arXiv subjects

Alexander Schied

Publications and source records attributed to Alexander Schied.

At least 37 records · Page 2Linked to original sources

arXiv2020

A central bank strategy for defending a currency peg

We consider a central bank strategy for maintaining a two-sided currency target zone, in which an exchange rate of two currencies is forced to stay between two thresholds. To keep the exchange rate from breaking the prescribed barriers, the central bank is generating permanent price impact and thereby accumulating inventory in the foreign currency. Historical examples of failed target zones illustrate that this inventory can become problematic, in particular when there is an adverse macroeconomic trend in the market. We model this situation through a continuous-time market impact model of Almgren--Chriss-type with drift, in which the exchange rate is a diffusion process controlled by the price impact of the central bank's intervention strategy. The objective of the central bank is to enforce the target zone through a strategy that minimizes the accumulated inventory. We formulate this objective as a stochastic control problem with random time horizon. It is solved by reduction to a singular boundary value problem that was solved by Lasry and Lions (1989). Finally, we provide numerical simulations of optimally controlled exchange rate processes and the corresponding evolution of the central bank inventory.

q-fin.TR↗

arXiv2020

On the $p^{\text{th}}$ variation of a class of fractal functions

The concept of the $p^{\text{th}}$ variation of a continuous function $f$ along a refining sequence of partitions is the key to a pathwise Itô integration theory with integrator $f$. Here, we analyze the $p^{\text{th}}$ variation of a class of fractal functions, containing both the Takagi--van der Waerden and Weierstraß functions. We use a probabilistic argument to show that these functions have linear $p^{\text{th}}$ variation for a parameter $p\ge1$, which can be interpreted as the reciprocal Hurst parameter of the function. It is shown moreover that if functions are constructed from (a skewed version of) the tent map, then the slope of the $p^{\text{th}}$ variation can be computed from the $p^{\text{th}}$ moment of a (non-symmetric) infinite Bernoulli convolution. Finally, we provide a recursive formula of these moments and use it to discuss the existence and non-existence of a signed version of the $p^{\text{th}}$ variation, which occurs in pathwise Itô calculus when $p\ge3$ is an odd integer.

math.PR↗

arXiv2020

Step roots of Littlewood polynomials and the extrema of functions in the Takagi class

We give a new approach to characterizing and computing the set of global maximizers and minimizers of the functions in the Takagi class and, in particular, of the Takagi--Landsberg functions. The latter form a family of fractal functions $f_α:[0,1]\to\mathbb R$ parameterized by $α\in(-2,2)$. We show that $f_α$ has a unique maximizer in $[0,1/2]$ if and only if there does not exist a Littlewood polynomial that has $α$ as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi--Landsberg functions with $α\in(-2,1/2]\cup(1,2)$. For $(1/2,1]$, we show that the step roots are dense in that interval. If $α\in (1/2,1]$ is a step root, then the set of maximizers of $f_α$ is an explicitly given perfect set with Hausdorff dimension $1/(n+1)$, where $n$ is the degree of the minimal Littlewood polynomial that has $α$ as its step root. In the same way, we determine explicitly the minima of all Takagi--Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to $[-2,-1/2]\cup[1/2,2]$.

math.CA↗

arXiv2019

An FBSDE approach to market impact games with stochastic parameters

We analyze a market impact game between $n$ risk averse agents who compete for liquidity in a market impact model with permanent price impact and additional slippage. Most market parameters, including volatility and drift, are allowed to vary stochastically. Our first main result characterizes the Nash equilibrium in terms of a fully coupled system of forward-backward stochastic differential equations (FBSDEs). Our second main result provides conditions under which this system of FBSDEs has indeed a unique solution, which in turn yields the unique Nash equilibrium. We furthermore obtain closed-form solutions in special situations and analyze them numerically

q-fin.TR↗

arXiv2019

Constructing functions with prescribed pathwise quadratic variation

We construct rich vector spaces of continuous functions with prescribed curved or linear pathwise quadratic variations. We also construct a class of functions whose quadratic variation may depend in a local and nonlinear way on the function value. These functions can then be used as integrators in Föllmer's pathwise Itō calculus. Our construction of the latter class of functions relies on an extension of the Doss--Sussman method to a class of nonlinear Itō differential equations for the Föllmer integral. As an application, we provide a deterministic variant of the support theorem for diffusions. We also establish that many of the constructed functions are nowhere differentiable.

math.PR↗

arXiv2019

On (signed) Takagi-Landsberg functions: $p^{\text{th}}$ variation, maximum, and modulus of continuity

