Calibration of the exponential Ornstein–Uhlenbeck process when spot prices are visible through the maximum log-likelihood method. Example with gold prices

Abstract

The purpose of this paper is to present a methodological procedure to estimate the parameters of the exponential Ornstein–Uhlenbeck process, also known as the Schwartz (J. Finance 52(3):923–973, 1997) one-factor model, in situations where the spot price of the commodity is observable. The proposal consists of looking at the probability function of the process as a function of the unknown parameters in discrete time, known as the likelihood function. Then the logarithm of that expression is calculated as it is easier to work with it. Finally, the problem of determining the values of the parameters that maximize the sum of the individual log-likelihoods (joint log-likelihood function) is solved to obtain the estimation equations explicitly. In that sense, this work is relevant because as spot prices are available, it is possible to estimate the parameters directly without the necessity of using more elaborate approaches like the Kalman filter. Finally, the paper applies this methodology to the concrete case of one precious metal that has an observable spot price and for which some empirical and theoretical studies suggest that it presents a mean-reverting pattern, gold. The estimated parameters are consistent with previous works and with the original data and the least squares method.