Generalized Autoregressive Conditional Heteroscedasticity (garch) and Stochastic Volatility (sv) Models

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PR Participation Ratio. 45, 48

SV Stochastic Volatility. 1, 5{8, 10, 11, 16, 17, 19, 20, 37, 50, 54, 56, 58, 60, 61

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Chapter 1

Introduction

The mathematics of the nancial market has always been a topic that arouses interest and imagination, and with no doubt, has been studied from many as-pects and in many di erent ways.

Central to all these studies are the concepts of probability distributions and correlations. The values of stocks, futures, options, etc. are stochastic in nature and are governed by the laws of stochastic processes, which are expressed in terms of probability distributions and correlations.

Meanwhile, the observables of the market are prices, volumes, turn-over (the amount of money paid in a trade), names of the brokers, and time of the trades. These quantities don't make much sense by themselves but do reveal the probabilistic dynamics of the market when put together and turned into statistics.

The dynamics of an asset is a ected by its own history as well as by the histories and current values of other assets in the market. The in uence from the asset's own history is termed autocorrelations, i.e. correlations in time, while the in uence from other assets are termed cross correlations.

SV models built on historical data describe the time evolution of the afore-mentioned observables in terms of probability distributions, autocorrelations and cross correlations. They may have some predictive power and help deter-mine a fair price of a given asset.

Quite often, instead of an asset's price, the relative price change, i.e. the return, is studied. Analytically solvable models often assume Gaussian return distribution, although data suggest fatter tails. In numerical return models, realistic reproduction of historical data may be achieved.

In addition to the model of a single asset, the correlation between a group of assets is also of great interest. For example, in principle component analysis, one wishes to identify a number of factors that \drive" the price evolution of a group of assets in the sense that each of the assets' returns can be expressed as a linear combination of the factors' returns. In this scenario, the eigenvalues of the covariance matrix are the variances of the factors' returns and their corresponding orthogonalized eigenvectors give the composition of the factors,

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i.e. the coe cients with which the factors are contructed as a linear combination of the assets.

Therefore, in this thesis we also study the elements and the eigenvalues distribution, as well as the eigenvectors composition of the covariance matrix. When the matrix is constructed from returns with simple Gaussian distribution, the matrix is termed a Wishart matrix and has been studied extensively in the literature. If the returns have Levy distributions, the matrix is termed Wishart-Levy and has been studied to some extent, particularly regarding its eigenvalue distribution [1].

However, it is understood that real stock/index returns are much more com-plicated than a straight-forward Gaussian or Levy distribution can describe | instead, one needs structured models. For this reason, we are particularly in-terested in a covariance matrix obtained for realistic return models. The so called GARCH(1,1) model is a realistic return model proven to have regularly varying tails [2]. So we study properties of eigenvalues and eigenvectors of such covariance matrices. Moreover, we also study how auto-correlations in the re-turns in uence the aforementioned properties. Such auto-correlations, known as second-order auto-correlations decay exponentially but may still leave footprints in the covariance matrix.

This report is organized as follows: Chapter 2 reviews some of the most in uential return model. Parameters of the models are tted to a few intraday return series and the predictive power of the models is compared. A calcula-tion of the unconditional distribution functions of SV models, especially in the case where the residual of the log-volatility and the innovation of the return are correlated normal variates, is also presented. In chapter 3 we investigate distributions of eigenvalues of the Wishart matrix, and study the in uence of auto-correlated returns. In chapter 4 distributions of elements and eigenvalues of the covariance matrix of identically speci ed GARCH(1,1) series are stud-ied. Finally, chapter 5 summarizes the results. Supplementary materials are provided in the appendices.

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Chapter 2

Return Models

In this chapter we review some of the discrete-time return models and t them to intraday returns. The intention is to compare these models in terms of fore-cast accuracy and to understand their statistical properties. In the following we rst describe these models brie y, then in section 2.1, section 2.2 and appendix A we describe how the GARCH and the SV models are tted to intraday re-turns and compare their forecasts. In section 2.3 we calculate the unconditional distribution functions of the SV model.

- Gaussian Distribution

The justi cation of modeling return series as independent, identically dis-tributed Gaussian variates comes from imagining the price process S(t) as

a geometric Brownian motion whose increment dw at each time step is p t

[pic 1]

independent and scales as dt. Then, by adding a drift term dt that represents some deterministic trend in the price process, one can express the price S(t) as a stochastic di erential equation:

dS = S dt + S dwt

Then by It^o's lemma, a stochastic di erential equation for the logarithmic price ln S can be obtained

d(ln S) = (

1

2)dt + dwt