Optimal execution with dynamic risk adjustment

Abstract

This article considers the problem of optimal liquidation of a position in a risky security quoted in a financial market, where price evolution are risky and trades have an impact on price as well as uncertainty in the filling orders. The problem is formulated as a continuous time stochastic optimal control problem aiming at maximising a generalised risk-adjusted profit and loss function. The expression of the risk adjustment is derived from the general theory of dynamic risk measures and is selected in the class of g-conditional risk measures. The resulting theoretical framework is nonclassical since the target function depends on backward components. We show that, under a quadratic specification of the driver of a backward stochastic differential equation, it is possible to find a closed form solution and an explicit expression of the optimal liquidation policies. In this way, it is immediate to quantify the impact of risk adjustment on the profit and loss and on the expression of the optimal liquidation policies.

Keywords:

• Optimal execution
• price impact
• dynamic risk measures
• trading strategies
• stochastic optimal control

Notes

Acknowledgements

We wish to thank two anonymous referees and the editors for careful scrutiny and helpful suggestions. We thanks Claudio Tebaldi for valuable discussions and for carefully reading the draft of this article. Holger Kraft, Athena Picarelli, and Emanuela Rosazza-Gianin deserve special mention for their useful comments. Finally, we are grateful to all participants to seminars and conferences where the work was presented. The usual disclaimer applies.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The relation is verified in Appendix B, p. 13.

2 As previously mentioned, see, e.g. Yong and Zhou (1999, Theorem 6.3, p. 42) for the forward part and Briand and Hu (2008) for the backward component.