Non-linear dependences in finance

TL;DR

This thesis explores non-linear dependencies in finance through statistical tools, models cross-sectional stock return dependencies, and investigates temporal dependencies like volatility clustering using copulas and self-exciting models.

Contribution

It introduces new statistical tests for dependence with fat-tailed data, proposes a calibration of a new factor model, and generalizes models of volatility clustering with copulas and self-exciting processes.

Findings

Extended goodness-of-fit tests for dependent, fat-tailed data

Calibration of a new factor model for stock returns

Copula-based framework for analyzing time series dependencies

Abstract

The thesis is composed of three parts. Part I introduces the mathematical and statistical tools that are relevant for the study of dependences, as well as statistical tests of Goodness-of-fit for empirical probability distributions. I propose two extensions of usual tests when dependence is present in the sample data and when observations have a fat-tailed distribution. The financial content of the thesis starts in Part II. I present there my studies regarding the "cross-sectional" dependences among the time series of daily stock returns, i.e. the instantaneous forces that link several stocks together and make them behave somewhat collectively rather than purely independently. A calibration of a new factor model is presented here, together with a comparison to measurements on real data. Finally, Part III investigates the temporal dependences of single time series, using the same tools and measures of correlation. I propose two contributions to the study of the origin and description of "volatility clustering": one is a generalization of the ARCH-like feedback construction where the returns are self-exciting, and the other one is a more original description of self-dependences in terms of copulas. The latter can be formulated model-free and is not specific to financial time series. In fact, I also show here how concepts like recurrences, records, aftershocks and waiting times, that characterize the dynamics in a time series can be written in the unifying framework of the copula.