Ulrich Horst

Humboldt-Universität zu Berlin | HU Berlin · Department of Mathematics

We consider a microstructure foundation for rough volatility models driven by Poisson random measures. In our model the volatility is driven by self-exciting arrivals of market orders as well as self-exciting arrivals of limit orders and cancellations. The impact of market order on future order arrivals is captured by a Hawkes kernel with power law...

We prove that the long-run behavior of Hawkes processes is fully determined by the average number and the dispersion of child events. For subcritical processes we provide FLLNs and FCLTs under minimal conditions on the kernel of the process with the precise form of the limit theorems depending strongly on the dispersion of child events. For a criti...

We consider an optimal liquidation problem with instantaneous price impact and stochastic resilience for small instantaneous impact factors. Within our modelling framework, the optimal portfolio process converges to the solution of an optimal liquidation problem with general semimartingale controls when the instantaneous impact factor converges to...

We consider a novel class of portfolio liquidation games with market drop‐out (“absorption”). More precisely, we consider mean‐field and finite player liquidation games where a player drops out of the market when her position hits zero. In particular, round‐trips are not admissible. This can be viewed as a no statistical arbitrage condition. In a m...

We establish the weak convergence of the intensity of a nearly-unstable Hawkes process with heavy-tailed kernel. Our result is used to derive a scaling limit for a financial market model where orders to buy or sell an asset arrive according to a Hawkes process with power-law kernel. After suitable rescaling the price-volatility process converges we...

We consider a mean-field control problem with càdlàg semimartingale strategies arising in portfolio liquidation models with transient market impact and self-exciting order flow. We show that the value function depends on the state process only through its law, and we show that it is of linear-quadratic form and that its coefficients satisfy a coupl...

This paper provides and extends second-order versions of several fundamental theorems on first-order regularly varying functions such as Karamata's theorem/representation and Tauberian's theorem. Our results are used to establish second-order approximations for the mean and variance of Hawkes processes with general kernels. Our approximations provi...

We consider extended mean-field control problems with multi-dimensional singular controls. A key challenge when analysing singular controls are jump costs. When controls are one-dimensional, jump costs are most naturally computed by linear interpolation. When the controls are multi-dimensional the situation is more complex, especially when the mode...

We establish a first and second-order approximation for an infinite dimensional limit order book model (LOB) in a single (''critical'') scaling regime where market and limit orders arrive at a common time scale. With our choice of scaling we obtain non-degenerate first-order and second-order approximations for the price and volume dynamics. While t...

We consider a novel class of portfolio liquidation games with market drop-out ("absorption"). More precisely, we consider mean-field and finite player liquidation games where a player drops out of the market when her position hits zero. In particular round-trips are not admissible. This can be viewed as a no statistical arbitrage condition. In a mo...

We provide a general probabilistic framework within which we establish scaling limits for a class of continuous-time stochastic volatility models with self-exciting jump dynamics. In the scaling limit, the joint dynamics of asset returns and volatility is driven by independent Gaussian white noises and two independent Poisson random measures that c...

We analyze novel portfolio liquidation games with self‐exciting order flow. Both the N‐player game and the mean‐field game (MFG) are considered. We assume that players' trading activities have an impact on the dynamics of future market order arrivals thereby generating an additional transient price impact. Given the strategies of her competitors ea...

We consider a mean-field control problem with c\`adl\`ag semimartingale strategies arising in portfolio liquidation models with transient market impact and self-exciting order flow. We show that the value function depends on the state process only through its law, and that it is of linear-quadratic form and that its coefficients satisfy a coupled s...

This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a linear-quadratic stochastic control problem with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous viscosity sub...

This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a linear-quadratic stochastic control problem with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous viscosity sub...

We study a class of deterministic mean field games on finite and infinite time horizons arising in models of optimal exploitation of exhaustible resources. The main characteristic of our game is an absorption constraint on the players' state process. As a result of the state constraint the optimal time of absorption becomes part of the equilibrium....

We consider an optimal liquidation problem with instantaneous price impact and stochastic resilience for small instantaneous impact factors. Within our modelling framework, the optimal portfolio process converges to the solution of an optimal liquidation problem with with general semi-martingale controls when the instantaneous impact factor converg...

This paper establishes a functional law of large numbers and a functional central limit theorem for marked Hawkes point measures and their corresponding shot noise processes. We prove that the normalized random measure can be approximated in distribution by the sum of a Gaussian white noise process plus an appropriate lifting map of a correlated on...

