- Financial mathematics
- Optimal stochastic control
- Market microstructure
- Program trading

- Optimal trade execution and price manipulation in order books with time-varying liquidity [Link to
SSRN] (with Antje Fruth
and Mikhail Urusov)

In financial markets, liquidity is not constant over time but exhibits strong seasonal patterns. In this article 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 trading dependent 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. - Optimal Liquidation in Dark Pools [Link to
SSRN] (with Peter Kratz)

We consider a finite time horizon, multi-asset optimal liquidation problem in discrete time for an investor having access to both a traditional 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. We prove existence and uniqueness of optimal trading strategies for risk averse mean-variance traders and find that dark pools change optimal trading strategies and can significantly reduce trading costs. Their effect can be reduced by adverse selection and trading restrictions. - Adaptive Basket Liquidation [Link to
SSRN]

We consider the infinite time-horizon optimal basket portfolio liquidation problem for a von Neumann-Morgenstern investor in a multi-asset extension of the liquidity model of Almgren (2003) with cross-asset impact. Using a stochastic control approach, we establish a "separation theorem": the sequence of portfolios held during an optimal liquidation depends only on the (co-)variance and (cross-asset) market impact of the assets, while the speed with which these portfolios are attained depends only on the utility function of the trader. We derive partial differential equations for both the sequence of attained portfolios and the trading speed. - Optimal basket liquidation with finite time horizon for CARA investors
[PDF]
(with Alexander Schied)

Now available with extended results as "Optimal basket liquidation for CARA investors is deterministic". - Optimal Portfolio Liquidation for CARA Investors [PDF]
(with Alexander Schied)

Now in extended form as "Optimal basket liquidation for CARA investors is deterministic". - Liquidation in the Face of Adversity: Stealth Vs. Sunshine Trading [Link to
SSRN] (with Alexander
Schied; WHU Finance Award for best submission to the Campus For
Finance Research Conference 2008)

We consider a multi-player situation in an illiquid market in which one player tries to liquidate a large portfolio in a short time span, while some competitors know of the seller’s intention and try to make a profit by trading in this market over a longer time horizon. We show that the liquidity characteristics, the number of competitors in the market and their trading time horizons determine the optimal strategy for the competitors: they either provide liquidity to the seller, or they prey by simultaneous selling. Depending on the expected competitor behavior, it might be sensible for the seller to pre-announce a trading intention (“sunshine trading”) or to keep it secret (“stealth trading”).

- Mean Reversion Pays, but Costs (RISK,
February 2011, p. 96–101) [Link to arXiv] (with Richard
Martin)

A mean-reverting financial instrument is optimally traded by buying it when it is sufficiently below the estimated 'mean level' and selling it when it is above. In the presence of linear transaction costs, a large amount of value is paid away crossing bid-offers unless a strategy is developed to overcome them in the form of a 'buffer' through which the price must move before a trade is done. In this paper we derive the optimal strategy and conclude that the buffer width is for low costs proportional to the cube root of the transaction cost, determining the proportionality constant explicitly. - Optimal basket liquidation for CARA investors is deterministic (Applied Mathematical
Finance, Volume 17, Number 6, p. 471–489, December 2010) [Link to
SSRN] (with Alexander Schied
and Mike Tehranchi)

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, multi-asset 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), 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. - A guided tour of new results on "Trade execution in illiquid markets"
(Blaetter
der DGVFM, Volume 31, Number 1, p. 79-90, April 2010) [PDF]

We give an overview of the dissertation "Trade execution in illiquid markets: Optimal stochastic control and multi-agent equilibria" (Schoeneborn (2008)). The dissertation focuses on two questions in the field of optimal trade execution strategies: First, how should traders best sell an illiquid asset position if they want to maximise the expected utility of liquidation proceeds? And second, in a situation where one market participant needs to liquidate a position, what is the effect of other market participants obtaining advance information of this impending liquidation? - Risk aversion and the dynamics of optimal liquidation strategies in
illiquid markets (Finance
and Stochastics, Volume 13, Number 2, April 2009) [PDF] (with Alexander Schied)

