arXiv subjects

Alexander Schied

Publications and source records attributed to Alexander Schied.

50 records · Page 3Linked to original sources

arXiv2015

Optimal Portfolio Liquidation in Target Zone Models and Catalytic Superprocesses

We study optimal buying and selling strategies in target zone models. In these models the price is modeled by a diffusion process which is reflected at one or more barriers. Such models arise for example when a currency exchange rate is kept above a certain threshold due to central bank intervention. We consider the optimal portfolio liquidation problem for an investor for whom prices are optimal at the barrier and who creates temporary price impact. This problem will be formulated as the minimization of a cost-risk functional over strategies that only trade when the price process is located at the barrier. We solve the corresponding singular stochastic control problem by means of a scaling limit of critical branching particle systems, which is known as a catalytic superprocess. In this setting the catalyst is a set of points which is given by the barriers of the price process. For the cases in which the unaffected price process is a reflected arithmetic or geometric Brownian motion with drift, we moreover give a detailed financial justification of our cost functional by means of an approximation with discrete-time models.

q-fin.PM↗

arXiv2015

Quasi-Hadamard differentiability of general risk functionals and its application

We apply a suitable modification of the functional delta method to statistical functionals that arise from law-invariant coherent risk measures. To this end we establish differentiability of the statistical functional in a relaxed Hadamard sense, namely with respect to a suitably chosen norm and in the directions of a specifically chosen "tangent space". We show that this notion of quasi-Hadamard differentiability yields both strong laws and limit theorems for the asymptotic distribution of the plug-in estimators. Our results can be regarded as a contribution to the statistics and numerics of risk measurement and as a case study for possible refinements of the functional delta method through fine-tuning the underlying notion of differentiability

math.ST↗

arXiv2014

Price manipulation in a market impact model with dark pool

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 pool. To this end, we focus on a class of dark-pool models whose market impact at the exchange is described by an Almgren--Chriss model. Conditions for the absence of price manipulation for all Almgren--Chriss models include the absence of temporary cross-venue impact, the presence of full permanent cross-venue impact, and the additional penalization of orders executed in the dark pool. When a particular Almgren--Chriss model has been fixed, we show by a number of examples that the regularity of the dark-pool model hinges in a subtle way on the interplay of all model parameters and on the liquidation time constraint. The paper can also be seen as a case study for the regularity of market impact models in general.

q-fin.TR↗

arXiv2014

Model-free CPPI

We consider Constant Proportion Portfolio Insurance (CPPI) and its dynamic extension, which may be called Dynamic Proportion Portfolio Insurance (DPPI). It is shown that these investment strategies work within the setting of Föllmer's pathwise Itô calculus, which makes no probabilistic assumptions whatsoever. This shows, on the one hand, that CPPI and DPPI are completely independent of any choice of a particular model for the dynamics of asset prices. They even make sense beyond the class of semimartingale sample paths and can be successfully defined for models admitting arbitrage, including some models based on fractional Brownian motion. On the other hand, the result can be seen as a case study for the general issue of robustness in the face of model uncertainty in finance.

q-fin.PM↗

arXiv2014

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 space. This concept of robustness 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 that are of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for ψ-weak convergence.

q-fin.RM↗

arXiv2013

A control problem with fuel constraint and Dawson-Watanabe superprocesses

We solve a class of control problems with fuel constraint by means of the log-Laplace transforms of $J$-functionals of Dawson-Watanabe superprocesses. This solution is related to the superprocess solution of quasilinear parabolic PDEs with singular terminal condition. For the probabilistic verification proof, we develop sharp bounds on the blow-up behavior of log-Laplace functionals of $J$-functionals, which might be of independent interest.

math.PR↗

arXiv2013

Probabilistic aspects of finance

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 developments and point to some areas that might deserve further investigation. We start by reviewing the basics of arbitrage pricing theory, with special emphasis on incomplete markets and on the different roles played by the "real-world" probability measure and its equivalent martingale measures. We then focus on the issue of model ambiguity, also called Knightian uncertainty. We present two case studies in which it is possible to deal with Knightian uncertainty in mathematical terms. The first case study concerns the hedging of derivatives, such as variance swaps, in a strictly pathwise sense. The second one deals with capital requirements and preferences specified by convex and coherent risk measures. In the final two sections we discuss mathematical issues arising from the dramatic increase of algorithmic trading in modern financial markets.

