Generalized Autoregressive Conditional Heteroscedasticity (garch) and Stochastic Volatility (sv) Models
Essay by dmd123456 • December 23, 2015 • Research Paper • 7,594 Words (31 Pages) • 1,207 Views
Essay Preview: Generalized Autoregressive Conditional Heteroscedasticity (garch) and Stochastic Volatility (sv) Models
Abstract
Return models and covariance matrices of return series have been studied. In particular, Generalized Autoregressive Conditional Heteroscedasticity (GARCH) and Stochastic Volatility (SV) models are compared with respect to their fore-casting accuracy when applied to intraday return series. SV models are found to be considerably more accurate and more consistent in accuracy in forecasting.
Covariance matrices formed from Gaussian and GARCH return series, and in particular, return series auto-correlated as an AR(1) process, have been studied. In the case of Gaussian returns, the largest eigenvalue is found to approximately follow a gamma distribution also when the returns are auto-correlated. Expres-sions relating the mean and the variance of the asymptotic Gaussian distribution of the matrix elements are derived. In the case of GARCH returns, both the largest and the smallest eigenvalues of the correlation matrix are seen to increase with increasing auto-correlation. The matrix elements are found to follow Levy distributions with di erent Levy indices for the diagonal and the non-diagonal elements.
Localization of eigenvectors of correlation matrices of returns from GARCH processes has been investigated. It is found that the localization is reduced as the auto-correlation is increased. Quantitatively, the number of localized eigenvectors decreases approximately as a quadratic function with the auto-correlation strength, i.e. the autoregressive coe cient of the AR(1) process.
Contents
1 | Introduction | 5 | ||
2 | Return Models | 7 | ||
2.1 | Case Study: Nordea 15-minute Returns . . . . . . . . . . . . . . | 10 | ||
2.1.1 | GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . | 11 | ||
2.1.2 | Stochastic Volatility Model . . . . . . . . . . . . . . . . . | 12 | ||
2.1.3 Comparison of the Forecasts . . . . . . . . . . . . . . . . | 16 | |||
2.2 | Case study: Volvo 30-minute Returns . . . . . . . . . . . . . . . | 18 | ||
2.3 | Unconditional Distribution Functions of SV Models . . . . . . . . | 20 | ||
2.3.1 | The Simpli ed model . . . . . . . . . . . . . . . . . . . . | 21 | ||
2.3.2 | The General model . . . . . . . . . . . . . . . . . . . . . . | 23 | ||
2.3.3 Relation to Conditional Distribution Functions . . . . . . | 28 | |||
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Acronyms
ACF Auto-correlation Function. 11
AR Autoregressive. 10, 31, 33, 51
ARIMA Integrated Autoregressive Moving Average. 9, 12, 20, 28
ARMA Autoregressive Moving Average. 10, 20
CDF Cummulative Distribution Function. 26, 38
GARCH Generalized Autoregressive Conditional Heteroscedasticity. 1, 6{8, 10, 11, 16{20, 32, 37, 38, 46, 50, 51, 53, 54, 56, 58{61
IPR Inverse Participation Ratio. 44
MGF Moment Generating Function. 25, 26
MLE Maximum Likelihood Estimate. 12, 23, 26
PDF Probability Density Function. 21, 25, 26, 28, 31, 38, 44{48, 65, 67, 71, 72
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