On Correlated Measurement Errors in the Schwartz-Smith Two-Factor Model

Abstract

The Schwartz–Smith two-factor model is commonly used for pricing of derivatives in commodity markets. For estimating and forecasting the term structures of futures prices, the logarithm of commodity spot price is represented as the sum of short- and long-term factors being the unobservable state variables. The futures prices derived as functions of the spot price lead to the simultaneous set of measurement equations, which is used for joint estimation of unobservable state variables and the model parameters through a filtering procedure. We propose a modified model where the error terms in the measurement equations are assumed to be serially correlated. In addition, for comparative analysis, the modelling of the logarithmic returns of futures prices is also considered. Out-of-sample prediction performances of two proposed models were illustrated using European Unit Allowances (EUA) futures prices from January 2017 to April 2021. Historically, this period corresponds to the second half of Phase III, and the beginning of Phase IV of the European Union Emission Trading System (EU-ETS).

Keywords: Kalman filter, pricing, futures, commodity, CO2 emission allowances, correlation, maximum likelihood estimation, linear state-space model

JEL Classification: C32, C52, C53