This thesis includes three chapters on electricity and natural gas prices. In the first chapter, we give a brief introduction to the characteristics of power prices and propose a mean reversion jump diffusion model, in which jump intensity depends on temperature data and overall system load, to model electricity prices. Compared to the models used in the literature, we find the model proposed in this chapter is better to capture the tail behavior in the electricity prices. In the second chapter, we use the model proposed in the first chapter to simulate the spark spread option and value the power generations. In order to simulate power generation, we first propose and estimate mean reversion jump diffusion model for natural gas prices, in which jump intensity is defined as a function of temperature and storage. Combing the model with the electricity models in chapter 1, we find that the value of power generation is closer to the real value of the power plants as reflected in the recent market transaction than one obtains from many other models used in literature. The third chapter investigates extremal dependence among the energy market. We find a tail dependence that exceeds the Pearson correlation ρ, which means the traditional Pearson correlation is not appropriate to model tail behavior of oil, natural gas and electricity prices. However, asymptotic dependence is rejected in all pairs except Henry Hub gas return and Houston Ship Channel gas return. We also find that extreme value dependence in energy market is stronger in bull market than that in bear market due to the special characteristics in energy market, which conflicts the accepted wisdom in equity market that tail correlation is much higher in periods of volatile markets from previous literature.

The purpose of this paper is to explore the information content in a trading, confirm the relationship between information-trading probability (PIN) and asset returns, and apply PIN to construct an investing strategy on a point of uninformed trader¡¦s view. I develop a decision marking model about trading decision between under order-driven market which is combined on the decision tree of the concept of D. Easley et al. (1997) and Merton (1976) jump diffusion model for modifying the PIN model to apply to order-driven market. As a result, the daily PIN were positive relatively with return, and the investing strategy which was based my model could make profit significantly in the sample period at TWSE in 2003, this investing strategy can earn profit in down and up market condition both. This result supports that hedging against information asymmetric risk is potential.

The main contribution of this thesis is to derive the properties and present a closed from solution of the exotic options under some specific types of Levy processes, such as American put options, American call options, British put options, British call options and American knock-out put options under either double exponential jump-diffusion processes or one-sided exponential jump-diffusion processes. Compared to the geometric Brownian motion, exponential jump-diffusion processes can better incorporate the asymmetric leptokurtic features and the volatility smile observed from the market. Pricing the option with early exercise feature is the optimal stopping problem to determine the optimal stopping time to maximize the expected options payoff. Due to the Markovian structure of the underlying process, the optimal stopping problem is related to the free-boundary problem consisting of an integral differential equation and suitable boundary conditions. By the local time-space formula for semi-martingales, the closed form solution for the options value can be derived from the free-boundary problem and we characterize the optimal stopping boundary as the unique solution to a nonlinear integral equation arising from the early exercise premium (EEP) representation. Chapter 2 and Chapter 3 discuss American put options and American call options respectively. When pricing options with early exercise feature under the double exponential jump-diffusion processes, a non-local integral term will be found in the infinitesimal generator of the underlying process. By the local time-space formula for semi-martingales, we show that the value function and the optimal stopping boundary are the unique solution pair to the system of two integral equations. The significant contributions of these two chapters are to prove the uniqueness of the value function and the optimal stopping boundary under less restrictive assumptions compared to previous literatures. In the degenerate case with only one-sided jumps, we find that the results are in line with the geometric Brownian motion models, which extends the analytical tractability of the Black-Scholes analysis to alternative models with jumps. In Chapter 4 and Chapter 5, we examine the British payoff mechanism under one-sided exponential jump-diffusion processes, which is the first analysis of British options for process with jumps. We show that the optimal stopping boundaries of British put options with only negative jumps or British call options with only positive jumps can also be characterized as the unique solution to a nonlinear integral equation arising from the early exercise premium representation. Chapter 6 provides the study of American knock-out put options under negative exponential jump-diffusion processes. The conditional memoryless property of the exponential distribution enables us to obtain an analytical form of the arbitrage-free price for American knock-out put options, which is usually more difficult for many other jump-diffusion models.

We utilise several asset pricing models that allow for discontinuities in the returns and volatility time series in order to obtain estimates of Value-at-Risk (VaR). The first class of model that we use mixes a continuous diffusion process with discrete jumps at random points in time (Poisson Jump Diffusion Model). We also apply a purely discontinuous model that does not contain any continuous component at all in the underlying distribution (Variance Gamma Model). These models have been shown to have some success in capturing certain characteristics of return distributions, a few being leptokurtosis and skewness. Calibrating these models onto the returns of an index of Australian stocks (All Ordinaries Index), we then use the resulting parameters to obtain daily estimates of VaR. In order to obtain the VaR estimates for the Poisson Jump Diffusion Model and the Variance Gamma Model, we introduce the use of an innovation from option pricing techniques, which concentrates on the more tractable characteristic functions of the models. Having then obtained a series of VaR estimates, we then apply a variety of criteria to assess how each model performs and also evaluate these models against the traditional approaches to calculating VaR, such as that suggested by J.P. Morgan???s RiskMetrics. Our results show that whilst the Poisson Jump Diffusion model proved the most accurate at the 95% VaR level, neither the Poisson Jump Diffusion or Variance Gamma models were dominant in the other performance criteria examined. Overall, no model was clearly superior according to all the performance criteria analysed, and it seems that the extra computational time required to calibrate the Poisson Jump Diffusion and Variance Gamma models for the purposes of VaR estimation do not provide sufficient reward for the additional effort than that currently employed by Riskmetrics.