We analyze novel portfolio liquidation games with self-exciting order flow. Both the N-player game and the mean-field game are considered. We assume that players' trading activities have an impact on the dynamics of future market order arrivals thereby generating an additional transient price impact. Given the strategies of her competitors each pla...

We derive an explicit solution for deterministic market impact parameters in the Graewe and Horst (2017) portfolio liquidation model. The model allows to combine various forms of market impact, namely instantaneous, permanent and temporary. We show that the solutions to the two benchmark models of Almgren and Chris (2001) and of Obizhaeva and Wang...

We provide a general probabilistic framework within which we establish scaling limits for a class of continuous-time stochastic volatility models with self-exciting jump dynamics. In the scaling limit, the joint dynamics of asset returns and volatility is driven by independent Gaussian white noises and two independent Poisson random measures that c...

We study a multi-dimensional optimal execution problem in illiquid markets with both instantaneous and persistent price impact and stochastic resilience. In our model the value function can be described by a multi-dimensional backward stochastic Riccati differential equations (BSRDE) with a singular terminal condition in one component. We prove the...

We study an optimal liquidation problem under the ambiguity with respect to price impact parameters. Our main results show that the value function and the optimal trading strategy can be characterized by the solution to a semi-linear PDE with superlinear gradient, monotone generator and singular terminal value. We also establish an asymptotic analy...

This paper establishes a functional law of large numbers and a functional central limit theorem for marked Hawkes point measures and their corresponding shot noise processes. We prove that the normalized random measure can be approximated in distribution by the sum of a Gaussian white noise process plus an appropriate lifting map of a correlated on...

We use a model with agency frictions to analyze the structure of a dealer market that faces competition from a crossing network. Traders are privately informed about their types (e.g. their portfolios), which is something the dealer must take into account when engaging his counterparties. Instead of participating in the dealer market, the traders m...

This paper derives a diffusion approximation for a sequence of discrete-time one-sided limit order book models with non-linear state dependent order arrival and cancellation dynamics. The discrete time sequences are specified in terms of an R+-valued best bid price process and an Lloc²-valued volume process. It is shown that under suitable assumpti...

In this paper we derive a second order approximation for an infinite dimensional limit order book model, in which the dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator (e.g. the volume standing at the top of the book). We study the fluctuations of the price and volume process relat...

We analyze linear McKean-Vlasov forward-backward SDEs arising in leader-follower games with mean-field type control and terminal state constraints on the state process. We establish an existence and uniqueness of solutions result for such systems in time-weighted spaces as well as a {convergence} result of the solutions with respect to certain pert...

This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a Markovian linear-quadratic control problems with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous viscosity sub...

We consider a mean field game (MFG) of optimal portfolio liquidation under asymmetric information. We prove that the solution to the MFG can be characterized in terms of a FBSDE with possibly singular terminal condition on the backward component or, equivalently, in terms of a FBSDE with finite terminal value, yet singular driver. Extending the met...

We consider a mean field game (MFG) of optimal portfolio liquidation under asymmetric information. We prove that the solution to the MFG can be characterized in terms of a FBSDE with possibly singular terminal condition on the backward component or, equivalently, in terms of a FBSDE with finite terminal value, yet singular driver. Extending the met...

We study a multi-dimensional optimal execution problem in illiquid markets with both instantaneous and persistent price impact and stochastic resilience. In our model the value function can be described by a multi-dimensional backward stochastic Riccati differential equations (BSRDE) with a singular terminal condition in one component. We prove the...

We consider generalized sender–receiver games in which the sender also has an action to choose, but this action is payoff-relevant only to himself. We study “cooperate and talk” equilibria (CTE) in which, before sending his message, the sender can commit to delegate his decision to the receiver. CTE are beneficial to the receiver (with respect to n...

We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). Our model is flexible enough to allow for a dependence of the price dynamics on volumes. For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model conv...

In this paper we derive a second order approximation for an infinite dimensional limit order book model, in which the dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator (e.g.~the volume standing at the top of the book). We study the fluctuations of the price and volume process relat...

This paper establishes the existence of relaxed solutions to mean field games (MFGs for short) with singular controls. As a by-product, we obtain an existence of relaxed solutions results for McKean- Vlasov stochastic singular control problems. Finally, we prove that the solutions to a particular class of MFGs with singular controls can be approxim...

