This article examines the hedging positions derived from the Black-Scholes(B-S) model and the GARCH(1,1) models, respectively, when the log returns of underlying asset exhibits GARCH(1,1) process. The result shows that Black-Scholes and GARCH options deltas, one of the hedging parameters, are similar for near-the-money options, and Black-Scholes options delta is higher then GARCH delta in absolute terms when the options are deep out-of-money, and Black-Scholes options delta is lower then GARCH delta in absolute terms when the options are deep in-the-money. Simulation study of hedging procedure of GARCH(1,1) and B-S models are performed, which also support the above findings.

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We analyze four alternative energy mutual funds using a multi-factor capital asset pricing model with generalized autoregressive conditionally heteroskedastic errors (CAPM-GARCH). Our findings will help portfolio managers and others who seek to predict the return on investment in alternative energy firms. We find that alternative energy firms tend to be riskier than the general US stock market, have a low, but significant and positive response to oil prices, and have a significantly high and negative response to the value of the dollar relative to other currencies. Our results also suggest that alternative energy firms should hedge against currency exchange rate fluctuation.

This thesis investigates the predictability of trading strategies in the European and American stock market from 2001 to 2013. More specific, we examine the indices CAC40, DAX, FTSE100, NASDAQ and S&P500 first with the simple moving averages, then with trading rules based on the forecasts of time – series models and finally with the combination of the technical trading rules and time –series models. The significance of the examined trading rules tested with standard t – tests. The standard tests results show that technical trading rules are the most profitable strategy, second follows the combined and then the time – series rules as the least profitable trading strategy related to buy – and – hold strategy. / Σκοπός της παρούσας εργασίας είναι η διερεύνηση της προβλεπτικής δυνατότητας στρατηγικών επενδύσεων που εφαρμόζονται στην Ευρωπαϊκή και Αμερικάνικη χρηματιστηριακή αγορά, για τη χρονική περίοδο 2001-2013. Πιο συγκεκριμένα θα εξετασθούν οι δείκτες CAC 40, DAX, FTSE 100, NASDAQ και S&P 500, με κανόνες κινητών μέσων όρων, με κανόνες που βασίζονται σε μοντέλα πρόβλεψης χρονολογικών σειρών και με κανόνες συνδυαστικών των δύο ανωτέρω. Οι παραπάνω στρατηγικές θα συγκριθούν με την στρατηγική διακράτησης (Buy-and-Hold), που έχει ορισθεί ως benchmark στρατηγική και η σημαντικότητα των αποτελεσμάτων θα εξετασθεί με στατιστικούς ελέγχους t-statistics.

The global foreign exchange market is the largest financial market with turnover in this market often outstripping the GDP of countries in which they are located. The dynamics in the foreign exchange market, especially price dynamics, have huge implications for financial asset values, financial returns and volatility in the international financial system. It is therefore an important area of study. Exchange rates have often departed significantly from the level implied by fundamentals and exhibit excessive volatility. This reality creates a role for central bank intervention in this market to keep the rate in line with economic fundamentals and the overall policy mix, to stabilize market expectations and to calm disorderly markets. Studies that attempt to measure the effectiveness of intervention in the foreign exchange market in terms of exchange rate trends and volatility have had mixed results. This, in many cases, reflects the unavailability of data and the weaknesses in the empirical frameworks used to measure effectiveness. This thesis utilises the most recent data available and some of the latest methodological advances to measure the effectiveness of central bank intervention in the foreign exchange markets of a variety of countries. It therefore makes a contribution in the area of applied empirical methodologies for the measurement of the dynamics of intervention in the foreign exchange market. It demonstrates that by using high frequency data and more robust and appropriate empirical methodologies central bank intervention in the foreign exchange market can be effective. Moreover, a framework that takes account of the interactions between different central bank policy instruments and price dynamics, the reaction function of the central bank, different states of the market, liquidity in the market and the profitability of the central bank can improve the effectiveness of measuring the impact of central bank policy in the foreign exchange market and provide useful information to policy makers.

