren-Chriss calibration reasoning, and separate models are trained and tested for several Hurst exponents.
The shape of the improvement differs on the two sides of the Hurst exponent. In the persistent regime the excess return comes mainly from the tails of the payoff distribution, while in the anti-persistent regime the advantage is obtained on most paths. Trajectory-level visualizations show trend-following with short positions learned in the persistent regime and inventory-holding to capture mean reversion in the anti-persistent regime.
Perspective
The demonstrated scope is a simulated liquidation problem with a fractional-Brownian price and linear market impact, with parameters set by the Almgren-Chriss calibration reasoning; each Hurst exponent needs its own trained model.
Whether real market prices follow fractional Brownian motion is still debated; when the Hurst exponent is close to 0 or 1, the learning procedure can still be unstable because rewards fluctuate strongly.