In this paper we provide a general mathematical framework for distributional transforms, which allows for many examples that are used extensively in the literature of finance, economics and optimization. We put a special focus on the class of probability distortions, which is a fundamental tool...

We consider a Nash equilibrium between two high-frequency traders in a simple market impact model with transient price impact and additional quadratic transaction costs. Extending a result by Schöneborn (2008), we prove existence and uniqueness of the Nash equilibrium and show that for small...

For a market impact model, price manipulation and related notions play a role that is similar to the role of arbitrage in a derivatives pricing model. Here, we give a systematic investigation into such regularity issues when orders can be executed both at a traditional exchange and in a dark...

We consider a stochastic game between a trader and a central bank in a target zone market with a lower currency peg. This currency peg is maintained by the central bank through the generation of permanent price impact, thereby aggregating an ever increasing risky position in foreign reserves. We...

Assuming geometric Brownian motion as unaffected price process S⁰, Gathertal and Schied (2011) derived a strategy for optimal order execution that reacts in a sensible manner on market changes but can still be computed in closed form. Here we will investigate the robustness of this strategy with...

With an alternative choice of risk criterion, we solve the HJB equation explicitly to find a closed-form solution for the optimal trade execution strategy in the Almgren-Chriss framework assuming the underlying unaffected stock price process is geometric Brownian motion

The viability of a market impact model is usually considered to be equivalent to the absence of price manipulation strategies in the sense of Huberman & Stanzl (2004). By analyzing a model with linear instantaneous, transient, and permanent impact components, we discover a new class of...

We continue the analysis of optimal execution strategies in the model for a limit order book with nonlinear price impact and exponential resilience that was considered in Alfonsi, Schied, and Fruth (2009). We now allow for non-homogeneous resilience rates and arbitrary trading dates and consider...

In the past decades, advanced probabilistic methods have had significant impact on the field of finance, both in academia and in the financial industry. Conversely, financial questions have stimulated new research directions in probability. In this survey paper, we review some of these...

Market impact models describe the feedback of trading strategies on the underlying asset prices. The resulting feedback effects lead to interesting questions of stability and regularity, which are closely related to the existence and behavior of strategies for optimal order execution. In this...