<p>I denna uppsats jämförs hur väl tre historiskt baserade konditionella GARCH-modeller samt en naiv historisk modell predikterar volatiliteten för Standard & Poors 500 Composite index år 2006, jämfört med den optionsbaserade volatiliteten från Chicago Board Option Exchange volatilitetsindex VIX. Både de konditionella, den naiva historiska- och VIX volatilitetsprognoser utvärderas mot den historiskt observerade volatiliteten för urvalsperioden. För att utvärdera respektive modells prediktionsförmåga används tre utvärderingsmått; Regressionsanalys med Walds koefficienttest, det genomsnittliga prognosfelet modellerat med Root Mean Square Error (RMSE), samt Theil’s U-statistikan.</p><p>Undersökningen visar att det är tröskel-GARCH-modellen TGARCH som bäst predikterar volatiliteten för S&P 500 men att en kombination av TGARCH och VIX tillför ytterligare förklaringsgrad till modellen.</p>

I denna uppsats jämförs hur väl tre historiskt baserade konditionella GARCH-modeller samt en naiv historisk modell predikterar volatiliteten för Standard &amp; Poors 500 Composite index år 2006, jämfört med den optionsbaserade volatiliteten från Chicago Board Option Exchange volatilitetsindex VIX. Både de konditionella, den naiva historiska- och VIX volatilitetsprognoser utvärderas mot den historiskt observerade volatiliteten för urvalsperioden. För att utvärdera respektive modells prediktionsförmåga används tre utvärderingsmått; Regressionsanalys med Walds koefficienttest, det genomsnittliga prognosfelet modellerat med Root Mean Square Error (RMSE), samt Theil’s U-statistikan. Undersökningen visar att det är tröskel-GARCH-modellen TGARCH som bäst predikterar volatiliteten för S&amp;P 500 men att en kombination av TGARCH och VIX tillför ytterligare förklaringsgrad till modellen.

This dissertation contains five chapters. An introduction and a summary of the research are given in Chapter 1. The other four chapters present theoretical results on the moment structure of GARCH processes. Some chapters also contain empirical examples in order to illustrate applications of the theory. The focus, however, is mainly on statistical theory. Chapter 2 considers the moments of a family of first-order GARCH processes. First, a general condition of the existence of any integer moment of the absolute values of the observations is given. Second, a general expression of this moments as a function of lower-order moments is derived. Third, the kurtosis and the autocorrelation function of the squared and absolute-valued observations are derived. The results apply to a host of different GARCH parameterizations. Finally, the existence, or the lack of it, of the theoretical counterpart to the so-called Taylor effect for some members of this GARCH family is discussed. The asymmetric power ARCH model is a recent addition to time series models that may be used for predicting volatility. Its performance is compared with that of standard models of conditional heteroskedasticity such as GARCH. This has previously been done empirically. In Chapter 3 the same issue is studied theoretically using unconditional fractional moments for the A-PARCH model that are derived for the purpose. The role of the heteroskedasticity parameter of the A-PARCH process is highlighted and compared with corresponding empirical results involving autocorrelation functions of power-transformed absolute-valued return series.In Chapter 4, a necessary and sufficient condition for the existence of the unconditional fourth moment of the GARCH(p,q) process is given as well as an expression for the moment itself. Furthermore, the autocorrelation function of the centred and squared observations of this process is derived. The statistical theory is further illustrated by a few special cases such as the GARCH(2,2) process and the ARCH(q) process.Nonnegativity constraints on the parameters of the GARCH(p,q) model may be relaxed without giving up the requirement of the conditional variance remaining nonnegative with probability one. Chapter 5 looks into the consequences of adopting these less severe constraints in the GARCH(2,2) case and its two second-order special cases, GARCH(2,1) and GARCH(1,2). This is done by comparing the autocorrelation function of squared observations under these two sets of constraints. The less severe constraints allow more flexibility in the shape of the autocorrelation function than the constraints restricting the parameters to be nonnegative. The theory is illustrated by an empirical example. / Revised versions of chapters 2-5 have been published as:He, C. and T. Teräsvirta, "Properties of moments of a amily of GARCH processes" in Journal of Econometrics, Vol. 92, No. 1, 1999, pp173-192.He, C. and T. Teräsvirta, "Statistical Properties of the Asymmetric Power ARCH Process" in R.F. Engle and H. White (eds) Cointegration, causality, and forecasting. Festschrift in honour of Clive W.J. Granger, chapter 19, pp 462-474, Oxford University Press, 1999.He, C. and T. Teräsvirta, "Fourth moment structure of the GARCH(p,q) process" in Econometric Theory, Vol. 15, 1999, pp 824-846.He, C. and T. Teräsvirta, "Properties of the autocorrelation function of squared observations for second order GARCH processes under two sets of parameter constraints" in Journal of Time Series Analysis, Vol. 20, No. 1, January 1999, pp 23-30.

