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. When deciding the liquidation strategy, 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, remarkably, 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 (pre-committed) policies. The relevant improvement depends only on the level of risk aversion, and grows without bound as the trader becomes more risk neutral. 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 (to minimize risk), and is slowed when price movements are unfavorable (to minimize transaction costs). Overall, the optimal dynamic policies exhibit much better performance in the right tail of worst outcomes, relative to optimal static policies.
From a mathematical perspective, our analysis exploits the dual representation of CVaR to convert the problem to a continuous-time, zero sum dynamic game. In this setting, we leverage the idea of the state-space augmentation, recently applied to certain discrete-time Markov decision processes with a CVaR objective. We 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 continuous-time linear-quadratic-Gaussian control problem with a CVaR objective, these results may be interesting in broader settings.