High-dimensional multivariate multifractal (HD-MMF) volatility models: regularized estimation,forecasting and risk management applications with realistically large portfolios of assets

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Modelling and forecasting of volatilities and correlations of asset returns play an important role in risk management, portfolio selection and derivative pricing. Volatility co-movements between assets or asset classes also shed light on the transmission of shocks between financial markets and national economies. In the wake of the global financial crisis, understanding these mechanisms has also grown in importance for market regulators. However, recent financial crises show that current volatility models leave considerable room for improvement facing two central problems: Problem P.1: Market regulators/investors need models which can forecast returns fluctuations and cross-correlations more accurately. Problem P.2: Investors need models which can handle realistically large portfolios of assets. Currently, state-of-the-art multivariate volatility models are mostly restricted to low-dimensional settings due to computational constraints. This is unsatisfactory because, according to the Basel Accords, the modelling of uncertainty, e.g., for the purpose of Value-at-Risk reporting, is a standard requirement for the risk management of financial institutions, which typically manage large portfolios of hundreds or thousands of assets. The major goal of this project is the introduction of a new class of models, the high-dimensional multivariate multifractal (HD-MMF) volatility models, which can overcome this gap and can be used for large asset portfolios. HD-MMF models are a unified solution to problems P.1 and P.2, combining recent advances in two different areas: 1. multivariate multifractal volatility models which can capture different degrees of long-term dependence in various powers of returns and in their correlations – a property pervasively found in empirical financial data, 2. regularized estimation techniques from the area of machine learning, which can overcome the underlying computational problem and provide an avenue for efficient estimation of high-dimensional models. Our estimation approach has three key benefits: First, we account for general temporal dependency typical for time series data. Second, we model the complete covariance matrix of asset returns explicitly under the assumption that it is a sparse matrix. Third, we use for the first time non-linear moment equations for the purpose of regularized GMM estimation, which could serve as a pilot study for a large number of other applications. By the end of this project, we expect the following deliverables: • A new analytical approach for the multivariate modelling and forecasting of multifractal volatility, • Estimation procedures specifically designed for HD-MMF settings with 100 to 1000 assets, • Optimal portfolios based on our models have lower volatility compared to competing models, • We will have explored the boundary of model sizes which can be handled and the trade-off between forecast performance and computational costs.