# Optimal execution with dynamic risk adjustment

## TL;DR

This paper develops a dynamic risk-adjusted optimal liquidation model in financial markets, deriving explicit solutions under quadratic risk measures to quantify the impact of risk preferences on trading strategies.

## Contribution

It introduces a novel continuous-time stochastic control framework incorporating dynamic risk measures for optimal liquidation, with explicit solutions under quadratic specifications.

## Key findings

- Closed-form solutions for optimal liquidation policies.
- Quantification of risk-adjustment impact on P&L.
- Framework based on g-conditional risk measures.

## Abstract

This paper considers the problem of optimal liquidation of a position in a risky security 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 maximizing a generalized 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.