In classic GARCH models for financial returns the innovations are usually assumed to be normally distributed. However, it is generally accepted that a non-normal innovation distribution is needed in order to account for the heavier tails often encountered in financial returns. Since the structure of the normal inverse Gaussian (NIG) distribution makes it an attractive alternative innovation distribution for this purpose, we extend the normal GARCH model by assuming that the innovations are NIG-distributed. We use the normal variance mixture interpretation of the NIG distribution to show that a NIG innovation may be interpreted as a normal innovation coupled with a multiplicative random impact factor adjustment of the ordinary GARCH volatility. We relate this new volatility estimate to realised volatility and suggest that the random impact factors are due to a news noise process influencing the underlying returns process. This GARCH model with NIG-distributed innovations leads to more accurate parameter estimates than the normal GARCH model. In order to obtain even more accurate parameter estimates, and since we expect an information gain if we use more data, we further extend the model to cater for high, low and close data, as well as full intraday data, instead of only daily returns. This is achieved by introducing the Brownian inverse Gaussian (BIG) process, which follows naturally from the unit inverse Gaussian distribution and standard Brownian motion. Fitting these models to empirical data, we find that the accuracy of the model fit increases as we move from the models assuming normally distributed innovations and allowing for only daily data to those assuming underlying BIG processes and allowing for full intraday data. However, we do encounter one problematic result, namely that there is empirical evidence of time dependence in the random impact factors. This means that the news noise processes, which we assumed to be independent over time, are indeed time dependent, as can actually be expected. In order to cater for this time dependence, we extend the model still further by allowing for autocorrelation in the random impact factors. The increased complexity that this extension introduces means that we can no longer rely on standard Maximum Likelihood methods, but have to turn to Simulated Maximum Likelihood methods, in conjunction with Efficient Importance Sampling and the Control Variate variance reduction technique, in order to obtain an approximation to the likelihood function and the parameter estimates. We find that this time dependent model assuming an underlying BIG process and catering for full intraday data fits generated data and empirical data very well, as long as enough intraday data is available. / Thesis (Ph.D. (Risk Analysis))--North-West University, Potchefstroom Campus, 2006.

The objective of this dissertation is to model the volatility of financial time series data using ARCH, GARCH and stochastic volatility models. It is found that the ARCH and GARCH models are easy to fit compared to the stochastic volatility models which present problems with respect to the distributional assumptions that need to be made. For this reason the ARCH and GARCH models remain more widely used than the stochastic volatility models. The ARCH, GARCH and stochastic volatility models are fitted to four data sets consisting of daily closing prices of gold mining companies listed on the Johannesburg stock exchange. The companies are Anglo Gold Ashanti Ltd, DRD Gold Ltd, Gold Fields Ltd and Harmony Gold Mining Company Ltd. The best fitting ARCH and GARCH models are identified along with the best error distribution and then diagnostics are performed to ensure adequacy of the models. It was found throughout that the student-t distribution was the best error distribution to use for each data set. The results from the stochastic volatility models were in agreement with those obtained from the ARCH and GARCH models. The stochastic volatility models are, however, restricted to the form of an AR(1) process due to the complexities involved in fitting higher order models. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

