Cross-Sectional and Temporal Dependence in Complex Data
Lead Research Organisation:
University College London
Department Name: Mathematics
Abstract
Understanding the temporal random nature of data sequences is central to many areas of research, be it academic or industry-based. The financial industry, for example, is particularly interested in inferring information from large data sets and thus discovering any patterns revealed by recognising a signal. There is an additional dimension: not only is advancement in the statistical analysis and the Mathematics of random time series needed (temporal analysis), but also their cross-sectional properties need to be better understood. That is, one would also like to know more about the dependence structures across several time series. This is a problem found in financial portfolio analysis, among others, where the price dynamics and the risk profile of a portfolio of financial assets may be determined by hundreds of time series.
This research programme treats multivariate generalised diffusions and multivariate Lévy processes from the perspective of structural and driving noise-based dependence structures. Such an investigation includes the study of concordance measures induced by endogenous factors, observed in the market or an economy, which may influence the dynamics of price processes and hedging strategies central in risk management.
Progress in these areas of Mathematics and Statistics could have an immediate impact on the analysis of market efficiency, the pricing and hedging of risks, model risk, and the wider concept of no-arbitrage in financial markets. Although these areas are of particular interest to the financial industry, it is a fact that these are important questions in the Mathematics of Data and Information, in general.
This research programme treats multivariate generalised diffusions and multivariate Lévy processes from the perspective of structural and driving noise-based dependence structures. Such an investigation includes the study of concordance measures induced by endogenous factors, observed in the market or an economy, which may influence the dynamics of price processes and hedging strategies central in risk management.
Progress in these areas of Mathematics and Statistics could have an immediate impact on the analysis of market efficiency, the pricing and hedging of risks, model risk, and the wider concept of no-arbitrage in financial markets. Although these areas are of particular interest to the financial industry, it is a fact that these are important questions in the Mathematics of Data and Information, in general.
People |
ORCID iD |
| Holly Brannelly (Student) |
 http://orcid.org/0000-0001-8103-0305
|
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509577/1 | 01/10/2016 | 24/03/2022 | |||
| 1939295 | Studentship | EP/N509577/1 | 01/10/2017 | 31/12/2021 | Holly Brannelly |
| EP/R513143/1 | 01/10/2018 | 30/09/2023 | |||
| 1939295 | Studentship | EP/R513143/1 | 01/10/2017 | 31/12/2021 | Holly Brannelly |
| Description | The main outcome thus far, from the work funded through this award, is the construction of two novel classes of stochastic quantile processes, in particular quantile diffusion processes, which govern the stochastic dynamics of quantiles in continuous time. We call these classes the random-level and the function-valued quantile processes, respectively. The modelling of quantiles in the discrete time-series setting is an expansive area of literature, due to the many advantages of modelling quantiles directly-especially in areas such as finance or insurance when tail risk is the central interest. The framework produces the mathematical theory necessary for the construction of the aforementioned stochastic quantile models, some analogous to quantile models in a discrete time-series setting. A sub-class of the quantile processes developed allows for parameterisations directly linked to, and thus interpretable in terms of, higher-order moments (e.g., skewness and kurtosis), and which in addition can accommodate autocorrelation structures and tail behaviours. This work provides extensive modelling flexibility especially useful in practice, for instance in risk modelling and management as required in finance and insurance, but also in environmental risk analysis and mitigation. There are currently three research papers in production. The first contains the development of the mathematical theory of the constructed classes of quantile processes, while the focus is placed especially on quantile diffusions. Here one finds, in addition, explicit families of quantile diffusions designed to capture skewness and kurtosis in data sets. As a central application, we show how quantile diffusions can induce distortions of probability measures and hint to their use in Financial and Insurance Mathematics. The second paper focuses on the simulation of quantile diffusions for which a closed-form expression cannot be obtained, and it includes the analysis of error propagation by means of divergence measures. The third work is about pricing and hedging financial assets and insurance liabilities, where we extend the application to general Markovian quantile processes. One aspect in the context of the constructed quantile processes is an alternative method to Edgeworth expansions as a means to analyse the tail behaviour of option prices. Another current investigation is the development of a complementary extension to the Black-Scholes-Merton asset pricing framework based on continuous-time quantile processes with the aim of inducing distributional tail behaviours by means other than stochastic or rough volatility models. The herewith considered approach, based on stochastic pricing kernels, appears to be suitable for building asset price dynamics (and associated risk management) in a multivariate setting, thus addressing questions about the modelling of cross-sectional dependences in portfolio assets and financial markets. Another exciting avenue is the development of hybrid quantile processes based on what we term the ``false-law'' construction. We see this subclass of quantile processes as an avenue towards stochastic models with non-trivial, temporal autocorrelation. |
| Exploitation Route | The multifaceted applicability of the obtained research results stems from the extensive flexibility of the constructed quantile processes. The most promising pathway, due to the one-to-one connection between quantiles and risk, is in risk modelling and management as needed, e.g., in the finance and insurance industries. The developed dynamical quantile models enable one to capture and parameterise model risk, which is an important issue in pricing and hedging problems, and which can stem from the model uncertainty with respect to the characteristics of given data sets. Routes to deployment in industry, but also in a governmental and regulatory setting, may pass through operations research encompassing a wide range of risk quantification and assessments. The solution of allocation problems, which rely on pricing mechanisms or sharing principles geared towards the inclusion of - or are susceptible to - tail risk, may rely on quantiles. The scenario generation, or risk assessment, of the impact emerging from a policy may be based on the stochastic evolution of quantiles. Beyond a financial or insurance context, the outcomes of this research could assist with the understanding, measurement, and the assessment of risk that derives from climate change, and help with the design of sustainable policies to mitigate the strain on natural assets and the loss of biodiversity. |
| Sectors | Aerospace, Defence and Marine,Digital/Communication/Information Technologies (including Software),Energy,Environment,Financial Services, and Management Consultancy,Healthcare |