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Multi-asset optimal trade execution with stochastic cross-effects: An Obizhaeva-Wang-type framework

Julia Ackermann, Thomas Kruse, Mikhail Urusov

TL;DR

This work develops a rigorous multi-asset optimal trade execution framework with stochastic cross-effects in price impact, resilience, and risk. By extending finite-variation controls to progressively measurable strategies, the authors recast the problem as a linear-quadratic stochastic control problem and solve it via Riccati BSDEs and related linear BSDEs, yielding a feedback form for the optimal strategy. The analysis reveals cross-hedging phenomena where trading in an asset with zero initial exposure can be optimal due to cross-effects, and provides a multi-asset Obizhaeva–Wang variant as a subsetting. General results include existence, uniqueness, and explicit constructions of optimal strategies under zero and general targets, with a suite of illustrative examples.

Abstract

We analyze a continuous-time optimal trade execution problem in multiple assets where the price impact and the resilience can be matrix-valued stochastic processes that incorporate cross-impact effects. In addition, we allow for stochastic terminal and running targets. Initially, we formulate the optimal trade execution task as a stochastic control problem with a finite-variation control process that acts as an integrator both in the state dynamics and in the cost functional. We then extend this problem continuously to a stochastic control problem with progressively measurable controls. By identifying this extended problem as equivalent to a certain linear-quadratic stochastic control problem, we can use established results in linear-quadratic stochastic control to solve the extended problem. This work generalizes [Ackermann, Kruse, Urusov; FinancStoch'24] from the single-asset setting to the multi-asset case. In particular, we reveal cross-hedging effects, showing that it can be optimal to trade in an asset despite having no initial position. Moreover, as a subsetting we discuss a multi-asset variant of the model in [Obizhaeva, Wang; JFinancMark'13].

Multi-asset optimal trade execution with stochastic cross-effects: An Obizhaeva-Wang-type framework

TL;DR

This work develops a rigorous multi-asset optimal trade execution framework with stochastic cross-effects in price impact, resilience, and risk. By extending finite-variation controls to progressively measurable strategies, the authors recast the problem as a linear-quadratic stochastic control problem and solve it via Riccati BSDEs and related linear BSDEs, yielding a feedback form for the optimal strategy. The analysis reveals cross-hedging phenomena where trading in an asset with zero initial exposure can be optimal due to cross-effects, and provides a multi-asset Obizhaeva–Wang variant as a subsetting. General results include existence, uniqueness, and explicit constructions of optimal strategies under zero and general targets, with a suite of illustrative examples.

Abstract

We analyze a continuous-time optimal trade execution problem in multiple assets where the price impact and the resilience can be matrix-valued stochastic processes that incorporate cross-impact effects. In addition, we allow for stochastic terminal and running targets. Initially, we formulate the optimal trade execution task as a stochastic control problem with a finite-variation control process that acts as an integrator both in the state dynamics and in the cost functional. We then extend this problem continuously to a stochastic control problem with progressively measurable controls. By identifying this extended problem as equivalent to a certain linear-quadratic stochastic control problem, we can use established results in linear-quadratic stochastic control to solve the extended problem. This work generalizes [Ackermann, Kruse, Urusov; FinancStoch'24] from the single-asset setting to the multi-asset case. In particular, we reveal cross-hedging effects, showing that it can be optimal to trade in an asset despite having no initial position. Moreover, as a subsetting we discuss a multi-asset variant of the model in [Obizhaeva, Wang; JFinancMark'13].

Paper Structure

This paper contains 27 sections, 33 theorems, 229 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The multi-asset trade execution problem for finite-variation strategies
  3. Notation
  4. The stochastic liquidity processes
  5. The finite-variation stochastic control problem
  6. On the price impact and the resilience
  7. Alternative representations for the costs and for the deviation
  8. Continuous extension to progressively measurable strategies
  9. Progressively measurable execution strategies
  10. Continuous extension of the cost functional
  11. Solution of the trade execution problem
  12. An equivalent LQ stochastic control problem
  13. The case of zero targets $\xi=0$ and $\zeta=0$
  14. General targets
  15. An equivalent LQ stochastic control problem without cross-terms
  16. ...and 12 more sections

Key Result

Proposition 2.7

Let $t \in [0,T]$ and $x,d \in \mathbb{R}^n$. Let $X=(X(s))_{s\in[t,T]}$ be an $\mathbb{R}^n$-valued càdlàg finite-variation process with $X(t-)=x$ and with associated process $D^X=(D^X(s))_{s\in[t,T]}$ defined by eq:deviationdynmultivariate. Then it holds that and

Figures (1)

  • Figure 1: The deviation in the first asset after a block trade $\Delta X(0)=(3,1)^\top$ (topleft), $\Delta X(0)=(1,3)^\top$ (topright), $\Delta X(0)=(3,-1)^\top$ (bottomleft), and $\Delta X(0)=(1,-3)^\top$ (bottomright) at the time $0$. The setting is $t_1=5$, $\rho_1=2$, $\rho_3=1$, and $\gamma(0)=I_2$ within \ref{['ex:resilience_effect']}. The brown lines indicate the deviation in the first asset without cross-resilience.

Theorems & Definitions (81)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Remark 2.5
  • Example 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Corollary 3.1
  • Remark 3.2
  • ...and 71 more