Electricity prices in competitive markets are extremely volatile with salient features such as mean-reversion and jumps and spikes. Modeling electricity spot prices is essential for asset and project valuation as well as risk management. I introduce the mean-reversion feature into a classical variance gamma model to model the electricity price dynamics as a mean-reverting variance gamma (MRVG) process. Derivative pricing formulae are derived through transform analysis and model parameters are estimated by the generalized method of moments and the Markov Chain Monte Carlo method. A real option approach is proposed to value a tolling contract incorporating operational characteristics of the generation asset and contractual constraints. Two simulation-based methods are proposed to solve the valuation problem. The effects of different electricity price assumptions on the valuation of tolling contracts are examined. Based on the valuation model, I also propose a heuristic scheme for hedging tolling contracts and demonstrate the validity of the hedging scheme through numerical examples. Autoregressive Conditional Heteroscedasticity (ARCH) and Generalized ARCH (GARCH) models are widely used to model price volatility in financial markets. Considering a GARCH model with heavy-tailed innovations for electricity price, I characterize the limiting distribution of a Value-at-Risk (VaR) estimator of the conditional electricity price distribution, which corresponds to the extremal quantile of the conditional distribution of the GARCH price process. I propose two methods, the normal approximation method and the data tilting method, for constructing confidence intervals for the conditional VaR estimator and assess their accuracies by simulation studies. The proposed approach is applied to electricity spot price data taken from the Pennsylvania-New Jersey-Maryland market to obtain confidence intervals of the empirically estimated Value-at-Risk of electricity prices. Several directions that deserve further investigation are pointed out for future research.

This thesis includes three chapters regarding influence functions, higher moments, and futures hedging. In Chapter 2, the objective is to use an influence function to better understand semi-kurtosis for use in analyzing peakedness and tail heaviness on one side of a distribution. Also, it is shown that both the right side semi-kurtosis and left side semi-kurtosis summed together, equal kurtosis, so the ratio of semi-kurtosis to kurtosis can be used to analyze asymmetry, as an alternative to skewness. In Chapter 3, the objective is to analyze higher moments of daily, weekly, and monthly stock market returns using large stocks, technology stocks, and small cap stocks. Kurtosis is found to be positive (greater than 3) and statistically significant for all of the daily and weekly stock market returns, indicating peakedness and fat tails. Similar to kurtosis, the left side semi-fourth moment (semi-kurtosis) is also found to be positive (greater than 1.5) for all of daily and weekly returns, indicating peakedness and fat tails on the left sides of the distributions. Skewness is found to be both positive and negative in the daily stock returns data, indicating asymmetry but with no consistent patterns. The fifth moment is also used to analyze asymmetry, as an alternative to skewness. The fifth moment and skewness (third moment) sometimes indicate opposite asymmetry results, as evidenced by different signs for the two moments. This is because the exponent of five for the fifth moment amplifies observations further from the mean, more so than the exponent of three for skewness. In Chapter 4, the objective is to analyze research on futures hedging and to identify the major factors affecting the use of futures hedging by commodity producers. A multifactor conceptual model is developed that explains the factors and subfactors that are likely to affect the commodity producers’ hedging decisions. Factors include industry characteristics, business operation characteristics, management characteristics, futures hedging costs, and substitute risk management instruments. This model provides a more complete understanding of the factors and subfactors affecting futures hedging, and should be of interest to academics and practitioners working with hedging models.

This thesis includes three chapters regarding influence functions, higher moments, and futures hedging. In Chapter 2, the objective is to use an influence function to better understand semi-kurtosis for use in analyzing peakedness and tail heaviness on one side of a distribution. Also, it is shown that both the right side semi-kurtosis and left side semi-kurtosis summed together, equal kurtosis, so the ratio of semi-kurtosis to kurtosis can be used to analyze asymmetry, as an alternative to skewness. In Chapter 3, the objective is to analyze higher moments of daily, weekly, and monthly stock market returns using large stocks, technology stocks, and small cap stocks. Kurtosis is found to be positive (greater than 3) and statistically significant for all of the daily and weekly stock market returns, indicating peakedness and fat tails. Similar to kurtosis, the left side semi-fourth moment (semi-kurtosis) is also found to be positive (greater than 1.5) for all of daily and weekly returns, indicating peakedness and fat tails on the left sides of the distributions. Skewness is found to be both positive and negative in the daily stock returns data, indicating asymmetry but with no consistent patterns. The fifth moment is also used to analyze asymmetry, as an alternative to skewness. The fifth moment and skewness (third moment) sometimes indicate opposite asymmetry results, as evidenced by different signs for the two moments. This is because the exponent of five for the fifth moment amplifies observations further from the mean, more so than the exponent of three for skewness. In Chapter 4, the objective is to analyze research on futures hedging and to identify the major factors affecting the use of futures hedging by commodity producers. A multifactor conceptual model is developed that explains the factors and subfactors that are likely to affect the commodity producers’ hedging decisions. Factors include industry characteristics, business operation characteristics, management characteristics, futures hedging costs, and substitute risk management instruments. This model provides a more complete understanding of the factors and subfactors affecting futures hedging, and should be of interest to academics and practitioners working with hedging models.