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. - Trade execution in illiquid markets. Optimal stochastic control and
multi-agent equilibria [PDF]

(Ph.D. thesis, TU Berlin, May 2008, 176 pages; GAUSS prize (First prize) of the DAV / DGVFM)

In the classical models of financial mathematics, it is assumed that arbitrarily large positions of assets can be traded at the current market price without affecting this price. This does not reflect reality for large transactions: First, a price premium must be paid for large positions. Second, large transactions do have a long-lasting effect on future prices. The purpose of this dissertation is to find optimal execution strategies in such an “illiquid market”.

In a first part, we analyze the situation of a single trader, who wants to liquidate a portfolio. The trader is facing a dilemma: on the one hand, a quick liquidation results in a strong adverse influence on the market price and thus reduces the liquidation proceeds. On the other hand, a slow execution results in a large risk, since the market price can move significantly during the liquidation time period due to exogenous events. In the first part of the dissertation, we determine the optimal trade-off in this dilemma. We use different modeling approaches with a special focus on utility maximization. The Hamilton-Jacobi-Bellman equation for this problem is a completely non-linear, degenerate partial differential equation. To solve it, we pursue the unusual approach of first obtaining the optimal control as a solution of a partial differential equation and subsequently constructing a solution to the Hamilton-Jacobi-Bellman equation by using the optimal control. For the liquidation of a portfolio consisting of several assets our approach allows us to reduce the high-dimensional Hamilton-Jacobi-Bellman equation to a two-dimensional problem if the market is “homogeneous” in a certain sense.

In the second part of this dissertation, we consider several market participants, who trade the same asset in an illiquid market. Every participant trades at the market price, which is influenced by the transactions of all participants in the same fashion. This leads to an interaction of the market participants. We investigate in particular the situation of a trader who needs to liquidate an asset position in a short time while other market participants are aware of her trading intentions. In a first market model, we can derive the optimal strategies for all agents in a complex closed form. We analyze the interaction of the market participants by reviewing examples and limit cases and find an explanation for the coexistence of cooperative and competitive behavior. For a second market model we show inductively that the value function for all market participants is of a special polynomial form. We thus obtain the optimal trading strategies as linear functions with coefficients which can be calculated by an explicit backward recursion. In this second market model, a quick sequence of buy and sell orders is optimal; by considering different limit cases, we discover that this phenomenon is related to the costs of round trip transactions.

- Optimal portfolio liquidation: market impact models and optimal control
(Oberwolfach
Report 4/2008) (with Alexander
Schied)

- The Topological Tverberg Theorem and Winding Numbers [PDF]
(with Günter M.
Ziegler)

(Journal of Combinatorial Theory Series A, Volume 112 , Issue 1 (October 2005), Pages: 82 - 104)

The Topological Tverberg Theorem claims that any continuous map of a (q - 1) (d + 1)-simplex to Rd identifies points from q disjoint faces. (This has been proved for affine maps, for d≤1, and if q is a prime power, but not yet in general.)The Topological Tverberg Theorem can be restricted to maps of the d-skeleton of the simplex. We further show that it is equivalent to a "Winding Number Conjecture" that concerns only maps of the (d - 1)-skeleton of a (q - 1) (d + 1)-simplex to Rd. "Many Tverberg partitions" arise if and only if there are "many q-winding partitions."The d = 2 case of the Winding Number Conjecture is a problem about drawings of the complete graphs K3q - 2 in the plane. We investigate graphs that are minimal with respect to the winding number condition.

- On the Topological Tverberg Theorem [ArXiv]

(Diploma thesis, 2004) - The 3-manifold geometrisation conjecture of W. P. Thurston

(Thesis for Part III of the Mathematical Tripos, 2002)

E-mail: schoeneborn * at * math * dot * tu-berlin * dot * de

Last updated: 13 September 2011