Optimal trade execution for Gaussian signals with power-law resilience

We characterize the optimal signal-adaptive liquidation strategy for an agent subject to power-law resilience and zero temporary price impact with a Gaussian signal, which can include e.g an OU process or fractional Brownian motion. We show that the optimal selling speed (Formula presented.) is a Gaussian Volterra process of the form (Formula presented.) on (Formula presented.), where (Formula presented.) and (Formula presented.) satisfy a family of (linear) Fredholm integral equations of the first kind which can be solved in terms of fractional derivatives. The term (Formula presented.) is the (deterministic) solution for the no-signal case given in Gatheral et al. [Transient linear price impact and Fredholm integral equations. Math. Finance, 2012, 22, 445–474], and we give an explicit formula for (Formula presented.) for the case of a Riemann-Liouville price process as a canonical example of a rough signal. With non-zero linear temporary price impact, the integral equation for (Formula presented.) becomes a Fredholm equation of the second kind. These results build on the earlier work of Gatheral et al. [Transient linear price impact and Fredholm integral equations. Math. Finance, 2012, 22, 445–474] for the no-signal case, and complement the recent work of Neuman and Voß[Optimal signal-adaptive trading with temporary and transient price impact. Preprint, 2020]. Finally we show how to re-express the trading speed in terms of the price history using a new inversion formula for Gaussian Volterra processes of the form (Formula presented.), and we calibrate the model to high frequency limit order book data for various NASDAQ stocks.