Stochastic models in finance

Abstract

This thesis reviews the properties of time varying variance or volatility models in finance and reviews the statistical properties of the returns of financial time series. The ability of these models to estimate the volatility of the returns of financial time series is also investigated. It is found that the returns of financial time series have some unique statistical properties. The mean of the returns varies over time around zero, and more importantly, the variance or volatility of the returns also varies over time. The distribution of the returns is symmetric, heavy tailed (leptokurtic) and linearly uncorrelated. However the returns are not independent as they are correlated in non-linear transforms of themselves, especially in the squares and in the logarithm of the squares. The returns also exhibit clustering in the sense that large (small) returns are usually followed by large (small) returns of either sign. The returns also exhibit persistence in the sense that the clustering of returns tend to persist for long time periods. It is found that the time varying mean, while very small in comparison to the returns, is never-the-less significant and that this should be accounted for when one is modelling returns. A non-parametric estimate of the volatility is found by applying a lowess filter to the data. This estimate is used to benchmark the ability of formal models to estimate the volatility of financial returns. The properties of two classes of models that model the variance an a time varying process are looked at. These are the Stochastic Variance (SV) class of models and the Autoregressive Conditional Heteroskedasticity (ARCH) class of models. It is found that the SV classes of models are able to explain many of the statistical properties of financial time series. The model is mean zero, symmetric, exhibits excess kurtosis and hence has heavy tails and is uncorrelated. The model also induces clustering and persistence (in the same sense as stated earlier) in simulated returns, however these simulated returns are more homogeneous than the actual data. The logarithm of the squared returns behave like an ARMA process, and under certain conditions so do the squared returns. The most important feature of the SV model is that the volatility of the returns is modelled as a time varying stochastic process. Because of this the model is an unseen time series component model, with the volatility being the unseen component. Moreover the volatility of the model is not dependent on past returns alone. Therefore to estimate the volatility and parameters of the model one must apply the Kalman filter to the state space form and use Quasi Maximum likelihood (QML). Another way is to use a Monte Carlo Markov Chain (MCMC) technique to simulate values of the volatility and parameters whose distributions converge to the likelihood of the returns. The estimated volatilities are considered to be estimates of the long run volatility as they are updated as more information comes to hand and hence they give a smooth estimate of the volatility. The ARCH model is also able to explain the statistical properties of financial time series. The main difference between the two models is that the ARCH model, models the volatility of the returns as a deterministic process, which is dependent directly on the previous returns. Therefore the estimated volatility from an ARCH model is a filtered estimate and hence it is more variable than the estimate from an SV model. Moreover the estimate of volatility from an ARCH model is considered to be a short term estimate of the volatility. Finally the models are applied to daily returns from share market data. It is found that both the models are able to adequately account for the heteroskedasticity, correlation, clustering and persistence in the data. However neither model is able to adequately account for the kurtosis in the data indicating that the specification of a Gaussian likelihood is not correct. The estimated volatility is found to follow a stationary process for each series examined. The SV model however appears to outperform the ARCH model in estimating volatility as the SV estimate of volatility is closer to the non-parametric benchmark of volatility than the ARCH estimate is. The SV model is better equipped to explain the heavy tails in the data than the ARCH model is, as the kurtosis statistic of the corrected returns is smaller for the SV model. Moreover the SV model provides better long run forecasts than the ARCH model, however in the short term the better forecaster appears to be dependent on the data set. Therefore it is concluded, for the data sets used, that the SV model is better able to explain the statistical properties of the returns of financial time series than the ARCH model. However it seems that the choice of model is data dependent and only time and hard work will tell which model is superior, if any.