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.

<p>The purpose of this study is to identify an impact on an option pricing within NASDAQ OMX Stockholm Market, if the underlying</p><p>asset prices include jumps. The current financial crisis, when jumps are much more evident than ever, makes this issue very actual and important in the global sense for the portfolio hedging and other risk management applications for example for the banking sector. Therefore, an investigation is based on OMXS30 Index and SEB A Bank. To detect jumps the Barndorff-Nielsen and Shephard non-parametric bipower variation test is used. First it is examined on simulations, to be finally implemented on the real data. An affirmation of a jumps occurrence requires to apply an appropriate model for the option pricing. For this purpose the Kou model, a double exponential jump-diffusion one, is proposed, as it incorporates essential stylized facts not available for another models. Th parameters in the model are estimated by a new approach - a combined cumulant matching with lambda taken from the Barrndorff-Nielsen and Shephard test. To evaluate how the Kou model manages on the option pricing, it is compared to the Black-Scholes model and to the real prices of European call options from the Stockholm Stock Exchange. The results show that the Kou model outperforms the latter.</p>

The purpose of this study is to identify an impact on an option pricing within NASDAQ OMX Stockholm Market, if the underlying asset prices include jumps. The current financial crisis, when jumps are much more evident than ever, makes this issue very actual and important in the global sense for the portfolio hedging and other risk management applications for example for the banking sector. Therefore, an investigation is based on OMXS30 Index and SEB A Bank. To detect jumps the Barndorff-Nielsen and Shephard non-parametric bipower variation test is used. First it is examined on simulations, to be finally implemented on the real data. An affirmation of a jumps occurrence requires to apply an appropriate model for the option pricing. For this purpose the Kou model, a double exponential jump-diffusion one, is proposed, as it incorporates essential stylized facts not available for another models. Th parameters in the model are estimated by a new approach - a combined cumulant matching with lambda taken from the Barrndorff-Nielsen and Shephard test. To evaluate how the Kou model manages on the option pricing, it is compared to the Black-Scholes model and to the real prices of European call options from the Stockholm Stock Exchange. The results show that the Kou model outperforms the latter.

<p>In this thesis the double exponential jump-diffusion model is considered and the Laplace transform is used as a method for pricing both plain vanilla and path-dependent options. The evolution of the underlying stock prices are assumed to follow a double exponential jump-diffusion model. To invert the Laplace transform, the Euler algorithm is used. The thesis includes the programme code for European options and the application to the real data. The results show how the Kou model performs on the NASDAQ OMX Stockholm Market in the case of the SEB stock.</p>

In this thesis the double exponential jump-diffusion model is considered and the Laplace transform is used as a method for pricing both plain vanilla and path-dependent options. The evolution of the underlying stock prices are assumed to follow a double exponential jump-diffusion model. To invert the Laplace transform, the Euler algorithm is used. The thesis includes the programme code for European options and the application to the real data. The results show how the Kou model performs on the NASDAQ OMX Stockholm Market in the case of the SEB stock.