Alexander Schied

Optimal Execution Strategies in Limit Order Books with General Shape Functions

We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (2005) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modelling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e. of the volume of the LOB, or the exponential recovery of the bid–ask spread. We consider both of these resilience modes and, in each case, derive explicit optimal execution strategies in discrete time. Applying our results to a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy of a risk-neutral investor, which explicitly solves the recursive scheme given in Obizhaeva and Wang (2005). We also provide some evidence for the robustness of optimal strategies with respect to the choice of the shape function and the resilience-type.

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.

Robust Preferences and Convex Measures of Risk

We prove robust representation theorems for monetary measures of risk in a situation of uncertainty, where no probability measure is given a priori. They are closely related to a robust extension of the Savage representation of preferences on a space of financial positions which is due to Gilboa and Schmeidler. We discuss the problem of computing the monetary measure of risk induced by the subjective loss functional which appears in the robust Savage representation.

Optimal Trade Execution under Geometric Brownian Motion in the Almgren and Chriss Framework

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.

Optimal investments for risk- and ambiguity-averse preferences: a duality approach

Ambiguity, also called Knightian or model uncertainty, is a key feature in financial modeling. A recent paper by Maccheroni et al. (preprint, 2004) characterizes investor preferences under aversion against both risk and ambiguity. Their result shows that these preferences can be numerically represented in terms of convex risk measures. In this paper we study the corresponding problem of optimal investment over a given time horizon, using a duality approach and building upon the results by Kramkov and Schachermayer (Ann. Appl. Probab. 9, 904–950, 1999; Ann. Appl. Probab. 13, 1504–1516, 2003).

Transient Linear Price Impact and Fredholm Integral Equations

We consider the linear-impact case in the continuous-time market impact model with transient price impact proposed by Gatheral (2008). In this model, the absence of price manipulation in the sense of Huberman and Stanzl (2004) can easily be characterized by means of Bochners theorem. This allows us to study the problem of minimizing the expected liquidation costs of an asset position under constraints on the trading times. We prove that optimal strategies can be characterized as measure-valued solutions of a generalized Fredholm integral equation of the first kind and analyze several explicit examples. We also prove theorems on the existence and nonexistence of optimal strategies. We show in particular that optimal strategies always exist and are nonalternating between buy and sell trades when price impact decays as a convex function of time. This is based on and extends a recent result by Alfonsi, Schied, and Slynko (2009) on the nonexistence of transaction-triggered price manipulation. We also prove some qualitative properties of optimal strategies and provide explicit expressions for the optimal strategy in several special cases of interest.

Optimal Trade Execution and Absence of Price Manipulations in Limit Order Book Models

We analyze the existence of price manipulation and optimal trade execution strategies in a model for an electronic limit order book with nonlinear price impact and exponential resilience. Our main results show that, under general conditions on the shape function of the limit order book, placing deterministic trade sizes at trading dates that are homogeneously spaced is optimal within a large class of adaptive strategies with arbitrary trading dates. This extends results from our earlier work with A. Fruth. Perhaps even more importantly, our analysis yields as a corollary that our model does not admit price manipulation strategies. This latter result contrasts the recent findings of Gatheral [Quant. Finance, to appear], where, in a related but different model, exponential resilience was found to give rise to price manipulation strategies when price impact is nonlinear.

On the Neyman-Pearson problem for law-invariant risk measures and robust utility functionals

Motivated by optimal investment problems in mathematical finance, we consider a variational problem of Neyman-Pearson type for law-invariant robust utility functionals and convex risk measures. Explicit solutions are found for quantile-based coherent risk measures and related utility functionals. Typically, these solutions exhibit a critical phenomenon: If the capital constraint is below some critical value, then the solution will coincide with a classical solution; above this critical value, the solution is a superposition of a classical solution and a less risky or even risk-free investment. For general risk measures and utility functionals, it is shown that there exists a solution that can be written as a deterministic increasing function of the price density.

Robust Preferences and Robust Portfolio Choice

This chapter focuses on the problems of robust preferences and robust portfolio choice. The problem of portfolio choice consists in choosing, among all the available positions, a position that is affordable, given the investors wealth w , and which is optimal with respect to the investors preferences. In its classical form, the problem of portfolio choice involves preferences of von Neumann-Morgenstern type, and a position X is affordable if its price does not exceed the initial capital w . More precisely, preferences are described by a utility functional E Q [ U ( X )], where U is a concave utility function and Q is a probability measure on the set of scenarios, which models the investors expectations. Recent research on the problem of portfolio choice has taken a much wider scope. On the one hand, the increasing role of derivatives and of dynamic hedging strategies has led to a more flexible notion of affordability. On the other hand, there is, nowadays, a much higher awareness of model uncertainty, and this has led to a robust formulation of preferences beyond the von Neumann–Morgenstern paradigm of expected utility. The chapter reviews the theory of robust preferences as developed by Schmeidler; Gilboa and Schmeidler; and Marinacci, Rustichini, and Marinacci. The chapter considers several approaches to the optimal investment problem for an economic agent who uses a robust utility functional and who can choose between risky and riskless investment opportunities in a financial market.

Comparative and qualitative robustness for law-invariant risk measures

When estimating the risk of a P&L from historical data or Monte Carlo simulation, the robustness of the estimate is important. We argue here that Hampel’s classical notion of qualitative robustness is not suitable for risk measurement, and we propose and analyze a refined notion of robustness that applies to tail-dependent law-invariant convex risk measures on Orlicz spaces. This concept captures the tradeoff between robustness and sensitivity and can be quantified by an index of qualitative robustness. By means of this index, we can compare various risk measures, such as distortion risk measures, in regard to their degree of robustness. Our analysis also yields results of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for ψ-weak convergence.