Asymptotic Impulse Control of Mean-Reverting Interest Rates with a Slowly Varying Stochastic Volatility

Abstract

This paper studies the optimal central bank intervention of interest rate problem, where the interest rate process is modelled by an Ornstein—Uhlenbeck (mean-reverting) process with a slowly varying stochastic volatility. The objective of the central bank is to maintain the interest rate close to a target level, subject to fixed and proportional costs of interventions. The problem is formulated as an impulse control problem, which is being converted to a free boundary problem by adopting an ansatz of a band policy. Due to the complexity introduced by the stochastic volatility, there is no analytical solution to the free boundary value problem in the literature. This paper applies a regular perturbation approach to derive an asymptotic solution to the value function and the optimal impulse control (intervention). We rigorously prove that the zeroth-order approximation of the optimal impulse control is associated with the first-order approximation of the value function. Moreover, we show that this zeroth-order suboptimal impulse control is asymptotically optimal in a specific family of impulse controls.