q-fin.PR↗

arXiv2013

Robust Strategies for Optimal Order Execution in the Almgren-Chriss Framework

Assuming geometric Brownian motion as unaffected price process $S^0$, Gatheral & 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 respect to misspecification of the law of $S^0$. We prove the surprising result that the strategy remains optimal whenever $S^0$ is a square-integrable martingale. We then analyze the optimization criterion of Gatheral & Schied (2011) in the case in which $S^0$ is any square-integrable semimartingale and we give a closed-form solution to this problem. As a corollary, we find an explicit solution to the problem of minimizing the expected liquidation costs when the unaffected price process is a square-integrable semimartingale. The solutions to our problems are found by stochastically solving a finite-fuel control problem without assumptions of Markovianity.

q-fin.TR↗

arXiv2013

Drift dependence of optimal trade execution strategies under transient price impact

We give a complete solution to the problem of minimizing the expected liquidity costs in presence of a general drift when the underlying market impact model has linear transient price impact with exponential resilience. It turns out that this problem is well-posed only if the drift is absolutely continuous. Optimal strategies often do not exist, and when they do, they depend strongly on the derivative of the drift. Our approach uses elements from singular stochastic control, even though the problem is essentially non-Markovian due to the transience of price impact and the lack in Markovian structure of the underlying price process. As a corollary, we give a complete solution to the minimization of a certain cost-risk criterion in our setting.

q-fin.TR↗

arXiv2013

Capacitary measures for completely monotone kernels via singular control

We give a singular control approach to the problem of minimizing an energy functional for measures with given total mass on a compact real interval, when energy is defined in terms of a completely monotone kernel. This problem occurs both in potential theory and when looking for optimal financial order execution strategies under transient price impact. In our setup, measures or order execution strategies are interpreted as singular controls, and the capacitary measure is the unique optimal control. The minimal energy, or equivalently the capacity of the underlying interval, is characterized by means of a nonstandard infinite-dimensional Riccati differential equation, which is analyzed in some detail. We then show that the capacitary measure has two Dirac components at the endpoints of the interval and a continuous Lebesgue density in between. This density can be obtained as the solution of a certain Volterra integral equation of the second kind.

math.OC↗

arXiv2010

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 modeling 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, 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.

q-fin.TR↗

arXiv2004

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.

math.PR↗

arXiv1998

Existence and regularity for a class of infinite-measure $(ξ,ψ,K)$-superprocesses

We extend the class of $(ξ,ψ,K)$-superprocesses known so far by applying a simple transformation induced by a \lq\lq weight function\rq\rq\ for the one-particle motion. These transformed superprocesses may exist under weak conditions on the branching parameters, and their state space automatically extends to a certain space of possibly infinite Radon measures. It turns out that a number of superprocesses which were so far not included in the general theory fall into this class. For instance, we are able to extend the hyperbolic branching catalyst of Fleischmann and Mueller to the case of $β$-branching. In the second part of this paper, we discuss regularity properties of our processes. Under the assumption that the one-particle motion is a Hunt process, we show that our superprocesses possess right versions having \cadlag\ paths with respect to a natural topology on the state space. The proof uses an approximation with branching particle systems on Skorohod space.

math.PR↗

arXiv1998

Rademacher's theorem on configuration spaces and applications

We consider an $L^2$-Wasserstein type distance $ρ$ on the configuration space $Γ_X$ over a Riemannian manifold $X$, and we prove that $ρ$-Lipschitz functions are contained in a Dirichlet space associated with a measure on $Γ_X$ satisfying some general assumptions. These assumptions are in particular fulfilled by a large class of tempered grandcanonical Gibbs measures with respect to a superstable lower regular pair potential. As an application we prove a criterion in terms of $ρ$ for a set to be exceptional. This result immediately implies, for instance, a quasi-sure version of the spatial ergodic theorem. We also show that $ρ$ is optimal in the sense that it is the intrinsic metric of our Dirichlet form.

math.PR↗