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

资产收益波动性和相关性的建模和预测在风险管理、投资组合选择和衍生品定价中起着重要作用。资产或资产类别之间的波动联动也有助于了解金融市场和国家经济之间的冲击传递。在全球金融危机之后，了解这些机制对市场监管机构也变得越来越重要。然而，最近的金融危机表明，目前的波动率模型有很大的改进空间，面临两个核心问题：问题P.1：市场监管机构/投资者需要模型，可以更准确地预测回报波动和交叉相关性。问题P.2：投资者需要能够处理现实的大型资产组合的模型。目前，最先进的多元波动模型大多限于低维设置，由于计算的限制。这是不能令人满意的，因为根据巴塞尔协定，不确定性的建模，例如，对于风险价值报告而言，这是金融机构风险管理的标准要求，金融机构通常管理数百或数千资产的大型投资组合。该项目的主要目标是引入一类新的模型，即高维多元多重分形（HD-MMF）波动率模型，它可以克服这一差距，并可用于大型资产投资组合。HD-MMF模型是问题P.1和P.2的统一解决方案，结合了两个不同领域的最新进展：1。 多元多重分形波动率模型，可以捕捉不同程度的长期依赖于各种权力的回报和他们的相关性-财产普遍发现的经验金融数据，2。 来自机器学习领域的正则化估计技术，它可以克服潜在的计算问题，并为高维模型的有效估计提供途径。我们的估计方法有三个关键的好处：首先，我们考虑到一般的时间依赖性典型的时间序列数据。其次，我们明确地假设资产收益率的完全协方差矩阵是一个稀疏矩阵。第三，我们第一次使用非线性矩方程的正则化GMM估计的目的，这可以作为一个试点研究，为大量的其他应用。在此项目结束时，我们预计将交付以下成果：· 一种新的分析方法的多变量建模和预测的多重分形波动，· 专门为100至1000个资产的HD-MMF设置设计的估计程序，· 与竞争模型相比，基于我们模型的最优投资组合具有更低的波动性，· 我们将探讨模型大小的边界，可以处理和预测性能和计算成本之间的权衡。

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.