Imputation of Missing Data with Application to Commodity Futures

Abstract: In recent years additional requirements have been imposed on financial institutions, including Central Counterparty clearing houses (CCPs), as an attempt to assess quantitative measures of their exposure to different types of risk. One of these requirements results in a need to perform stress tests to check the resilience in case of a stressed market/crisis. However, financial markets develop over time and this leads to a situation where some instruments traded today are not present at the chosen date because they were introduced after the considered historical event. Based on current routines, the main goal of this thesis is to provide a more sophisticated method to impute (fill in) historical missing data as a preparatory work in the context of stress testing. The models considered in this paper include two methods currently regarded as state-of-the-art techniques, based on maximum likelihood estimation (MLE) and multiple imputation (MI), together with a third alternative approach involving copulas. The different methods are applied on historical return data of commodity futures contracts from the Nordic energy market. By using conventional error metrics, and out-of-sample log-likelihood, the conclusion is that it is very hard (in general) to distinguish the performance of each method, or draw any conclusion about how good the models are in comparison to each other. Even if the Student’s t-distribution seems (in general) to be a more adequate assumption regarding the data compared to the normal distribution, all the models are showing quite poor performance. However, by analysing the conditional distributions more thoroughly, and evaluating how well each model performs by extracting certain quantile values, the performance of each method is increased significantly. By comparing the different models (when imputing more extreme quantile values) it can be concluded that all methods produce satisfying results, even if the g-copula and t-copula models seems to be more robust than the respective linear models.