It is well established that stock market volatility has a memory of the past, moreover it is found that volatility correlations are long ranged. As a consequence, volatility cannot be characterized by a single correlation time in general. Recent empirical work suggests that the volatility correlation functions of various assets actually decay as a power law. Moreover it is well established that the distribution functions for the returns do not obey a Gaussian distribution, but follow more the type of distributions that incorporate what are commonly known as fat–tailed distributions. As a result, if one is to model the evolution of the stock price, stock market or any financial derivative, then standard Brownian motion models are inaccurate. One must take into account the results obtained from empirical studies and work with models that include realistic features observed on the market. In this thesis we show that it is possible to derive the path integral for a non-Gaussian option pricing model that can capture fat–tails. However we find that the path integral technique can only be used on a very small set of problems, as a number of situations of interest are shown to be intractable. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1378473 / Thesis (Ph.D.) -- University of Adelaide, School of Electrical and Electronic Engineering, 2010

It is well established that stock market volatility has a memory of the past, moreover it is found that volatility correlations are long ranged. As a consequence, volatility cannot be characterized by a single correlation time in general. Recent empirical work suggests that the volatility correlation functions of various assets actually decay as a power law. Moreover it is well established that the distribution functions for the returns do not obey a Gaussian distribution, but follow more the type of distributions that incorporate what are commonly known as fat–tailed distributions. As a result, if one is to model the evolution of the stock price, stock market or any financial derivative, then standard Brownian motion models are inaccurate. One must take into account the results obtained from empirical studies and work with models that include realistic features observed on the market. In this thesis we show that it is possible to derive the path integral for a non-Gaussian option pricing model that can capture fat–tails. However we find that the path integral technique can only be used on a very small set of problems, as a number of situations of interest are shown to be intractable. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1378473 / Thesis (Ph.D.) -- University of Adelaide, School of Electrical and Electronic Engineering, 2010

The thesis aims to investigate some of the approaches to modelling portfolio returns with heavy-tailed risk factors. It first elaborates on the univariate time series models, and compares the benchmark model (GARCH with Student t innovations or its GJR extension) predictive performance with its two competitors, the EVT-GARCH model and the Markov-Switching Multifractal (MSM) model. The motivation of EVT extension of GARCH specification is to use a more proper distribution of the innovations, based on the empirical distribution function. The MSM is one of the best performing models in the multifractal literature, a markov-switching model which is unique by its parsimonious specification and variability. The performance of these models is assessed with Mincer-Zarnowitz regressions as well as by comparison of quality of VaR and expected shortfall predictions, and the empirical analysis shows that for the risk management purposes the EVT-GARCH dominates the benchmark as well as the MSM. The second part addresses the dependence structure modelling, using the Gauss and t-copula to model the portfolio returns and compares the result with the classic variance-covariance approach, concluding that copulas offer a more realistic estimates of future extreme quantiles.

We consider tail empirical processes for long memory stochastic volatility models with heavy tails and leverage. We show a dichotomous behaviour for the tail empirical process with fixed levels, according to the interplay between the long memory parameter and the tail index; leverage does not play a role. On the other hand, the tail empirical process with random levels is not affected by either long memory or leverage. The tail empirical process with random levels is used to construct a family of estimators of the tail index, including the famous Hill estimator and harmonic moment estimators. The limiting behaviour of these estimators is not affected by either long memory or leverage. Furthermore, we consider estimators of risk measures such as Value-at-Risk and Expected Shortfall. In these cases, the limiting behaviour is affected by long memory, but it is not affected by leverage. The theoretical results are illustrated by simulation studies.

Statistical inference is a tool for creating models that can accurately display real-world events. Special importance is given to the financial methods that model risk and large price movements. A parameter that describes tail heaviness, and risk overall, is α. This research finds a representative distribution that models α. The absolute value of standardized stock returns from the Center for Research on Security Prices are used in this research. The inference is performed using R. Approximations for α are found using the ptsuite package. The GAMLSS package employs maximum likelihood estimation to estimate distribution parameters using the CRSP data. The distributions are selected by using AIC and worm plots. The Skew t family is found to be representative for the parameter α based on subsets of the CRSP data. The Skew t type 2 distribution is robust for multiple subsets of values calculated from the CRSP stock return data.

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