# Junwei Xu

I am currently a PhD student at the LSE.

Supervisors:
Dr. Arne Løkka and Prof. Mihail Zervos

Completed Education:
MSc Financial Mathematics, London School of Economics, 2013
BSc Mathematics, University College London, 2012

Contact details:
j.xu19@lse.ac.uk
Department of Mathematics, Columbia House, London School of Economics, Houghton Street, London WC2A 2AE, UK

Research Interests

stochastic analysis, optimal control, optimal execution, contract theory, principal-agent problems

Working Papers

• Optimal liquidation in an Almgren-Chriss type model with Lévy processes and finite time horizons, with Arne Løkka

We consider an Almgren-Chriss type liquidation model and aim to maximise the expected exponential utility of the cash position at a given finite time. The unaffected asset price follows a Lévy process which may provide a good statistical fit to observed asset price data for short time horizons. The temporary price impact is described by a general function, satisfying some reasonable conditions. We reduce the problem to a deterministic optimisation problem and we derive the optimal liquidation strategy and the corresponding value function in closed forms. It turns out that, if the unaffected asset price has a positive drift, then it might be optimal to wait for a while during selling, or it might be optimal to buy back at the beginning of trading, and price manipulation is allowed in the case of positive drift. We solve the deterministic optimisation problem using calculus of variations. To this end, the Beltrami identity approach doesn't apply in a classical sense because the integrand in the objective functional is not sufficiently smooth. Nonetheless, we establish necessary and sufficient conditions for the optimiser in a fairly general setting. In particular, we characterise the optimiser using the Beltrami identity, which is a first order ordinary differential equation. This characterisation allows us to get a closed-form solution.

• Optimal liquidation in a general one-sided limit order book for a risk averse investor, with Arne Løkka

In a general one-sided limit order book with a general resilience function where the unaffected price process follows a Lévy process, we consider the problem of optimal liquidation for an investor with constant absolute risk aversion. Since liquidation normally takes place within a short time interval, modelling the risk as a Lévy process should provide a good statistical fit to observed market data. Our formulation also allows for a discontinuous limit order book which can provide a reasonable approximation for a limit order book with a discrete shape in practice. We derive an explicit expression for the value function. The optimal intervention boundary completely characterises the optimal liquidation strategy. In particular, this problem provides an example of a solvable two-dimensional singular Markovian optimal control problem with an optimal intervention boundary which could be discontinuous.

• Optimal liquidation trajectories for the Almgren-Chriss model with Lévy processes, with Arne Løkka

We consider an optimal liquidation problem with infinite horizon in the Almgren-Chriss framework, where the unaffected asset price follows a Lévy process. The temporary price impact is described by a general function which satisfies some reasonable conditions. We consider an investor with constant absolute risk aversion, who wants to maximise the expected utility of the cash received from the sale of his assets, and show that this problem can be reduced to a deterministic optimisation problem which we are able to solve explicitly. In order to compare our results with exponential Lévy models, which provide very good statistical fit with observed asset price data for short time horizons, we derive the (linear) Lévy process approximation of such models. In particular we derive expressions for the Lévy process approximation of the exponential Variance-Gamma Lévy process, and study properties of the corresponding optimal liquidation strategy. We find that for the power-law temporary impact function, the optimal strategy is to liquidate so quickly that it may be practically infeasible. We therefore study the case where the temporary impact function follows a power-law for small liquidation speeds, but tends faster to infinity than a power-law as the liquidation speed tends to infinity. In particular, we obtain an explicit expression for the connection between the temporary impact function for the Lévy model and the temporary impact function for the Brownian motion model, for which the optimal liquidation strategies from the two models coincide.