We consider the problem faced by an investor who must liquidate a given basket of assets over a finite time horizon. The investor's goal is to maximize the expected utility of the sales revenues over a class of adaptive strategies. We assume that the investor's utility has constant absolute risk aversion (CARA) and that the asset prices are given by a very general continuous-time, multiasset price impact model. Our main result is that (perhaps surprisingly) the investor does no worse if he narrows his search to deterministic strategies. In the case where the asset prices are given by an extension of the nonlinear price impact model of Almgren [(2003) Applied Mathematical Finance, 10, pp. 1-18], we characterize the unique optimal strategy via the solution of a Hamilton equation and the value function via a nonlinear partial differential equation with singular initial condition.Market impact modelling, illiquid markets, optimal liquidation, optimal trade execution, algorithmic trading, utility maximization, Hamilton-Jacobi-Bellman equation, finite fuel control,

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.

In financial markets, liquidity is not constant over time but exhibits strong seasonal patterns. In this paper, 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 tradingdependent 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.

We consider a large trader seeking to liquidate a portfolio using both a transparent trading venue and a dark pool. Our model captures the price impact of trading in transparent traditional venues as well as the execution uncertainty of trading in a dark pool. The unique optimal execution strategy uses both venues continuously. The order size in the dark pool can over-or underrepresent the portfolio size depending on adverse selection and the correlation structure of the assets in the portfolio. Introduction a dark pool results in delayed trading at the traditional venue. The appeal of the dark pool is increased by liquidity but reduced by adverse selection. By pushing up prices at the traditional venue and parallel selling in the dark pool, a trader might generate profits; we provide sufficient conditions to rule out such profitable price manipulation strategies.

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. K E Y W O R D Slimit order book, market impact model, optimal order execution, profitable round trip trading strategies, resilience, stochastic order book depth, timevarying liquidity Mathematical Finance. 2019;29:507-541.wileyonlinelibrary.com/journal/mafi