Parameter Estimation for the Ornstein–Uhlenbeck Process from Extreme Values

    The objective of the work is to derive a maximum likelihood estimator for the parameters of an Ornstein-Uhlenbeck (OU) process based on the extreme values of the process. The OU process is a stochastic process extensively used to model physical and financial phenomena. For a Brownian motion the so-called OHLC estimators are commonly used, which employ not only the opening and closing values of the process, but also its maximum (high) and minimum (low). However, for the OU process, no such analogue has been proposed. In this work, we benefit from the recently found joint law of an OU process and its supremum, and propose a maximum likelihood estimator based on the observed opening, closing, and maixmum values of the process.

    In this work, we derive the joint probability density of the OU process and its supremum, which - to our best knowledge - has not yet been reported in the literature. We then use this density to construct the OHC estimator of parameters: mean μ, reversion rate k and volatility σ, based on the observed opening, closing, and maixmum values of the process. We also derive the analogous estimator for an OU process stopped at a random time. We validate our estimators by applying it simulated trajectories with known parameters. We compare these estimators with the maximum likelihood estimator based only on the opening and closing values and find that the new estimator performs noticeably better. Finally, we apply our estimators to empirical data.