This dissertation consists of three essays. In the first essay, I estimate a high dimensional covariance matrix of returns for 88 individual stocks from the S&P 100 index, using daily return data for 1995-2005. This study applies the two-step estimator of the dynamic conditional correlation multivariate GARCH model, proposed by Engle (2002b) and Engle and Sheppard (2001) and applies variations of this model. This is the first study estimating variances and covariances of returns using a large number of individual stocks (e.g., Engle and Sheppard (2001) use data on various aggregate sub-indexes of stocks). This avoids errors in estimation of GARCH models with contemporaneous aggregation of stocks (e.g. Nijman and Sentana 1996; Komunjer 2001). Second, this is the first multivariate GARCH adopting a systematic general-to-specific approach to specification of lagged returns in the mean equation. Various alternatives to simple GARCH are considered in step one univariate estimation, and econometric results favour an asymmetric EGARCH extension of Engle and Sheppard’s model. In essay two, I aggregate a variance-covariance matrix of return risk (estimated using DCC-MVGARCH in essay one) to an aggregate index of return risk. This measure of risk is compared with the standard approach to measuring risk from a simple univariate GARCH model of aggregate returns. In principle the standard approach implies errors in estimation due to contemporaneous aggregation of stocks. The two measures are compared in terms of correlation and economic values: measures are not perfectly correlated, and the economic value for the improved estimate of risk as calculated here is substantial. Essay three has three parts. The major part is an empirical study of the aggregate risk return tradeoff for U.S. stocks using daily data. Recent research indicates that past risk-return studies suffer from inadequate sample size, and this suggests using daily rather than monthly data. Modeling dynamics/lags is critical in daily models, and apparently this is the first such study to model lags correctly using a general to specific approach. This is also the first risk return study to apply Wu tests for possible problems of endogeneity/measurement error for the risk variable. Results indicate a statistically significant positive relation between expected returns and risk, as is predicted by capital asset pricing models. Development of the Wu test leads naturally into a model relating aggregate risk of returns to economic variables from the risk return study. This is the first such model to include lags in variables based on a general to specific methodology and to include covariances of such variables. I also derive coefficient links between such models and risk-return models, so in theory these models are more closely related than has been realized in past literature. Empirical results for the daily model are consistent with theory and indicate that the economic and financial variables explain a substantial part of variation in daily risk of returns. The first section of this essay also investigates at a theoretical and empirical level several alternative index number approaches for aggregating multivariate risk over stocks. The empirical results indicate that these indexes are highly correlated for this data set, so only the simplest indexes are used in the remainder of the essay.

GARCH models have been widely used in finance to model volatility ever since the introduction of the ARCH model and its extension to the generalized ARCH (GARCH) model. Lately, there has been growing interest in modelling seasonal volatility, most recently with the introduction of the multiplicative seasonal GARCH models. As an application of the multiplicative seasonal GARCH model with real data, call prices from the major stock market index of India are calculated using estimated parameter values. It is shown that a multiplicative seasonal GARCH option pricing model outperforms the Black-Scholes formula and a GARCH(1,1) option pricing formula. A parametric bootstrap procedure is also employed to obtain an interval approximation of the call price. Narrower confidence intervals are obtained using the multiplicative seasonal GARCH model than the intervals provided by the GARCH(1,1) model for data that exhibits multiplicative seasonal GARCH volatility.

In classic GARCH models for financial returns the innovations are usually assumed to be normally distributed. However, it is generally accepted that a non-normal innovation distribution is needed in order to account for the heavier tails often encountered in financial returns. Since the structure of the normal inverse Gaussian (NIG) distribution makes it an attractive alternative innovation distribution for this purpose, we extend the normal GARCH model by assuming that the innovations are NIG-distributed. We use the normal variance mixture interpretation of the NIG distribution to show that a NIG innovation may be interpreted as a normal innovation coupled with a multiplicative random impact factor adjustment of the ordinary GARCH volatility. We relate this new volatility estimate to realised volatility and suggest that the random impact factors are due to a news noise process influencing the underlying returns process. This GARCH model with NIG-distributed innovations leads to more accurate parameter estimates than the normal GARCH model. In order to obtain even more accurate parameter estimates, and since we expect an information gain if we use more data, we further extend the model to cater for high, low and close data, as well as full intraday data, instead of only daily returns. This is achieved by introducing the Brownian inverse Gaussian (BIG) process, which follows naturally from the unit inverse Gaussian distribution and standard Brownian motion. Fitting these models to empirical data, we find that the accuracy of the model fit increases as we move from the models assuming normally distributed innovations and allowing for only daily data to those assuming underlying BIG processes and allowing for full intraday data. However, we do encounter one problematic result, namely that there is empirical evidence of time dependence in the random impact factors. This means that the news noise processes, which we assumed to be independent over time, are indeed time dependent, as can actually be expected. In order to cater for this time dependence, we extend the model still further by allowing for autocorrelation in the random impact factors. The increased complexity that this extension introduces means that we can no longer rely on standard Maximum Likelihood methods, but have to turn to Simulated Maximum Likelihood methods, in conjunction with Efficient Importance Sampling and the Control Variate variance reduction technique, in order to obtain an approximation to the likelihood function and the parameter estimates. We find that this time dependent model assuming an underlying BIG process and catering for full intraday data fits generated data and empirical data very well, as long as enough intraday data is available. / Thesis (Ph.D. (Risk Analysis))--North-West University, Potchefstroom Campus, 2006.