Analysis of financial time series using non-parametric Bayesian techniques
Keywords: information flow
renyi entropy
machine learning
gaussian processes
financial volatility

The overarching aim of this thesis is to show that Gaussian processes and Renyi entropy can be valuable non-parametric tools for forecasting intraday volatility for a wide range of financial time series.
In this thesis empirical volatility forecasting using Gaussian processes (GPs) is presented for stocks, market indices, forex and cryptocurrencies. Key innovations are presented in the application of GPs by using separated negative and positive returns in transformed log space, and the use of Renyi transfer entropy for incorporating information flow from other time series to modify volatility forecasts. Significant performance gains are demonstrated over strong benchmarks for volatility forecasting - the mcsGARCH model that is specially designed for intraday volatility, rolling average, and use of last observed value as next step forecast. A set of robust loss functions are used for assessing performance, and we establish the significance of all results at the 95% confidence interval.
This thesis demonstrates empirically that standalone GPs perform better than GARCH, EGARCH, GJR-GARCH in forecasting intraday financial volatility. This is done by presenting the results from the largest study done in literature to date, that uses GPs for this purpose. Approximately 50K experiments are carried out and over 18 million volatility forecasts are analysed, on 11 years worth of trading data from 50 market symbols sampled at 1-minute frequency to establish the significance of the results.
It is recognized that GARCH, EGARCH, GJR-GARCH have not been designed for intraday volatility forecasting therefore this comparison is not with the best in class. mcsGARCH is chosen as the best in class GARCH model specifically designed for forecasting intraday volatility. After this selection it is empirically established that when mcsGARCH is used for benchmarking the performance of GPs, it is found that plain GPs are substantially worse than mcsGARCH. This negative result helps establish the need for innovation in the application of GPs to volatility forecasting.
An innovation for the GP intraday volatility forecasting model is developed by regressing on the negative and positive returns envelopes separately in log-space. The forecasts generated from this method are found to significantly outperform all benchmarks including mcsGARCH in terms of their loss function performance. This superior performance against mcsGARCH successfully demonstrated that GPs can be a significant tool in the volatility practitioner's forecasting toolkit.
In order to incorporate information from other financial time series into the volatility forecast the concept of Renyi transfer entropy is introduced as a non-parametric measure for calculating the asymmetrical information flows between two financial time series with the ability to emphasize different parts of the event space. The effective Renyi transfer entropy for key market indices is calculated and information flows around key global events are visualized. It is found that very high flows of information occur around global events of financial importance but very little influence is found for politically important events.
A new approach is devised for folding the effective Renyi transfer entropy into the forecast made by envelope based GPs. This new approach of using the ERTE modified envelope GP forecasts is applied for predicting intraday volatility for 6 market indices and 12 forex pairs. The performance improvement from these modified forecasts is benchmarked and found to be significantly better than all benchmarks considered including the standalone envelope GP forecasts. The parameters used in calculating the ERTE forecasts are found empirically and values which provide the most consistent gains in the training data are used in the comparison against benchmarks.
The approach is extended further and applied to forecast the volatility in cryptocurrency markets by conducting experiments on 6 month long time series from 10 cryptocurrencies sampled at 5-minute frequency. It is found that envelope based GPs significantly outperform against all three benchmarks on this dataset. This is the largest volatility forecasting study done to date by using intraday frequency data from multiple cryptocurrencies.
An innovation is presented for folding the effective Renyi transfer entropy value from multiple time series into one value by extending the non-parametric formulation for ERTE. While this extension results in the need for a larger data set, its application for calculating ERTE remains computationally simple with a high degree of flexibility
The combined time series ERTE value is used for modifying the forecasts from envelope based GPs and applied to the case of using up to 9 secondary cryptocurrency time series for calculating the forecasts for 1 primary cryptocurrency. This technique is trialled for all 10 cryptocurrencies in the data and it is found that although this method still outperforms the three benchmarks, its performance against the standalone envelope based GP is not consistent. A potential reason for this is discussed by assessing the relationship between the discretization parameter in ERTE and the requirements it poses for the size of the dataset.
The thesis closes with a discussion of future directions and open questions.