Partial Information, Return Predictability and Identification of Linear Stochastic Differential Equations

Asset allocation and option pricing models are often formulated by means of linear stochastic differential equations. We show that this class of models is not identifiable from information contained in discrete-time data when the expected return process is unobservable. The indeterminacy arises because the discrete-time reduced form of the linear diffusion process can be mapped into a set of observationally equivalent continuous-time representations with different covariance structures. Despite the indeterminacy of the underlying stochastic model, we prove that portfolio strategies and option prices are asymptotically uniquely determined because they depend on invariant combinations of the unidentifiable continuous-time parameters. This implies that partially informed investors with different but admissible parameter estimates follow the same portfolio policy and compute the same option prices. Our results thus allow to re-conciliate the continuous-time formulation of the financial problem with the econometric inference of the model parameters based on discrete-time observations. Using financial data, it is shown that arbitrary or ad hoc parameter restrictions are not necessarily admissible, resulting in sub-optimal allocations and incorrect derivative prices