This dissertation investigates the volatility of the relationships between exchange rates and interest rates. The first part of the paper explores the transmission relationship between these two markets using a time-series model. Previous studies have assumed that covariance was constant in both markets. However, if the volatilities of the exchange rate and interest rate markets are correlated over time, the interaction and spillover effects between the two markets may be affected by time-varying covariance. Hence, this paper utilizes the BEKK-GARCH model developed by Engle and Kroner (1995) to capture the dynamic relationship between the exchange rates and interest rates. This study uses the returns data for G7 members¡¦ exchange rates and interest rates to test whether these markets exhibited volatilities spillover from 1978 to 2009. The results show bi-directional volatility spillovers in the markets of the UK, the Euro countries, and Canada, where the volatilities of the two markets were interrelated. The second part of the paper explores the relationship between exchange rates and interest rates using a jump diffusion model. Previous studies assumed that the dynamic processes of exchange rates and interest rates follow a diffusion process with a continuous time path, but an increasing number of empirical studies have shown that a continuous diffusion stochastic model does not capture the dynamic process of these variables. Thus, this paper investigates the discontinuous variables of exchange rates and interest rates and assumes that these variables follow a jump diffusion process. The UIRP model is employed to explore the relationship between both variables and to divide the systematic risk into systematic continuous risk and systematic jump risk. The returns data for G7 members¡¦ exchange rates and interest rates from 2005 to 2010 were analyzed to test whether the expected exchange rate is affected by jump components when the interest rate market experiences a jump. The results show that the jump diffusion model has more explanatory power than the pure diffusion model does, and, when the interest rate market experiences a jump risk, the systematic jump risk has a significant relationship with the expected exchange rates in some G7 countries.

In this study, we use the McKean's integral equation to evaluate the American option price for constant jump di

Most of the derivative pricing models are developed in the jump diffusion models, and many literatures assume those jumps are diversifiable. However, we find many risk cannot be avoided through diversification. In this paper, we extend the Jarrow and Yildirim model to consider the existence of systematic jump risk in nominal interest rate, real interest rate and inflation rate to derive the no-arbitrage condition by using Esscher transformation. In addition, this study also derives the value of TIPS and TIPS European call option. Furthermore, we use the econometric theory to decompose TIPS market price volatility into a continuous component and a jump component. We find the jump component contribute most of the TIPS market price volatility. In addition, we also use the TIPS yield index to obtain the systematic jump component and systematic continuous component to find the systematic jump beta and the systematic continuous beta. The results show that the TIPS with shorter time to maturity are more vulnerable to systematic jump risk. In contrast, the individual TIPS with shorter time to maturity is more vulnerable to systematic jump. Finally, the sensitive analysis is conducted to detect the impacts of jumps risk on the value of TIPS European call option.

We utilise several asset pricing models that allow for discontinuities in the returns and volatility time series in order to obtain estimates of Value-at-Risk (VaR). The first class of model that we use mixes a continuous diffusion process with discrete jumps at random points in time (Poisson Jump Diffusion Model). We also apply a purely discontinuous model that does not contain any continuous component at all in the underlying distribution (Variance Gamma Model). These models have been shown to have some success in capturing certain characteristics of return distributions, a few being leptokurtosis and skewness. Calibrating these models onto the returns of an index of Australian stocks (All Ordinaries Index), we then use the resulting parameters to obtain daily estimates of VaR. In order to obtain the VaR estimates for the Poisson Jump Diffusion Model and the Variance Gamma Model, we introduce the use of an innovation from option pricing techniques, which concentrates on the more tractable characteristic functions of the models. Having then obtained a series of VaR estimates, we then apply a variety of criteria to assess how each model performs and also evaluate these models against the traditional approaches to calculating VaR, such as that suggested by J.P. Morgan???s RiskMetrics. Our results show that whilst the Poisson Jump Diffusion model proved the most accurate at the 95% VaR level, neither the Poisson Jump Diffusion or Variance Gamma models were dominant in the other performance criteria examined. Overall, no model was clearly superior according to all the performance criteria analysed, and it seems that the extra computational time required to calibrate the Poisson Jump Diffusion and Variance Gamma models for the purposes of VaR estimation do not provide sufficient reward for the additional effort than that currently employed by Riskmetrics.

In this thesis, we investigate two stock price models, the Black-Scholes (BS) model and the Merton Jump-Diffusion (MJD) model. Comparing the logarithmic return of the BS model and the MJD model with empirical stock price data, we conclude that the Merton Jump-Diffusion Model is substantially more suitable for the stock market. This is concluded visually not only by comparing the density functions but also by analyzing mean, variance, skewness and kurtosis of the log-returns. One technical contribution to the thesis is a suggested decision rule for initial guess of a maximum likelihood estimation of the MJD-modeled parameters.

The thesis focuses on pricing complex options using Monte Carlo simulations. Due to the versatility of the Monte Carlo method, we are able to evaluate option prices with various underlying asset models: jump diffusion models, illiquidity models, stochastic volatility and so on. Both European options and Bermudan options are studied in this thesis.For the jump diffusion model in Merton (1973), we demonstrate European and Bermudan option pricing by the Monte Carlo scheme and extend this to multiple underlying assets; furthermore, we analyse the effect of stochastic volatility.For the illiquidity model in the spirit of Glover (2008), we model the illiquidity impact on option pricing in the simulation study. The four models considered are: the first order feedback model with constant illiquidity and stochastic illiquidity; the full feedback model with constant illiquidity and stochastic illiquidity. We provide detailed explanations for the present of path failures when simulating the underlying asset price movement and suggest some measures to overcome these difficulties.