In this dissertation, we introduce novel exponential jump-diffusion models for pricing options. Firstly, the normal convolution gamma mixture jump-diffusion model is presented. This model generalizes Merton's jump-diffusion and Kou's double exponential jump-diffusion. We show that the normal convolution gamma mixture jump-diffusion model captures some economically important features of the asset price, and that it exhibits heavier tails than both Merton jump-diffusion and double exponential jump-diffusion models. Secondly, the normal convolution double gamma jump-diffusion model for pricing options is presented. We show that under certain configurations of both the normal convolution gamma mixture and the normal convolution double gamma jump-diffusion models, the latter exhibits a heavier left or right tail than the former. For both models, the maximum likelihood procedure for estimating the model parameters under the physical measure is fairly straightforward; moreover, the likelihood function is given in closed form thereby eliminating the need to embed a probability density function recovery procedure such as the fast Fourier transform or the Fourier-cosine expansion methods in the parameter estimation procedure. In addition, both models can reproduce the implied volatility surface observed in the options data and provide a good fit to the market-quoted European option prices.

Jump-diffusion models can under certain assumptions be expressed as partial integro-differential equations (PIDE). Such a PIDE typically involves a convection term and a nonlocal integral like for the here considered models of Merton and Kou. We transform the PIDE to eliminate the convection term, discretize it implicitly using finite differences and the second order backward difference formula (BDF2) on a uniform grid. The arising dense linear system is solved by an iterative method, either a splitting technique or a circulant preconditioned conjugate gradient method. Exploiting the Fast Fourier Transform (FFT) yields the solution in only $O(n\log n)$ operations and just some vectors need to be stored. Second order accuracy is obtained on the whole computational domain for Merton's model whereas for Kou's model first order is obtained on the whole computational domain and second order locally around the strike price. The solution for the PIDE with convection term can oscillate in a neighborhood of the strike price depending on the choice of parameters, whereas the solution obtained from the transformed problem is stabilized. / Master of Science

In this thesis, modelling with L&eacute / vy processes is considered in three parts. In the first part, the general geometric L&eacute / vy market model is examined in detail. As such markets are generally incomplete, it is shown that the market can be completed by enlarging with a series of new artificial assets called &ldquo / power-jump assets&rdquo / based on the power-jump processes of the underlying L&eacute / vy process. The second part of the thesis presents two different methods for pricing European options: the martingale pricing approach and the Fourier-based characteristic formula method which is performed via fast Fourier transform (FFT). Performance comparison of the pricing methods led to the fact that the fast Fourier transform produces very small pricing errors so the results of both methods are nearly identical. Throughout the pricing section jump sizes are assumed to have a particular distribution. The third part contributes to the empirical applications of L&eacute / vy processes. In this part, the stochastic volatility extension of the jump diffusion model is considered and calibration on Standard&amp / Poors (S&amp / P) 500 options data is executed for the jump-diffusion model, stochastic volatility jump-diffusion model of Bates and the Black-Scholes model. The model parameters are estimated by using an optimization algorithm. Next, the effect of additional stochastic volatility extension on explaining the implied volatility smile phenomenon is investigated and it is found that both jumps and stochastic volatility are required. Moreover, the data fitting performances of three models are compared and it is shown that stochastic volatility jump-diffusion model gives relatively better results.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Chih-Hang Wu / In recent years the stock markets have shown tremendous volatility with significant spikes and drops in the stock prices. Within the past decade, there have been numerous jumps in the market; one key example was on September 17, 2001 when the Dow industrial average dropped 684 points following the 9-11 attacks on the United States. These evident jumps in the markets show the inaccuracy of the Black-Scholes model for pricing options. Merton provided the first research to appease this problem in 1976 when he extended the Black-Scholes model to include jumps in the market. In recent years, Kou has shown that the distribution of the jump sizes used in Merton’s model does not efficiently model the actual movements of the markets. Consequently, Kou modified Merton’s model changing the jump size distribution from a normal distribution to the double exponential distribution. Kou’s research utilizes mathematical equations to estimate the value of an American put option where the underlying stocks follow a jump-diffusion process. The research contained within this thesis extends on Kou’s research using Monte Carlo simulation (MCS) coupled with least-squares regression to price this type of American option. Utilizing MCS provides a continuous exercise and pricing region which is a distinct difference, and advantage, between MCS and other analytical techniques. The aim of this research is to investigate whether or not MCS is an efficient means to pricing American put options where the underlying stock undergoes a jump-diffusion process. This thesis also extends the simulation to utilize copulas in the pricing of baskets, which contains several of the aforementioned type of American options. The use of copulas creates a joint distribution from two independent distributions and provides an efficient means of modeling multiple options and the correlation between them. The research contained within this thesis shows that MCS provides a means of accurately pricing American put options where the underlying stock follows a jump-diffusion. It also shows that it can be extended to use copulas to price baskets of options with jump-diffusion. Numerical examples are presented for both portions to exemplify the excellent results obtained by using MCS for pricing options in both single dimension problems as well as multidimensional problems.

Value-at-Risk (VaR) model fails to predict financial risk accurately especially during financial crises. This is mainly due to the model’s inability to calibrate new market information and the fact that the risk measure is characterised by poor tail risk quantification. An alternative approach which comprises of the Expected Shortfall measure and the Lognormal Jump-Diffusion (LJD) model has been developed to address the aforementioned shortcomings of VaR. This model is called the Simulated-Expected-Shortfall (SES) model. The Maximum Likelihood Estimation (MLE) approach is used in determining the parameters of the LJD model since it’s more reliable and authenticable when compared to other nonconventional parameters estimation approaches mentioned in other literature studies. These parameters are then plugged into the LJD model, which is simulated multiple times in generating the new loss dataset used in the developed model. This SES model is statistically conservative when compared to peers which means it’s more reliable in predicting financial risk especially during a financial crisis. / Statistics / M.Sc. (Statistics)

The jump phenomenons of many financial assets prices have been observed in many empirical papers. In this paper we extend the Heath-Jarrow-Morton model to include the jump component to derive the European-style pricing formula of the interest rate futures options. We use numerical method to simulate the options prices and analyze how each component of HJM model under jump-diffusion processes affects the interest rate futures options. Finally, we utilize LSM method which are presented by Longstaff and Schwartz to derive American options prices and compare it with European options.

<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.

Jump diffusion models are being used more and more often in financial applications. Consisting of a Brownian motion (with drift) and a jump component, such models have a number of parameters that have to be set at some level. Maximum Likelihood Estimation (MLE) turns out to be suitable for this task, however it is computationally demanding. For a complicated likelihood function it is seldom possible to find derivatives. The global maximum of a likelihood function defined for a jump diffusion model can however, be obtained by numerical methods. I chose to use the Bound Optimization BY Quadratic Approximation (BOBYQA) method which happened to be effective in this case. However, results of Maximum Likelihood Estimation (MLE) proved to be hard to interpret.

In the literature, many methods are proposed to value American options. However, due to computational difficulty, there are only approximate solution or numerical method to evaluate American options. It is not easy for general investors either to understand nor to apply. In this thesis, we build up an option pricing and virtual asset model system, which provides a friendly environment for general public to calculate early exercise boundary of an American option. This system modularize the well-handled pricing models to provide the investors an easy way to value American options without learning difficult financial theories. The system consists two parts: the first one is an option pricing system, the other one is an asset model simulation system. The option pricing system provides various option pricing methods to the users; the virtual asset model system generates virtual asset prices for different underlying models.