We study a class $\mathfrak X^H$ of signed Takagi-Landsberg functions with Hurst parameter $H\in(0,1)$. We first show that the functions in $\mathfrak X^H$ admit a linear $p^{\text{th}}$ variation along the sequence of dyadic partitions of $[0,1]$, where $p=1/H$. The slope of the linear increase can be represented as the $p^{\text{th}}$ absolute moment of the infinite Bernoulli convolution with parameter $2^{H-1}$. The existence of a continuous $p^{\text{th}}$ variation enables the use of the functions in $\mathfrak X^H$ as test integrators for higher-order pathwise Itô calculus. Our next results concern the maximum, the maximizers, and the modulus of continuity of the classical Takagi-Landsberg function for all $0<H<1$. Then we identify the uniform maximum, the uniform maximal oscillation, and a uniform modulus of continuity for the class $\mathfrak X^H$.

math.PR↗

arXiv2019

Nash equilibrium for risk-averse investors in a market impact game with transient price impact

We consider a market impact game for $n$ risk-averse agents that are competing in a market model with linear transient price impact and additional transaction costs. For both finite and infinite time horizons, the agents aim to minimize a mean-variance functional of their costs or to maximize the expected exponential utility of their revenues. We give explicit representations for corresponding Nash equilibria and prove uniqueness in the case of mean-variance optimization. A qualitative analysis of these Nash equilibria is conducted by means of numerical analysis.

q-fin.TR↗

arXiv2018

Mild and viscosity solutions to semilinear parabolic path-dependent PDEs

We study and compare two concepts for weak solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). The first is that of mild solutions as it appears, e.g., in the log-Laplace functionals of historical superprocesses. The aim of this paper is to show that mild solutions are also solutions in a viscosity sense. This result is motivated by the fact that mild solutions can provide value functions and optimal strategies for problems of stochastic optimal control. Since unique mild solutions exist under weak conditions, we obtain as a corollary a general existence result for viscosity solutions to semiilinear parabolic PPDEs.

math.PR↗

arXiv2018

On the minimizers of energy forms with completely monotone kernel

Motivated by the problem of optimal portfolio liquidation under transient price impact, we study the minimization of energy functionals with completely monotone displacement kernel under an integral constraint. The corresponding minimizers can be characterized by Fredholm integral equations of the second type with constant free term. Our main result states that minimizers are analytic and have a power series development in terms of even powers of the distance to the midpoint of the domain of definition and with nonnegative coefficients. We show moreover that our minimization problem is equivalent to the minimization of the energy functional under a nonnegativity constraint.

math.OC↗

arXiv2018

Model-free portfolio theory and its functional master formula

We use pathwise Itô calculus to prove two strictly pathwise versions of the master formula in Fernholz' stochastic portfolio theory. Our first version is set within the framework of Föllmer's pathwise Itô calculus and works for portfolios generated from functions that may depend on the current states of the market portfolio and an additional path of finite variation. The second version is formulated within the functional pathwise Itô calculus of Dupire (2009) and Cont \& Fournié (2010) and allows for portfolio-generating functionals that may depend additionally on the entire path of the market portfolio. Our results are illustrated by several examples and shown to work on empirical market data.

q-fin.PM↗

arXiv2018

The associativity rule in pathwise functional Itô calculus

In this paper we establish the associativity property of the pathwise Itô integral in a functional setting for continuous integrators. Here, associativity refers to the computation of the Itô differential of an Itô integral, by means of the intuitive cancellation of the Itô differential and integral signs.

math.PR↗

arXiv2017

High-frequency limit of Nash equilibria in a market impact game with transient price impact

We study the high-frequency limits of strategies and costs in a Nash equilibrium for two agents that are competing to minimize liquidation costs in a discrete-time market impact model with exponentially decaying price impact and quadratic transaction costs of size $θ\ge0$. We show that, for $θ=0$, equilibrium strategies and costs will oscillate indefinitely between two accumulation points. For $θ>0$, however, strategies, costs, and total transaction costs will converge towards limits that are independent of $θ$. We then show that the limiting strategies form a Nash equilibrium for a continuous-time version of the model with $θ$ equal to a certain critical value $θ^*>0$, and that the corresponding expected costs coincide with the high-frequency limits of the discrete-time equilibrium costs. For $θ\neqθ^*$, however, continuous-time Nash equilibria will typically not exist. Our results permit us to give mathematically rigorous proofs of numerical observations made in Schied and Zhang (2013). In particular, we provide a range of model parameters for which the limiting expected costs of both agents are decreasing functions of $θ$. That is, for sufficiently high trading speed, raising additional transaction costs can reduce the expected costs of all agents.

