An Inverse Problem for a Nonlinear Stochastic Differential Equation Modeling Price Dynamics

Abstract

Diffusion processes are widely used for mathematical modeling in finance e.g. in modeling foreign exchange rates. This paper presents a non-linear stochastic continuous time model that captures the main characteristics of price dynamics. The generalized mean reversion process discloses various features of observed price movements such as multi-modality of the distributions, multiple equilibria, and regime switching. The attractors depend substantially on the economic environment. The model reveals a significant connection between exchange rates and its fundamentals. Furthermore, it is consistent with traditional flexible exchange rate models. Stochastic differential equations describing diffusion processes are directly linked to the forward Kolmogorov equation. In order to calibrate the models, efficient algorithms identifying the system parameters are in demand. Taking into account nonlinear effects in volatility and drift and dependence on observed economic data, which are not directly modeled, one obtains problems which cannot be treated by standard numerical methods. The coefficients are rapidly oscillatory and strong instabilities may arise. To handle these problems we develop numerical methods, which are used to simulate the nonlinear dynamics of exchange rates depending on economic data.