An exchange rate model where the fundamentals follow a jump-diffusion process

This paper presents some models of exchange rate with jumps, namely jump diffusion exchange rate models. Jump diffusion models are quite common in computational and theoretical finance. It is known that exchange rates sometimes exhibit jumps during some time periods. Therefore, it is important to take into account the presence of these jumps in exchange rate modeling in general. However, even the simplest jump diffusion model introduces some analytical difficulty in terms of finding a solution to the model. The models we analyze in this paper make use of Approximation Theory in order to come up with closed form solutions to the underlying variables. This approach leads to the branch of differential equations called functional differential equations and more specifically the so-called delay differential equations. Our approach leads to a second order delay differential equation. Though, in principle, these types of functional differential equations can be solved analytically in some cases, the task, in general, is quite enormous. We circumvent this technical difficulty by deriving an approximate solution using a power series expansion of the second order. Therefore, we derive a complete solution to the models and also investigate the model's predictions of the exchange rate. We introduce two jump diffusion models. The first model examines the case where there are jumps with a constant magnitude. The second model considers the case of jumps of different sizes. These are relatively simpler cases to be analyzed. We will present some computational aspects in terms of the difficulty often encountered in estimating these types of models. The difficulty increases for the type of exchange rate models being considered in this paper. Taking advantage of the specification of the models we have estimated the parameters using a two-step M-estimation strategy that combines full information maximum likelihood estimation in the first step and the simulated method of moments in the second step.