On the singular control of exchange rates

Abstract

Consider the problem of a central bank that wants to manage the exchange rate between its domestic currency and a foreign one. The central bank can purchase and sell the foreign currency, and each intervention on the exchange market leads to a proportional cost whose instantaneous marginal value depends on the current level of the exchange rate. The central bank aims at minimizing the total expected costs of interventions on the exchange market, plus a total expected holding cost. We formulate this problem as an infinite time-horizon stochastic control problem with controls that have paths which are locally of bounded variation. The exchange rate evolves as a general linearly controlled one-dimensional diffusion, and the two nondecreasing processes giving the minimal decomposition of a bounded-variation control model the cumulative amount of foreign currency that has been purchased and sold by the central bank. We provide a complete solution to this problem by finding the explicit expression of the value function and a complete characterization of the optimal control. At each instant of time, the optimally controlled exchange rate is kept within a band whose size is endogenously determined as part of the solution to the problem. We also study the expected exit time from the band, and the sensitivity of the width of the band with respect to the model's parameters in the case when the exchange rate evolves (in absence of any intervention) as an Ornstein-Uhlenbeck process, and the marginal costs of controls are constant. The techniques employed in the paper are those of the theory of singular stochastic control and of one-dimensional diffusions.

The Optimal Control of Government Stabilization Funds

“…Harrison and Taksar [7] studied a two-sided stochastic singular control problem of a Brownian motion with drift, which is mathematically simpler than the above mean-reverting process, but did not present a full-blown algorithm to compute the boundary of the non-intervention region. Regarding the two-sided stochastic singular control problem of a mean-reverting process, more general settings than the present paper have been treated by Matomäki [13] and recently by Ferrari and Vargiolu [14]. We have solved the two-sided stochastic singular control problem of our paper not only for a symmetric but also for an asymmetric running cost function (see Remark 3 for details).…”

“…Applying the methods of Ferrari and Vargiolu [14], it is possible to prove that there exists a unique solution for the system (25)-(30). Solving the above system determines the candidate for value function v and the candidate for optimal stabilization fund band [a, b].…”

Stochastic recursive optimal control problem with obstacle constraint involving diffusion type control

“…Moreover, some kinds of singular control problems were connected with the optimal stopping problems by Dufour and Miller [20], and with the free boundary problems by Dai and Yi [21]. On the other hand, for mathematical finance, Oksendal and Sulem [22] modeled optimal portfolio problems with transaction costs in terms of singular control problems, whereas Cadenillas and Zapatero [17], Ferrari and Vargiolu [23] applied stochastic impulse control to the exchange rate problems. In addition, Wu and Zhang [24] were concerned with a utility maximization problem with the step-shaped consumption strategy by impulse controls.…”