# Tail decay arising in GARCH models and volatility risk

**Specialeforsvar** ved Jens Munch Larsen

**Titel:** Tail decay arising in GARCH models and volatility risk

**Abstract:** This thesis explores GARCH models as a form of modelling time dependence through volatility. The thesis has the GARCH(1,1) model as a baseline and it is shown that a requirement for stationarity of GARCH(1,1) processes is negativity of the term $E \log A$ where $A$ is the coefficient in the corresponding recurrence equation $X_{t+1} = A_t X_t + B_t$. Building on this it is shown that stationary volatility (in an infinite sense) has a limiting distribution that exhibits probability tail decay. This is shown through the notion of stochastic fixed point equations and drawing on the renewal theorem. Thus it is shown that the tail distribution of volatility in GARCH can be shown to be asymptotically exponentially decaying. The thesis then goes on to discuss how GARCH model parameters can be estimated and simulated. Especially it is explored how to simulate a stationary variable through using the stationarity condition. Furthermore it is illustrated how constants arising in the tail decay can be calculated and in extension it is assessed how well the tail decay approximation compares to Monte Carlo simulations. It is then explored how copulas can be used to model dependence of noise between GARCH(1,1) processes especially tail dependencies are considered thus enabling the us of Monte Carlo for risk assessment of mixed asset portfolios. Lastly it is discussed how to alter the model in order to allow the intertemporal volatility dependence to be more sophisticated. In particular the general GARCH(p,q) model is considered and the stationarity condition is discovered to be a matter of having a negative Lyapunov Exponent which is furthermore explored how to calculate through resampling as it is not particularly nice with multiple lags. The main parts of the thesis is built around Goldie's paper about tail decay of stochastic fixed point equations.

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**Vejleder:** Jeffrey Collamore**Censor:** Mette M Havning