Citation: Mastroeni, Loretta, and Vellucci Pierluigi. 2022. Construction of an SDE Model from Intraday Copper Futures Prices. Risks 10: 218. https://doi.org/10.3390/ risks10110218 Academic Editor: Mogens Steffensen Received: 22 August 2022 Accepted: 14 November 2022 Published: 17 November 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). risks Article Construction of an SDE Model from Intraday Copper Futures Prices Loretta Mastroeni *,† , Pierluigi Vellucci † Department of Economics, University of Roma Tre, 00145 Roma, Italy * Correspondence: lmastroeni@os.uniroma3.it † These authors contributed equally to this work. Abstract: This paper introduces a model for intraday copper futures prices based on a stochastic differential equation (SDE). In particular, we derive an SDE that fits the model to the data and that is based on the whitening filter approach, a method characterizing linear time-variant systems. This method is applied to construct a model able to simulate the trajectories of copper futures prices, statistically described by means of an empirical autocorrelation approach. We show that the predictability of copper futures prices is rather weak. In fact, the developed model produces trajectories close to the actual data only in the short term. Consequently, the investment risk for copper futures is high. We also show that the performance of the model improves significantly if the time series satisfy particular conditions, e.g., those with a determinism measure. Keywords: stochastic differential equations; autocorrelation; dynamical systems; determinism; time series analysis; copper; prices 1. Introduction Financial time series modeling is an essential aspect of forecasting and risk evaluation in financial markets. In this paper, we consider the case of copper, which, according to the NYMEX, is the third most used metal, which makes it important to assess the nature of its price fluctuations. Further, the likely pressure on copper prices due to its possible scarcity is a cause for concern Gordon et al. (2006); Tilton and Lagos (2007), especially in light of its importance for the growing network industry. Our previous works, Mastroeni and Vellucci (2019); Mastroeni et al. (2018), showed that a significant level of noise is usually present in the time series of copper futures intraday prices, supporting the conclusion that logarithmic returns have both a stochastic and deterministic nature. Hence, the use of stochastic models appears to be a natural choice. In this paper, we develop a novel stochastic differential equation (SDE) that models the data and is based on the whitening filter approach Wiener (1949), a method characterizing linear time-variant systems. From a statistical point of view, we assume that the time series of prices can be described by autocorrelation. This property is obtained by means of statistical analyses of the historical data recorded for the futures copper closing prices (HG1 ticker) as exchanged on the COMEX market (CMX). Starting from a model of the empirical autocorrelation shown by the time series of prices, we introduce an SDE, which is characterized by this autocorrelation, and fits the model to the data. To the best of our knowledge, there have been no papers following this approach. The purpose of the developed SDE model is to move one step further concerning the analysis started in our previous paper Mastroeni et al. (2018). In that paper, we showed that the time series of copper prices (the same studied in the present paper but for a shorter interval) exhibited both stochastic and chaotic features. At the same time, the recurrence plot of the time series revealed a pattern typical of intermittency phenomena. Risks 2022, 10, 218. https://doi.org/10.3390/risks10110218 https://www.mdpi.com/journal/risks