基于条件风险价值目标的风险敏感最优执行
Risk-Sensitive Optimal Execution via a Conditional Value-at-Risk Objective
摘要：
考虑一个清算问题，在这个问题中，一个风险厌恶的交易者试图在存在市场冲击和随机价格波动的情况下清算固定数量的资产。交易者需要在市场冲击所产生的交易成本和持有头寸的波动风险之间进行权衡。我们的公式开始于Almgren和Chriss（2000）的开创性模型的连续时间和无限水平变化，但我们将执行不足的条件风险价值（CVaR）定义为目标，并允许动态（自适应）交易策略。在这种情况下，我们能够推导出最优清算策略及其价值函数的封闭式表达式。我们的研究结果产生了一些重要的实际见解。我们能够量化自适应策略相对于优化静态策略的优势。相关的改进仅取决于风险厌恶的水平：对于中等水平的风险厌恶，最优动态策略优于最优静态策略5- 15%，优于最优成交量加权价（VWAP）策略15- 25%。这种改善是通过表现出“货币侵略性”的动态政策实现的：当价格走势有利时，交易加速，当价格走势不利时，交易放缓。从数学的角度来看，我们的分析利用CVaR的对偶表示转换的问题，连续时间，零和游戏。利用状态空间增广的思想，得到了描述最优值函数的偏微分方程，该方程是可分的，是Emden-Fowler方程的一个特例.这导致了一个封闭形式的解决方案。由于我们的问题是一个特殊情况下的线性二次高斯控制问题的CVaR目标，这些结果可能是有趣的，在更广泛的设置。
英文摘要：
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.
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