Mechanics of good trade execution in the framework of linear temporary market impact

Abstract

We define the concept of good trade execution and we construct explicit adapted good trade execution strategies in the framework of linear temporary market impact. Good trade execution strategies are dynamic, in the sense that they react to the actual realisation of the traded asset price path over the trading period; this is paramount in volatile regimes, where price trajectories can considerably deviate from their expected value. Remarkably, however, the implementation of our strategies does not require the full specification of an SDE evolution for the traded asset price, making them robust across different models. Moreover, rather than minimising the expected trading cost, good trade execution strategies minimise trading costs in a pathwise sense, a point of view not yet considered in the literature. The mathematical apparatus for such a pathwise minimisation hinges on certain random Young differential equations that correspond to the Euler–Lagrange equations of the classical Calculus of Variations. These Young differential equations characterise our good trade execution strategies in terms of an initial value problem that allows for easy implementations.

Keywords:

• Quantitative trading strategies
• Market microstructure
• Quantitative finance
• Quantitative finance techniques

JEL code:

• C61 – Optimization Techniques
• Programming Models
• Dynamic Analysis
• C6– Mathematical Methods and Programming
• C – Mathematical
• C60 – General
• C6 –Mathematical Methods and Programming
• C – Mathematical and Quantitative Methods

Acknowledgments

We are grateful to Eyal Neuman for discussion and suggestions that helped us improve the paper. We would also like to thank two anonymous referees for their comments and recommendations. C. Bellani gratefully acknowledges support from EPSRC studentship award 1943803.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 In the case of linear temporary market impact and quadratic inventory cost, a recent work by Belak et al. (2018) actually discusses techniques that can be more generally applied to the case of general semimartingales. In this case, there is no HJB equation; instead the authors rely on forward–backward stochastic differential equations. In section 2, we will review this general solution.

2 Recall that the 2-variation ${∥x∥}_{2,[0,T]}$ of a path $x:t\mapsto x_t\in {\mathbb{R}}^d$ is defined as $\begin{array}{rl}{∥x∥}_{2,[0,T]}& :=sup\left\{{\left(\sum _{t_i}{∣x_{t_{i+1}}-x_{t_i}∣}^2\right)}^{1/2}:\right.\\ 0& =t_0<t_1<\cdots <t_n=T\left.\right\},\end{array}$ where the supremum is taken over all the partitions of the interval $[0,T]$.

3 See definition A.2 in appendix for the definition of Caratheodory function. The function L in the statement of proposition 2.2 is assumed to be a Caratheodory function with the choices: 1. the open interval $(0,T)$ as the subset U of ${\mathbb{R}}^n$ in definition A.2; 2. the two-dimensional variable $(q,r)$ as the variable ξ in definition A.2.

4 See definitions A.2 and A.3.

5 This assumption on the form of the partial derivative ${\mathrm{\partial }}_{r}L$ is satisfied in all three examples that we consider in the paper, namely by the Lagrangians $F(t,S,q,r)=rS+L(t,S,q,r)$ in equations (22(22) $\begin{array}{r}F(t,S,q,r):=rS+c_1^2{r}^2+c_2^2{q}^2,\end{array}$(22) ), (31(31) $\begin{array}{r}F(t,S,q,r):=rS+c_1^2{r}^2+c_2^{2}t{q}^2,\end{array}$(31) ) and (35(35) $\begin{array}{r}F(t,S,q,r):=rS+c_1^2{r}^2+c_2^{2}qS,\end{array}$(35) ).