Curve following in illiquid markets

Naujokat, Felix; Westray, Nicholas

doi:10.1007/s11579-011-0042-5

Cited by 19 publications

(54 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Intuitively, an execution of the optimal dark pool order for asset i should bring the position in asset i to its optimal value given unchanged positions in all other assets

j \neq i

. The following proposition confirms this intuition (see also Naujokat and Westray for a similar result). Proposition Let

t \in [0, T]

x \in R^{n}

be the portfolio position at time t and

i = 1, \dots, n

.…”

Section: Properties Of the Value Function And The Optimal Strategysupporting

confidence: 64%

“…They carry out a verification argument for a candidate value function obtained by considering only deterministic strategies. Naujokat and Westray and Höschler () treat similar control problems with jumps. The focus of both texts is on trading with limit orders rather than with dark pools; they only treat single‐asset trading and obtain the single‐asset case of this paper as special cases of their respective settings.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Portfolio Liquidation in Dark Pools in Continuous Time

Kratz

Schöneborn

2013

Mathematical Finance

View full text Add to dashboard Cite

We consider an illiquid financial market where a risk averse investor has to liquidate a portfolio within a finite time horizon [0, T ] and can trade continuously at a traditional exchange (the "primary venue") and in a dark pool. At the primary venue, trading yields a linear price impact. In the dark pool, no price impact costs arise but order execution is uncertain, modeled by a multi-dimensional Poisson process. We characterize the costs of trading by a linear-quadratic functional which incorporates both the price impact costs of trading at the primary exchange and the market risk of the position. The liquidation constraint implies a singularity of the value function of the resulting minimization problem at the terminal time T . Via the HJB equation and a quadratic ansatz, we obtain a candidate for the value function which is the limit of a sequence of solutions of initial value problems for a matrix differential equation. We show that this limit exists by using an appropriate matrix inequality and a comparison result for Riccati matrix equations. Additionally, we obtain upper and lower bounds of the solutions of the initial value problems, which allow us to prove a verification theorem. If a single asset position is to be liquidated, the investor slowly trades out of her position at the primary venue, with the remainder being placed in the dark pool at any point in time. For multi-asset liquidations this is generally not the case; it can, e.g., be optimal to oversize orders in the dark pool in order to turn a poorly balanced portfolio into a portfolio bearing less risk.

show abstract

“…Intuitively, an execution of the optimal dark pool order for asset i should bring the position in asset i to its optimal value given unchanged positions in all other assets

j \neq i

. The following proposition confirms this intuition (see also Naujokat and Westray for a similar result). Proposition Let

t \in [0, T]

x \in R^{n}

be the portfolio position at time t and

i = 1, \dots, n

.…”

Section: Properties Of the Value Function And The Optimal Strategysupporting

confidence: 64%

Section: Introductionmentioning

confidence: 99%

Portfolio Liquidation in Dark Pools in Continuous Time

Kratz

Schöneborn

2013

Mathematical Finance

View full text Add to dashboard Cite

show abstract

“…This motivates us to let c = (c t ) 0≤t<T denote from now on an (F t ) 0≤t<Tadapted, càdlàg semimartingale with BSRDE dynamics (19) and terminal condition (20). In addition, we will assume that…”

Section: Connection Between Stochastic Lq Problems and Bsrdesmentioning

confidence: 99%

“…In particular, the dynamics in (19) are only required to hold on [0, T −ε] for every ε > 0, that is, strictly before T . So, more precisely, we will say that (c, N) is a supersolution of the BSRDE (19) with terminal condition η if (22) and (20) hold true.…”

Section: Connection Between Stochastic Lq Problems and Bsrdesmentioning

confidence: 99%

“…Otherwise, a penalization depending on the deviation of X u T from the target position Ξ T is implemented. We refer to, e.g., Rogers and Singh [23], Naujokat and Westray [19], Almgren and Li [1], Frei and Westray [10], Cartea and Jaimungal [8], Cai et al [7], Bank et al [4] and, for asymptotic considerations, to Chan and Sircar [9]. Note, however, that the above cited papers may neither allow for an arbitrary predictable target strategy ξ nor for stochastic price impact κ and stochastic volatility ν.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Linear Quadratic Stochastic Control Problems with Stochastic Terminal Constraint

Bank¹,

Voß²

2018

SIAM J. Control Optim.

View full text Add to dashboard Cite

We study a linear quadratic optimal control problem with stochastic coefficients and a terminal state constraint, which may be in force merely on a set with positive, but not necessarily full probability. Under such a partial terminal constraint, the usual approach via a coupled system of a backward stochastic Riccati equation and a linear backward equation breaks down. As a remedy, we introduce a family of auxiliary problems parametrized by the supersolutions to this Riccati equation alone. The target functional of these problems dominates the original constrained one and allows for an explicit description of both the optimal control policy and the auxiliary problem's value in terms of a suitably constructed optimal signal process. This suggests that, for the minimal supersolution of the Riccati equation, the minimizers of the auxiliary problem coincide with those of the original problem, a conjecture that we see confirmed in all examples understood so far.Mathematical Subject Classification (2010): 93E20, 60H10, 91G80Keywords: linear quadratic control, stochastic terminal constraint, backward stochastic Riccati differential equation, optimal tracking, optimal signal process * We are grateful to Paulwin Graewe, Ulrich Horst, Peter Imkeller and Alexandre Popier for insightful discussions. Our paper has also greatly benefited from the comments and suggestions of two anonymous referees.

show abstract

Reducing Obizhaeva–Wang-type trade execution problems to LQ stochastic control problems

Ackermann,

Kruse,

Urusov

2024

Finance Stoch

View full text Add to dashboard Cite

We start with a stochastic control problem where the control process is of finite variation (possibly with jumps) and acts as integrator both in the state dynamics and in the target functional. Problems of such type arise in the stream of literature on optimal trade execution pioneered by Obizhaeva and Wang (J. Financ. Mark. 16:1–32, 2013) (models with finite resilience). We consider a general framework where the price impact and the resilience are stochastic processes. Both are allowed to have diffusive components. First we continuously extend the problem from processes of finite variation to progressively measurable processes. Then we reduce the extended problem to a linear–quadratic (LQ) stochastic control problem. Using the well-developed theory on LQ problems, we describe the solution to the obtained LQ one and translate it back to the solution for the (extended) initial trade execution problem. Finally, we illustrate our results by several examples. Among other things, the examples discuss the Obizhaeva–Wang model with random (terminal and moving) targets, the necessity to extend the initial trade execution problem to a reasonably large class of progressively measurable processes (even going beyond semimartingales), and the effects of diffusive components in the price impact process and/or the resilience process.

show abstract

Curve following in illiquid markets

Cited by 19 publications

References 29 publications

Portfolio Liquidation in Dark Pools in Continuous Time

Portfolio Liquidation in Dark Pools in Continuous Time

Linear Quadratic Stochastic Control Problems with Stochastic Terminal Constraint

Reducing Obizhaeva–Wang-type trade execution problems to LQ stochastic control problems

Contact Info

Product

Resources

About