This thesis consists of five self-contained papers, which are all related to the financial markets in the three Baltic States, Estonia, Latvia and Lithuania.  Paper [I] studies the impact of news from the Moscow and New York stock exchanges on the returns and volatilities of the Baltic States' stock market indices using a time series model that accounts for asymmetries in the conditional mean and variance functions. We find that news from New York has stronger e¤ects on returns in Tallinn. High-risk shocks in New York have a stronger impact on volatility in Tallinn, whereas volatility in Vilnius is more in.uenced by high-risk shocks from Moscow. Riga does not seem to be affected by news arriving from abroad. Paper [II] suggests a nonlinear and multivariate time series model framework that enables the study of simultaneity in returns and in volatilities, as well as asymmetric effects arising from shocks and exogenous variables. The model is employed to study the three Baltic States' stock exchanges. Using daily data, we find recursive structures, with returns in Riga, directly depending on returns in Tallinn and Vilnius, and Tallinn on Vilnius. For volatilities, both Riga and Vilnius depend on Tallinn. Paper [III] studies the link between political news, and the returns and volatilities in the Baltic States' stock markets. We find that domestic and foreign non-Russian political news led, on average, to lower uncertainty in the stock markets of Riga and Tallinn in 2001-2003. At the same time, political risk from Russia increased the volatility of the stock market in Tallinn. There is a weak relationship between political risk and the stock market volatility in the Baltic countries in 2004-2007. Paper [IV] studies the impact of market jumps on the time varying return correlations between stock market indices in the Baltic countries. An EARJI-EGARCH model facilitating direct modeling of the time varying return correlations is introduced. The empirical results indicate that there are quite a large number of identified jumps in the emerging Baltic States' stock markets. Isolated market jumps in one of the markets generally have no or small e¤ects on the time-varying correlations. In contrast, simultaneous jumps of equal sign increase the average correlation, in some cases by as much as 100 percent. In Paper [V] the hypothesis that financial development promotes economic growth is tested for the three Baltic countries using a time series approach that allows for interactions between the countries. We find that economic growth is a positive function of financial development, proxied by the amount of bank credit to the private sector, in the long run. The results also show that there is long run interaction between the three Baltic countries.

Condfidence in the future is a signicant factor for business development. However frequently, accurate and specific purposes are spread over the market environment influence.Thus,it is necessary to make an appropriate consideration of instability, which is peculiar to the dynamic development. Volatility, variance and standard deviation are used to characterize the deviation of the investigated quantity from mean value. Volatility is one of the main instruments to measure the risk of the asset. The increasing availability of financial market data has enlarged volatility research potential but has also encouraged research into longer horizon volatility forecasts. In this paper we investigate stock index volatility with respect to capitalization with help of GARCH-modelling. There are chosen three indexes of OMX Nordic Exchange for our research. The Nordic list segment indexes comprising Nordic Large Cap, Mid Cap and Small Cap are based on the three market capitalization groups. We implement GARCH-modeling for considering indexes and compare our results in order to conclude which ones of the indexes is more volatile. The OMX Nordic list indexis quiet new(2002)and reorganized as late as October 2006. The current value is now about 300 and no options do exist. In current work we are also interested in estimation of the Heston model(SVmodel), which is popular in financial world and can be used in option pricing in the future. The results of our investigations show that Large Cap Index is more volatile then Middle and Small Cap Indexes.

