Controlling stochastic currency risk exposure optimally

Abstract

Investors operating in countries adopting different currencies face an additional risk in the form of currency exchange rates. This thesis aims at deriving the optimal hedging strategy for such investors through the use of futures and forwards. Having described the processes underlying the economic framework which affect the investor, the theory of stochastic optimal control will be used to formulate and solve this the problem mathematically. As an attempt to solve the problem analytically, the dynamic programming approach will first be employed. However, since the resulting Hamilton-Jacobi-Bellman equation involves a highly non-linear second order partial differential equation, such a solution is hard to obtain in closed form and so we resort to numerical techniques. To this end we shall employ the Markov chain approximation method, in which a sequence of optimal stochastic control problems for Markov chains will be solved via the dynamic programming approach. The latter will lead to a sequence of functional equations, which have to be solved for the approximating value function. An approximate solution to these functional equations will then be obtained numerically via the Implicit method which, provided the approximating Markov chains are locally consistent, converges to the original controlled stochastic integral equation. Furthermore under this local consistency, the solutions to the functional equations are also known to converge to the value function of the original stochastic optimal control problem.