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 $u^*_t $ut∗ is a Gaussian Volterra process of the form $u^*(t)=u^0(t)+\bar {u}(t)+\int _0^t k(u,t)\,{\rm d}W_u $u∗(t)=u0(t)+u¯(t)+∫0tk(u,t)dWu on $[0,T) $[0,T), where $k(\cdot ,\cdot ) $k(⋅,⋅) and $\bar {u} $u¯ satisfy a family of (linear) Fredholm integral equations of the first kind which can be solved in terms of fractional derivatives. The term $u^0(t) $u0(t) 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 $k(u,t) $k(u,t) 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 $k(u,t) $k(u,t) 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 $\int _0^t g(t-s) \,{\rm d}W_s $∫0tg(t−s)dWs, and we calibrate the model to high frequency limit order book data for various NASDAQ stocks.