We consider a central bank strategy for maintaining a two-sided currency target zone, in which an exchange rate of two currencies is forced to stay between two thresholds. To keep the exchange rate from breaking the prescribed barriers, the central bank is generating permanent price impact and thereby accumulating inventory in the foreign currency. Historical examples of failed target zones illustrate that this inventory can become problematic, in particular when there is an adverse macroeconomic trend in the market. We model this situation through a continuous-time market impact model of Almgren--Chriss-type with drift, in which the exchange rate is a diffusion process controlled by the price impact of the central bank's intervention strategy. The objective of the central bank is to enforce the target zone through a strategy that minimizes the accumulated inventory. We formulate this objective as a stochastic control problem with random time horizon. It is solved by reduction to a singular boundary value problem that was solved by Lasry and Lions (1989). Finally, we provide numerical simulations of optimally controlled exchange rate processes and the corresponding evolution of the central bank inventory.

中文翻译：

---

中央银行捍卫货币挂钩的策略

我们考虑央行维持双边货币目标区的策略，其中两种货币的汇率被迫保持在两个阈值之间。为了防止汇率突破规定的壁垒，中央银行正在产生永久性的价格影响，从而积累外币库存。目标区域失败的历史例子表明，这种库存可能会出现问题，特别是当市场出现不利的宏观经济趋势时。我们通过具有漂移的 Almgren--Chriss 型连续时间市场影响模型对这种情况进行建模，其中汇率是受央行干预策略的价格影响控制的扩散过程。中央银行的目标是通过最小化累积库存的策略来强制执行目标区域。我们将此目标表述为具有随机时间范围的随机控制问题。它是通过归约到 Lasry 和 Lions (1989) 解决的奇异边值问题来解决的。最后，我们提供了最优控制汇率过程的数值模拟以及中央银行库存的相应演变。