This paper establishes the existence of relaxed solutions to mean field games (MFGs for short) with singular controls. We also prove approximations of solutions results for a particular class of MFGs with singular controls by solutions, respectively control rules, for MFGs with purely regular controls. Our existence and approximation results strong...

We study an optimal execution problem in illiquid markets with both instantaneous and persistent price impact and stochastic resilience when only absolutely continuous trading strategies are admissible. In our model the value function can be described by a three-dimensional system of backward stochastic differential equations (BSDE) with a singular...

We study an optimal execution problem in illiquid markets with both instantaneous and persistent price impact and stochastic resilience when only absolutely continuous trading strategies are admissible. In our model the value function can be described by a three-dimensional system of backward stochastic differential equations (BSDE) with a singular...

This paper is concerned with the derivation of a functional scaling limit theorem for a certain class of discrete time Markov chains, each consisting of a one dimensional reference process and an $L^2_{loc}$-valued volume process, for which the conditional probability distributions of the increments are assumed to depend only on the current value o...

We propose a general discrete-time framework for deriving equilibrium prices of financial securities. It allows for heterogeneous agents, unspanned random endowments and convex trading constraints. We give a dual characterization of equilibria and provide general results on their existence and uniqueness. In the special case where all agents have p...

We use a principal-agent model to analyze the structure of a book-driven dealer market when the dealer faces competition from a crossing network or dark pool. The agents are privately informed about their types (e.g. their portfolios), which is something that the dealer must take into account when engaging his counterparties. Instead of trading wit...

This paper establishes a maximum principle for quasi-linear reflected backward stochastic partial differential equations (RBSPDEs for short). We prove the existence and uniqueness of the weak solution to RBSPDEs allowing for non-zero Dirichlet boundary conditions and, using a stochastic version of De Giorgi's iteration, establish the maximum princi...

This paper establishes a maximum principle for quasi-linear reflected backward stochastic partial differential equations (RBSPDEs for short). We prove the existence and uniqueness of the weak solution to RBSPDEs allowing for non-zero Dirichlet boundary conditions and, using a stochastic version of De Giorgi's iteration, establish the maximum princi...

We define feasible, posterior individually rational solutions for two-person Bayesian games with a single informed player. Such a solution can be achieved by direct signalling from the informed player and requires approval of both players after the signal has been sent. Without further assumptions on the Bayesian game, a solution does not necessari...

Order display is associated with benefits and costs. Benefits arise from increased execution-priority, while costs are due to adverse market impact. We analyze a structural model of optimal order placement that captures trade-off between the costs and benefits of order display. For a benchmark model of pure liquidity competition, we give a closed-f...

Order display is associated with benefits and costs. Benefits arise from increased execution-priority, while costs are due to adverse market impact. We analyze a structural model of optimal order placement that captures trade-off between costs and benefits of order display. For a benchmark model of pure liquidity competition, we give closed-form so...

We define a stochastic model of a two-sided limit order book in terms of its key quantities \textit{best bid [ask] price} and the \textit{standing buy [sell] volume density}. For a simple scaling of the discreteness parameters, that keeps the expected volume rate over the considered price interval invariant, we prove a limit theorem. The limit theo...

We analyze conditional optimization problems arising in discrete time Principal-Agent problems of delegated portfolio optimization. Applying tools from Conditional Analysis to the case of linear contracts we show that most results known in the literature for very specific instances of the problem carry over to translation invariant and time-consist...

We develop a model of an order-driven exchange competing for order flow with off-exchange trading mechanisms. Liquidity suppliers face a trade-off between benefits and costs of order exposure. If they display trading intentions, they attract additional trade demand. We show, in equilibrium, hiding trade intentions can induce mis-coordination betwee...

We study a constrained optimal control problem with possibly degenerate coefficients arising in models of optimal portfolio liquidation under market impact. The coefficients can be random in which case the value function is described by a degenerate backward stochastic partial differential equation (BSPDE) with singular terminal condition. For this...

We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges to a continuous-time limit when the order arrival rates tend to infinity, the impac...

In this paper the problem of optimal trading in illiquid markets is addressed when the deviations from a given stochastic target function describing, for instance, external aggregate client flow are penalized. Using techniques of singular stochastic control, we extend the results of [F. Naujokat and N. Westray, Math. Financ. Econ., 4 (2011), pp. 29...

