Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Harmonic Analysis and the Theory of Probability. By S. Bochner pp. 176. 35s. 1955. (California University Press and Cambridge University Press)

Risks embedded in asset price dynamics are taken to be accumulations of surprise jumps. A Markov pure jump model is formulated on making variance gamma parameters deterministic functions of the price level. Estimation is done by matrix exponentiation of the transition rate matrix for a continuous time finite state Markov chain approximation. The motion is decomposed into a space dependent drift and a space dependent martingale component. Though there is some local mean reversion in the drift, space dependence of the martingale component renders the dynamics to be of the momentum type. Local risk is measured using market calibrated measure distortions that introduce risk charges into the lower and upper prices of two price economies. These risks are compensated by the exponential variation of space dependent arrival rates. Estimations are conducted for the S&P 500 index (S
 P
 X), the exchange traded fund for the financial sector (X
 L
 F), J. P. Morgan stock prices (J
 P
 M), the ratio of JPM to XLF, and the ratio of XLF to SPX.

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Measure distorted arrival rate risks and their rewards

Numerical approximations of optimal portfolios in mispriced asymmetric Lévy markets

Asset price dynamics are taken to be accumulations of surprise jumps in the logarithm of prices. A Markov pure jump model is formulated on making variance gamma parameters deterministic functions of the price level. Estimation is done by matrix exponentiation of the transition rate matrix for a continuous time finite state Markov chain approximation. The motion is decomposed into a space dependent drift and a space dependent martingale component. Though there is some local mean reversion by and large the dynamics estimated is that of the momentum type. Risk compensation is seen by a linear relation between the exponential variation and measure distorted variations for the bid and ask prices of two price economies. Estimations are conducted for the S&P 500 index (SPX), the exchange traded fund for the financial sector (XLF), J. P. Morgan stock prices (JPM), the ratio of JPM to XLF and the ratio of XLF to SPX.

Arrival Rate Functions

In this dissertation we develop a spatially inhomogeneous Markov process as a model for financial asset prices. This model is called the Hunt variance gamma process. We define it via its infinitesimal generator and prove that this generator induces a unique measure on the space ofcádì ag functions. We next describe a procedure to do computations with this model by finding a continuous-time Markov chain approximation. This approximation is used to calibrate the model to fit the S&P 500 futures option surface. Next we investigate specific characteristics of the process, showing how it differs from both Lévy and Sato processes. We conclude by using the calibrated model to answer questions about properties of the risk-neutral distribution of future stock prices. We observe a more accurate fit to the risk-neutral term structure of volatility, skewness, and kurtosis, and the presence of mean-reversion in conditional probabilities involving large jumps. Dedication There is only one person to whom I can dedicate this thesis. And though she will probably never read more than a page of it, I couldn't have done it without her. Mary, thanks for everything. I love you. ii Acknowledgments There are so many people who have contributed either directly or indirectly to this dissertation. It is impossible for me to acknowledge everyone who has helped me in some manner during my time at the University of Maryland. Nevertheless, I would be negligent if I didn't at least make an effort to mention those who have contributed so much. To start, I have to mention my adviser, Dr. Dilip Madan. His enthusiasm and expertise piqued my interest in mathematical finance initially. As I learned more, I realized just how knowledgeable he was. But what impressed me most was his willingness to share that knowledge with his students in seminars and private conversations. Without his ideas, suggestions, and recommendations, I never would have reached this point. I would also like to acknowledge the members of my committee for their work in reviewing and improving my work. and Dr. Alt, thank you for taking time to help me. During the last few years, Dr. Balan has been especially kind in assisting me with a number of administrative details, for which I thank him. Along these lines, I am grateful to the entire Norbert Wiener Center for allowing me to attend conferences, classes, and seminars throughout my time in Maryland. It is fitting that …

Bryant Angelos

Papers

The hunt variance gamma process with applications to option pricing