q-fin.TR↗

arXiv2017

A market impact game under transient price impact

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 transaction costs the high-frequency traders engage in a "hot-potato game", in which the same asset position is sold back and forth. We then identify a critical value for the size of the transaction costs above which all oscillations disappear and strategies become buy-only or sell-only. Numerical simulations show that for both traders the expected costs can be lower with transaction costs than without. Moreover, the costs can increase with the trading frequency when there are no transaction costs, but decrease with the trading frequency when transaction costs are sufficiently high. We argue that these effects occur due to the need of protection against predatory trading in the regime of low transaction costs.

q-fin.TR↗

arXiv2016

Domains of weak continuity of statistical functionals with a view toward robust statistics

Many standard estimators such as several maximum likelihood estimators or the empirical estimator for any law-invariant convex risk measure are not (qualitatively) robust in the classical sense. However, these estimators may nevertheless satisfy a weak (Krätschmer/Schied/Zähle 2012, 2014) or a local (Zähle 2016) robustness property on relevant sets of distributions. One aim of our paper is to identify sets of local robustness, and to explain the benefit of the knowledge of such sets. For instance, we will be able to demonstrate that many maximum likelihood estimators are robust on their natural parametric domains. A second aim consists in extending the general theory of robust estimation to our local framework. In particular we provide a corresponding Hampel-type theorem linking local robustness of a plug-in estimator with a certain continuity condition.

math.ST↗

arXiv2016

Pathwise no-arbitrage in a class of Delta hedging strategies

We consider a strictly pathwise setting for Delta hedging exotic options, based on Föllmer's pathwise Itō calculus. Price trajectories are $d$-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix. The existence of Delta hedging strategies in this pathwise setting is established via existence results for recursive schemes of parabolic Cauchy problems and via the existence of functional Cauchy problems on path space. Our main results establish the nonexistence of pathwise arbitrage opportunities in classes of strategies containing these Delta hedging strategies and under relatively mild conditions on the local volatility matrix.

q-fin.MF↗

arXiv2015

Multivariate transient price impact and matrix-valued positive definite functions

We consider a model for linear transient price impact for multiple assets that takes cross-asset impact into account. Our main goal is to single out properties that need to be imposed on the decay kernel so that the model admits well-behaved optimal trade execution strategies. We first show that the existence of such strategies is guaranteed by assuming that the decay kernel corresponds to a matrix-valued positive definite function. An example illustrates, however, that positive definiteness alone does not guarantee that optimal strategies are well-behaved. Building on previous results from the one-dimensional case, we investigate a class of nonincreasing, nonnegative and convex decay kernels with values in the symmetric $K\times K$ matrices. We show that these decay kernels are always positive definite and characterize when they are even strictly positive definite, a result that may be of independent interest. Optimal strategies for kernels from this class are well-behaved when one requires that the decay kernel is also commuting. We show how such decay kernels can be constructed by means of matrix functions and provide a number of examples. In particular we completely solve the case of matrix exponential decay.

q-fin.TR↗

arXiv2015

On a class of generalized Takagi functions with linear pathwise quadratic variation

We consider a class $\mathscr{X}$ of continuous functions on $[0,1]$ that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function. Second, each function in $\mathscr{X}$ admits a linear pathwise quadratic variation and can thus serve as an integrator in Föllmer's pathwise Itō calculus. We derive several uniform properties of the class $\mathscr{X}$. For instance, we compute the overall pointwise maximum, the uniform maximal oscillation, and the exact uniform modulus of continuity for all functions in $\mathscr{X}$. Furthermore, we give an example of a pair $x,y\in\mathscr{X}$ such that the quadratic variation of the sum $x+y$ does not exist.

math.PR↗

arXiv2015

A state-constrained differential game arising in optimal portfolio liquidation

We consider $n$ risk-averse agents who compete for liquidity in an Almgren--Chriss market impact model. Mathematically, this situation can be described by a Nash equilibrium for a certain linear-quadratic differential game with state constraints. The state constraints enter the problem as terminal boundary conditions for finite and infinite time horizons. We prove existence and uniqueness of Nash equilibria and give closed-form solutions in some special cases. We also analyze qualitative properties of the equilibrium strategies and provide corresponding financial interpretations.

math.OC↗