We establish existence and regularity results for a class of backward stochastic partial differential equations with singular terminal condition. The equation describes the value function of a non-Markovian stochastic control optimal problem in which the terminal state of the controlled process is prespecified. The analysis of such control problems...

We establish existence and uniqueness of a classical solution to a semilinear parabolic partial differential equation with singular initial condition. This equation describes the value function of the control problem of a financial trader that needs to unwind a large asset portfolio within a short period of time. The trader can simultaneously submi...

We show that the excessive use of hidden orders causes artificial price pressures and abnormal asset returns. Using a simple game-theoretical setting, we demonstrate that this effect naturally arises from mis-coordination in trading schedules between traders, when suppliers of liquidity do not sufficiently disclose their trade intentions. As a resu...

We take advantage of a unique data set, NASDAQ ModelView, to empirically analyze the determinants and the impact of hidden liquidity on public exchanges. Our findings are as follows. First, the cross-sectional presence of hidden liquidity is well explained by observable and readily available stock characteristics. The spread captures most of the li...

We cross-sectionally analyze the presence of aggregated hidden depth and trade volume in the S&P 500 and identify its key determinants. We find that the spread is the main predictor for a stock’s hidden dimension, both in terms of traded and posted liquidity. Our findings moreover suggest that large hidden orders are associated with larger transact...

We develop a sequential trade model of Iceberg order execution in a limit order book. The Iceberg-trader has the freedom to expose his trading intentions or (partially) shield the true order size against other market participants. Order exposure can cause drastic market reactions (“market impact”) in the end leading to higher transaction costs. On...

We consider a class of generalized capital asset pricing models in continuous time with a finite number of agents and tradable securities. The securities may not be sufficient to span all sources of uncertainty. If the agents have exponential utility functions and the individual endowments are spanned by the securities, an equilibrium exists and th...

We consider general economies in which rational agents interact locally. The local aspect of the interactions is designed to capture in a simple abstract way social interactions, that is, socio-economic environments in which markets do not mediate all of agents' choices, and each agent's choice might be in part determined, for instance, by family,...

We consider a full equilibrium model in continuous time comprising a finite number of agents and tradable securities.We show that, if the agents’ endowments are spanned by the securities and if the agents have entropic utilities, an equilibrium exists and the agents’ optimal trading strategies are constant. Affine processes, and the theory of infor...

We provide results on the existence and uniqueness of equilibrium in dynamically incomplete financial markets in discrete time. Our framework allows for heterogeneous agents, unspanned random endowments and convex trading constraints. In the special case where all agents have preferences of the same type and all random endowments are replicable by...

In this paper we deal with the utility maximization problem with a general utility function. We derive a new approach in which we reduce the utility maximization problem with general utility to the study of a fully-coupled Forward-Backward Stochastic Differential Equation (FBSDE).

We develop a sequential trade model of Iceberg order execution in a limit order book. The Iceberg-trader has the freedom to expose his trading intentions or (partially) shield the true order size against other market participants. Order exposure can cause drastic market reactions ("market impact") in the end leading to higher transaction costs. On...

We develop a sequential trade model of Iceberg order execution in a limit order book. The Iceberg-trader has the freedom to expose his trading intentions or (partially) shield the true order size against other market participants. Order exposure can cause drastic market reactions (“market impact”) in the end leading to higher transaction costs. On...

In this article the problem of curve following in an illiquid market is addressed. Using techniques of singular stochastic control, we extend the results of [NW11] to a two- sided limit order market with temporary market impact and resilience, where the bid ask spread is now also controlled. We first show existence and uniqueness of an optimal cont...

In illiquid markets, option traders may have an incentive to increase their portfolio value by using their impact on the dynamics of the underlying. We provide a mathematical framework to construct optimal trading strategies under market impact in a multi-player framework by introducing strategic interactions into the model of Almgren [Appl. Math....

We consider a full equilibrium model in continuous time comprising a finite number of agents and tradable securities.We show that, if the agents’ endowments are spanned by the securities and if the agents have entropic utilities, an equilibrium exists and the agents’ optimal trading strategies are constant. Affine processes, and the theory of infor...

In this article the problem of curve following in an illiquid market is addressed. Using techniques of singular stochastic control, we extend the results of [NW11] to a twosided limit order market with temporary market impact and resilience, where the bid ask spread is now also controlled. We first show existence and uniqueness of an optimal contro...

