Optimal trade execution with transient relative price impact and directional views: A 2nd-order variational approach to a 3-dimensional non-convex free boundary problem

This talk is given by Prof. Dirk Becherer (Humboldt University-Berlin)

We solve the optimal execution problem to trade a large financial asset position within finite time in an illiquid market, where price impact is transient and possibly non-linear (in log-prices), like in models by J.P.Bouchaud, F.Lillo or J.Gatheral. We derive a complete solution for the optimal control problem of finite-fuel type, where mechanical price impact is

1.  multiplicative instead of additive as in the of seminal articles by Obizhaeva/Wang (2013), Predoiu/Shaikhet/Shreve (2011) or Schied et al. (2010), and
2. we admit for non-vanishing drift in the fundamental price process, that is directional views about short-term price trends (as in Almgren/Chriss 2000, ch.4).

Multiplicative impact in relative percentage terms is well established (cf. Bertsimas/Lo 1998, ch.3) and avoids the possibility of negative asset prices.

We obtain a complete characterization of the regular three-dimensional free boundary surface which separates the no-/action regions for the non-convex three-dimensional singular control problem by a family of characteristic curves, whose description is explicit up to the solution of ODEs. While the free boundary description is almost as explicit as in Obizhaeva/Wang, our analysis is more demanding for a lack of an apparent convexity structure to exploit. 

Yet, a key argument to prove global optimality turns out to show at first a local optimality for a candidate free boundary under smooth perturbations by 2nd-order variational arguments. 

For the optimal trading application , the results may shed light on phenomena, like for instance

a) how to profit from directional views (signals) about price trends by optimal (non-trivial) round-trips; 

b) down-or upward directional views do lead to respective front- or back-loading in the optimal execution of (sell) trading strategies;

c) optimal trading strategies are qualitatively different and their profitability can depend non-monotonically on the resilience (transience) parameters for the price impact. 

(This is joint work with Peter Frentrup)

If you are interested to join this talk, please contact Dr Miryana Grigorova at m.r.grigorova@leeds.ac.uk for the Zoom details.