  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.