In this paper the problem of optimal derivative design, profit maximization and risk minimization under adverse selection when multiple agencies compete for the business of a continuum of heterogenous agents is studied. In contrast with the principal-agent models that are extended within, here the presence of ties in the agents' best-response corre...

We propose an equilibrium framework within which to price financial securities written on non- tradable underlyings such as temperature indices. We analyze a financial market with a finite set of agents whose preferences are described by a convex dynamic risk measure generated by the solution of a backward stochastic differential equation. The agen...

In illiquid markets, option traders may have an incentive to increase their portfolio value by using their impact on the dynamics of the underlying. We provide a mathematical framework within which to value derivatives under market impact in a multi-player framework by introducing strategic interactions into the model of Almgren and Chriss (2001)....

We state conditions for existence and uniqueness of equilibria in evolutionary models with an infinity of locally and globally interacting agents. Agents face repeated discrete choice problems. Their utility depends on the actions of some designated neighbors and the average choice throughout the whole population. We show that the dynamics on the l...

In this paper, we establish a convergence result for equilibria in systems of social interactions with many locally and globally interacting players. Assuming spacial homogeneity and that interactions between different agents are not too strong, we show that equilibria of systems with finitely many players converge to the unique equilibrium of a be...

We consider the problem of Adverse Selection and optimal derivative design within a Principal–Agent framework. The principal’s income is exposed to non-hedgeable risk factors arising, for instance, from weather or climate phenomena. She evaluates her risk using a coherent and law invariant risk measure and tries minimize her exposure by selling der...

This survey reviews a dynamic multi-asset framework in which het-erogeneous agents with multi-period planning horizons interact. This framework distinguishes between temporary equilibrium maps describing the basic market mechanism of an asset market, forecasting rules which model the way in which expectations are formed, and a model for ex-ogenous...

In this paper we analyze a credit economy � la Kiyotaki and Moore [1997. Credit cycles. Journal of Political Economy 105, 211-248] enriched with learning dynamics, where both borrowers and lenders need to form expectations about the future price of the collateral. We find that under homogeneous learning, the MSV REE for this economy is E-stable and...

We investigate the asset prices dynamics and the long-run market shares of two competing financial mediators who are selected by consumers. We demonstrate that the social interaction among consumers constitutes an endogenous path-depending source of risk in a financial market. Depending on consumers’ evaluation of the mediator’s investment, asset p...

One approach to the analysis of stochastic fluctuations in market prices is to model characteristics of investor behaviour and the complex interactions between market participants, with the aim of extracting consequences in the aggregate. This agent-based viewpoint in finance goes back at least to the work of Garman (1976) and shares the philosophy...

We propose a method of pricing financial securities written on nontradable underlyings such as temperature or precipitation levels. To this end, we analyze a financial market where agents are exposed to financial and nonfinancial risk factors. The agents hedge their financial risk in the stock market and trade a risk bond issued by an insurance com...

Homo Economicus has progressed from an atomistic and self-interested individual in standard economics to a socially embedded agent in modern economics who is endowed with a particular social identity or with specific preferences for the latter. While this vision makes the economic agent more realistic, its representation by adding variables in an a...

We analyze an interactive model of credit ratings where external shocks spread by a contagious chain reaction to the entire economy. Counterparty relationships along with discrete adjustments of credit ratings generate a transition mechanism that allows the financial distress of one firm to spill over to its business partners. The spread of financi...

We study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time. It turns out that semi-Markov processes are tailor made for modelling inert investors. With a suitable scaling, we show that when the price is driven by the market imbalance, the l...

Although models of social interactions have been used extensively to explain a myriad of economic and social phenomena, there are few general theorems concerning the existence, uniqueness or the limit behavior of equilibria in the literature. In this paper we consider systems with local and global social interactions. Individuals’ utilities depend...

We consider general economies in which rational agents interact locally. The local aspect of the interactions is designed to represent in a simple abstract way social interactions, that is, socioeconomic environments in which markets do not mediate all of agents’ choices, which might be in part determined, for instance, by family, peer group, or et...

We identify possible long-run market shares and the long-run asset price dynamics of financial markets with heterogenous interacting agents. This involves stability conditions for a class of difference equation in a random environment, where the random environment is endogenously generated by agents' investment behavior. Depending on the evaluation...