Mikhail Urusov

Optimal Trade Execution and Price Manipulation in Order Books with Time‐Varying Liquidity

In financial markets, liquidity is not constant over time but exhibits strong seasonal patterns. In this article we consider a limit order book model that allows for time-dependent, deterministic depth and resilience of the book and determine optimal portfolio liquidation strategies. In a first model variant, we propose a trading dependent spread that increases when market orders are matched against the order book. In this model no price manipulation occurs and the optimal strategy is of the wait region - buy region type often encountered in singular control problems. In a second model, we assume that there is no spread in the order book. Under this assumption we find that price manipulation can occur, depending on the model parameters. Even in the absence of classical price manipulation there may be transaction triggered price manipulation. In specific cases, we can state the optimal strategy in closed form.

Optimal stopping via measure transformation: the Beibel–Lerche approach

Optimal stopping of diffusions and related processes is usually done by solving a free boundary problem. In this paper, we propagate an alternative way, which has already been described in two earlier papers of Beibel and Lerche; we call it the B–L approach. It can be viewed as optimal stopping via measure transformation. While we emphasized in Beibel and Lerche a rather algebraic view, we describe here more the analytic side of the approach. Finally, it is related to some recent Jamshidians results on a duality in optimal stopping.

On a class of optimal stopping problems for diffusions with discontinuous coefficients

In this paper, we introduce a modification of the free boundary problem related to optimal stopping problems for diffusion processes. This modification allows the application of this PDE method in cases where the usual regularity assumptions on the coefficients and on the gain function are not satisfied. We apply this method to the optimal stopping of integral functionals with exponential discount of the form

Deterministic criteria for the absence of arbitrage in one-dimensional diffusion models

E_x\int_0^{\tau}e^{-\lambda s}f(X_s) ds

Variance reduction for discretised diffusions via regression

,

Martingale Property of Generalized Stochastic Exponentials

\lambda\ge0

On the submartingale/supermartingale property of diffusions in natural scale

for one-dimensional diffusions

A note on delta hedging in markets with jumps

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Optimal trade execution in order books with stochastic liquidity

. We prove a general verification theorem which justifies the modified version of the free boundary problem. In the case of no drift and discount, the free boundary problem allows to give a complete and explicit discussion of the stopping problem.

A functional limit theorem for irregular SDEs

We obtain a deterministic characterisation of the no free lunch with vanishing risk, the no generalised arbitrage and the no relative arbitrage conditions in the one-dimensional diffusion setting and examine how these notions of no-arbitrage relate to each other.