# Risk-Sensitive Optimal Execution via a Conditional Value-at-Risk Objective

arXiv: https://arxiv.org/abs/2201.11962 · HTML: https://mlsanity.com/paper/2201.11962/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jan 2022.

## Built on by (most cited first, 2 of 2)

- [On Dynamic Programming Decompositions of Static Risk Measures in Markov Decision Processes](https://mlsanity.com/paper/2304.12477/) (2023)
- [Q-learning for Quantile MDPs: A Decomposition, Performance, and Convergence Analysis](https://mlsanity.com/paper/2410.24128/) (2024)

## Abstract (arXiv)

We consider a liquidation problem in which a risk-averse trader tries to liquidate a fixed quantity of an asset in the presence of market impact and random price fluctuations. The trader encounters a trade-off between the transaction costs incurred due to market impact and the volatility risk of holding the position. Our formulation begins with a continuous-time and infinite horizon variation of the seminal model of Almgren and Chriss (2000), but we define as the objective the conditional value-at-risk (CVaR) of the implementation shortfall, and allow for dynamic (adaptive) trading strategies. In this setting, we are able to derive closed-form expressions for the optimal liquidation strategy and its value function. Our results yield a number of important practical insights. We are able to quantify the benefit of adaptive policies over optimized static policies. The relevant improvement depends only on the level of risk aversion: for moderate levels of risk aversion, the optimal dynamic policy outperforms the optimal static policy by 5-15%, and outperforms the optimal volume weighted average price (VWAP) policy by 15-25%. This improvement is achieved through dynamic policies that exhibit "aggressiveness-in-the-money": trading is accelerated when price movements are favorable, and is slowed when price movements are unfavorable. From a mathematical perspective, our analysis exploits the dual representation of CVaR to convert the problem to a continuous-time, zero-sum game. We leverage the idea of the state-space augmentation, and obtain a partial differential equation describing the optimal value function, which is separable and a special instance of the Emden-Fowler equation. This leads to a closed-form solution. As our problem is a special case of a linear-quadratic-Gaussian control problem with a CVaR objective, these results may be interesting in broader settings.