Torsten Schöneborn

Risk Aversion and the Dynamics of Optimal Liquidation Strategies in Illiquid Markets

We consider the infinite-horizon optimal portfolio liquidation problem for a von Neumann-Morgenstern investor in the liquidity model of Almgren (2003). Using a stochastic control approach, we characterize the value function and the optimal strategy as classical solutions of nonlinear parabolic partial differential equations. We furthermore analyze the sensitivities of the value function and the optimal strategy with respect to the various model parameters. In particular, we find that the optimal strategy is aggressive or passive in-the-money, respectively, if and only if the utility function displays increasing or decreasing risk aversion. Surprisingly, only few further monotonicity relations exist with respect to the other parameters. We point out in particular that the speed by which the remaining asset position is sold can be decreasing in the size of the position but increasing in the liquidity price impact.

Optimal Liquidation in Dark Pools

We consider a large trader liquidating a portfolio using a transparent trading venue with price impact and a dark pool with execution uncertainty. The optimal execution strategy uses both venues continuously, with dark pool orders over-/underrepresenting the portfolio size depending on return correlations; trading at the traditional venue is delayed depending on dark liquidity. Pushing up prices at the traditional venue while selling in the dark pool might generate profits. If future returns depend on historical dark pool liquidity, then sending orders to the dark pool can be worthwhile simply to gather information.

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 portfolio liquidation for CARA investors

We consider the finite-time optimal portfolio liquidation problem for a von Neumann-Morgenstern investor with constant absolute risk aversion (CARA). As underlying market impact model, we use the continuous-time liquidity model of Almgren and Chriss (2000). We show that the expected utility of sales revenues, taken over a large class of adapted strategies, is maximized by a deterministic strategy, which is explicitly given in terms of an analytic formula. The proof relies on the observation that the corresponding value function solves a degenerate Hamilton-Jacobi-Bellman equation with singular initial condition.

Adaptive Basket Liquidation

Abstract We consider the infinite-horizon optimal basket portfolio liquidation problem for a von Neumann–Morgenstern investor in a multi-asset extension of the liquidity model of Almgren (Appl. Math. Finance 10:1–18, 2003) with cross-asset impact. Using a stochastic control approach, we establish a “separation theorem”: the sequence of portfolios held during an optimal liquidation depends only on the (co-)variance and (cross-asset) market impact of the assets, while the speed with which these portfolios are reached depends only on the utility function of the trader. We derive partial differential equations for both the sequence of portfolios reached during the execution and the trading speed.

PORTFOLIO LIQUIDATION IN DARK POOLS IN CONTINUOUS TIME: PORTFOLIO LIQUIDATION IN DARK POOLS IN CONTINUOUS TIME

We consider an illiquid financial market where a risk averse investor has to liquidate a portfolio within a finite time horizon [0, T] and can trade continuously at a traditional exchange (the “primary venue�?) and in a dark pool. At the primary venue, trading yields a linear price impact. In the dark pool, no price impact costs arise but order execution is uncertain, modeled by a multidimensional Poisson process. We characterize the costs of trading by a linear�?quadratic functional which incorporates both the price impact costs of trading at the primary exchange and the market risk of the position. The solution of the cost minimization problem is characterized by a matrix differential equation with singular boundary condition; by means of stochastic control theory, we provide a verification argument. If a single�?asset position is to be liquidated, the investor slowly trades out of her position at the primary venue, with the remainder being placed in the dark pool at any point in time. For multi�?asset liquidations this is generally not the case; for example, it can be optimal to oversize orders in the dark pool in order to turn a poorly balanced portfolio into a portfolio bearing less risk.

OPTIMAL LIQUIDATION AND ADVERSE SELECTION IN DARK POOLS

We consider an investor who has access both to a traditional venue and a dark pool for liquidating a position in a single asset. While trade execution is certain on the traditional exchange, she faces linear price impact costs. On the other hand, dark pool orders suffer from adverse selection and trade execution is uncertain. Adverse selection decreases order sizes in the dark pool while it speeds up trading at the exchange. For small orders, it is optimal to avoid the dark pool completely. Adverse selection can prevent profitable round†trip trading strategies that otherwise would arise if permanent price impact were included in the model.

Optimal trade execution in order books with stochastic liquidity

In financial markets, liquidity changes randomly over time. We consider such random variations of the depth of the order book and evaluate their influence on optimal trade execution strategies. If the stochastic structure of liquidity changes satisfies certain conditions, then the unique optimal trading strategy exhibits a conventional structure with a single wait region and a single buy region, and profitable round‐trip strategies do not exist. In other cases, optimal strategies can feature multiple wait regions and optimal trade sizes that can be decreasing in the size of the position to be liquidated. Furthermore, round‐trip strategies can be profitable depending on bid–ask spread assumptions. We illustrate our findings with several examples including the Cox–Ingersoll–Ross model for the evolution of liquidity.