In the context of the guarantee liability valuation, the sophisticated fund-of-funds structure, of some Canadian segregated fund products, often requires us to model multiple market indices simultaneously in order to benchmark the return of the underlying fund. In this thesis, we apply multivariate GARCH models with Gaussian and non-Gaussian noise to project the future investment scenarios of the fund. We further conduct a simulation study to investigate the difference, among the proposed multivariate models, in the valuation of the Guaranteed Minimum Maturity Benefit (GMMB) option. Based on the pre-data analysis, the proposed multivariate GARCH models are data driven. The goodness-of-fit for the models is evaluated through formal statistical tests from univariate and multivariate perspectives. The estimation and associated practical issues are discussed in details. The impact from the innovation distributions is addressed. More importantly, we demonstrate an actuarial approach to manage the guarantee liability for complex segregated fund products.

This paper aims to model and forecast the volatility of gold price with the help of other precious metals. The data applied for application part in the article involves three financial time series which are gold, silver and platinum daily spot prices. The volatility is modeled by univariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models including GARCH and EGARCH with different distributions such as normal distribution and student-t distribution. At the same time, comparisons of estimation and forecasting the volatility between GARCH family models have been done.

In the context of the guarantee liability valuation, the sophisticated fund-of-funds structure, of some Canadian segregated fund products, often requires us to model multiple market indices simultaneously in order to benchmark the return of the underlying fund. In this thesis, we apply multivariate GARCH models with Gaussian and non-Gaussian noise to project the future investment scenarios of the fund. We further conduct a simulation study to investigate the difference, among the proposed multivariate models, in the valuation of the Guaranteed Minimum Maturity Benefit (GMMB) option. Based on the pre-data analysis, the proposed multivariate GARCH models are data driven. The goodness-of-fit for the models is evaluated through formal statistical tests from univariate and multivariate perspectives. The estimation and associated practical issues are discussed in details. The impact from the innovation distributions is addressed. More importantly, we demonstrate an actuarial approach to manage the guarantee liability for complex segregated fund products.

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This study attempts to capture the behavior of volatility in the commodity futures market by importing the normal mixture GARCH Model and the NIG mixture GARCH model (Normal-inverse Gaussian Mixture GARCH Model). Normal mixture GARCH Model (what follows called NM-GARCH Model) is a model mixed by two to several normal distributions with a specific weight portfolio, and its variance abide by GAECH process. The ability of capturing the financial data with leptokurtosis and fat-tail of NM-GARCH Model is better than Normal GARCH Model and Student¡¦s t GARCH Model.¡CAlso¡AThe Variance of the factor with lower weight in NM-GARCH Model usually higher, and the volatility of the factor with higher weight is lower, which explains the situation happens in the real market that the probability of large fluctuations (shocks) is small, and the probability of small fluctuations are higher. Generally, the volatilities which keeping occurring in common cases are respectively flat, and the shocks usually bring large impacts but less frequent. NIG Mixture Distribution is a distribution mixed by two to several weighted distributions, and the distribution of every factor abides by NIG Distribution. Compare to Normal Mixture Distribution, NIG Mixture Distribution takes the advantages of NIG Distribution into account, which can not only explain leptokurtosis and the deviation of data, but describe the fat-tail phenomenon more complete as well, because of the both tails of NIG Distribution decreasing slowly. This study will apply the NM GARCH Model and NIG GARCH Model to the Volatility forecasting of the return rates in the crude oil futures market, and infer the predictive abilities of this two kinds of models are significantly better than other volatility model by implementing parameter estimation, forecasting, loss function and statistic significant test.