Nonlinear dependence and time dynamics in financial markets

In this project we have carried out research on the statistical modeling of financial markets. Of particular interest is the returns of financial assets, e.g. stocks and bonds, and the dependence between returns from different assets. Modelling of returns are of importance when calculating the risk of a financial position, and for selecting the optimal mix of assets in a portfolio. A much used assumption in practice is to model financial returns with a Gaussian distribution, due to its simplicity. In particular, the dependence between the returns from different assets will be completely described by the correlation coefficient. It is also easy to calculate the risk of a financial position if one assume Gaussianity. However, as many researchers have pointed out, one observe that in reality there is a higher probability of extreme events, i.e. losses, than the Gaussian distribution will give, and there is a higher dependence between financial assets when the market is falling. This means that asset returns are not Gaussian, at least for relatively short-horizon returns, say one day, and the dependence between returns not correctly described by the correlation coefficients. Based on these observations, we have earlier introduced a dependence measure that are able to model such asymmetric dependencies, called the Local Gaussian Correlation (LGC). In this project we have extend these previous results. In particular, by noting that the dependence structure between assets also varies in time. The main aim of this project have been to develop new methods for capturing the behaviour of asset returns, which in turn will improve financial risk measurement and asset allocation methods that are widely used in the financial services industry. The project has resulted in several publications in scientific journals.

Prosjektet har finansiert en rekke opphold av utenlandske forskere ved UiB, og prosjektet har finansiert flere reiser til konferanser, samt opphold ved utenlandske universiteter for prosjektets deltagere fra Norge. Dette har ført til økt internasjonalt samarbeid for gruppen i statistikk ved UiB, bl.a. med Waseda University, Japan og LUMSA, Italia. Spesielt vil vi nevne at prosjektet har ført til at UiB, Matematisk institutt, har rekruttert en prof. II i statistikk fra Italia. Prosjektet har resultert i ny kunnskap rundt modellering av avhengigheter mellom stokastiske variable, spesielt modeller for finansielle avkastninger. Prosjektet har resultert i flere vitenskapelige artikler. Potensielt har resultatene fra prosjektet praktisk relevans, da metodene utviklet i prosjektet kan benyttes for å beskrive avhengigheter mellom finansielle avkastninger på en mer presis måte, noe som vil kunne forbedre beregning av risiko og aktivaallokeringsmetoder, som benyttes i finansnæringen.

Key statistical features of asset returns are their nonlinear dependencies and the non-Gaussianity of their joint distribution. In this project, we will carry out further research on the statistical analysis of financial markets, based on earlier results from previous projects financed by the Finansmarkedsfondet, having in common the theory and application of the dependence measure termed local Gaussian correlation (LGC). The project is divided in two related sub-projects: i) Time dynamics of the dependence structure of asset returns. Earlier results by us suggest that the dependence structure between asset returns is time-varying, but this has not been explicitly modelled by the LGC. The first sub-project consist of an extension of the classical LGC setting by embedding it into a Hidden Markov Model (HMM) context, i.e., allow for regime switching models. By combining HMM framework with LGC, we intend to achieve a more realistic modeling of asset returns and their dependencies, thus improving both asset allocation methods as well as risk measurement calculations. The new model is called a Markov-switching LGC (MSLGC) model. ii) Modelling ultra-high frequency asset returns. The second part of the project is the application of the new Markov-switching LGC (MSLGC) to a) classical return series (daily/weekly/monthly frequencies) from major international stock markets, and b) ultra high-frequency (UHF) financial data. The latter is motivated by increased evidence of nonlinear dependence in UHF data, which has never been studied neither by the common LGC nor by MSLGC models, thus leaving more than sufficient potential for interesting empirical results. The models developed in both sub-projects will further be utilized in other settings, in particular, for asset allocation and risk measurement calculations. Also extensions towards the modelling of the number of trades in different assets will be considered.