On A New Class of Tempered Stable Distributions: Moments and Regular Variation

We extend the class of tempered stable distributions first introduced in Rosinski 2007. Our new class allows for more structure and more variety of tail behaviors. We discuss various subclasses and the relation between them. To characterize the possible tails we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs

Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein--Uhlenbeck processes

Properties of the law $\mu$ of the integral $\int_0^{\infty}c^{-N_{t-}}\,dY_t$ are studied, where $c>1$ and $\{(N_t,Y_t),t\geq0\}$ is a bivariate L\'{e}vy process such that $\{N_t\}$ and $\{Y_t\}$ are Poisson processes with parameters $a$ and $b$, respectively. This is the stationary distribution of some generalized Ornstein--Uhlenbeck process. The law $\mu$ is parametrized by $c$, $q$ and $r$, where $p=1-q-r$, $q$, and $r$ are the normalized L\'{e}vy measure of $\{(N_t,Y_t)\}$ at the points $(1,0)$, $(0,1)$ and $(1,1)$, respectively. It is shown that, under the condition that $p>0$ and $q>0$, $\mu_{c,q,r}$ is infinitely divisible if and only if $r\leq pq$. The infinite divisibility of the symmetrization of $\mu$ is also characterized. The law $\mu$ is either continuous-singular or absolutely continuous, unless $r=1$. It is shown that if $c$ is in the set of Pisot--Vijayaraghavan numbers, which includes all integers bigger than 1, then $\mu$ is continuous-singular under the condition $q>0$. On the other hand, for Lebesgue almost every $c>1$, there are positive constants $C_1$ and $C_2$ such that $\mu$ is absolutely continuous whenever $q\geq C_1p\geq C_2r$. For any $c>1$ there is a positive constant $C_3$ such that $\mu$ is continuous-singular whenever $q>0$ and $\max\{q,r\}\leq C_3p$. Here, if $\{N_t\}$ and $\{Y_t\}$ are independent, then $r=0$ and $q=b/(a+b)$.Comment: Published in at http://dx.doi.org/10.1214/08-AOP402 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Probability on Matrix-Cone Hypergroups

Recent investigations of M. Rösler [13] and M. Voit [16] provide examples of hypergroups with properties similar to the group- or vector space case and with a sufficiently rich structure of automorphisms, providing thus tools to investigate the theory limits of normalized random walks and the structure of the corresponding limit laws. The investigations are parallel to corresponding investigations for vector spaces and simply connected nilpotent Lie groups

A Lévy process for the GNIG probability law with 2nd order stochastic volatility and applications to option pricing

Here we derive the Lévy characteristic triplet for the GNIG probability law. This characterizes the corresponding Lévy process. In addition we derive equivalent martingale measures with which to price simple put and call options. This is done under two different equivalent martingale measures. We also present a multivariate Lévy process where the marginal probability distribution follows a GNIG Lévy process. The main contribution is, however, a stochastic process which is characterized by autocorrelation in moments equal and higher than two, here a multivariate specification is provided as well. The main tool for achieving this is to add an integrated Feller square root process to the dynamics of the second moment in a time-deformed Browninan motion. Applications to option pricing are also considered, and a brief discussion is held on the topic of estimation of the suggested process

Option Pricing Driven by Lévy Processes

The methodology of pricing financial derivatives, particularly stock options, was first introduced by Bachelier and developed by Black, Scholes and Merton, who gave the explicit formula for option pricing. Recent developed models such as jump-diffusion, Heston and Variance Gamma are also widely studied within the quantitative finance field and are proven to be applicable to a certain degree in real markets. A brief understanding of option pricing with stochastic processes is given in this thesis. Risk neutral valuation and notion of finding an equivalent martingale measure provide a framework under which derivatives are priced. Basic procedures of constructing a Brownian motion and stochastic integral from fundamental blocks are introduced. Infinitely divisible distributions and Lévy processes are detailedly discussed, including Lévy-Itô decomposition and the notion of subordination. Exponential-Lévy model and Fourier transform methods are presented to illustrate different approaches to option pricing. Simulation of AAPL stock prices based on estimated parameters from historical data under jump diffusion model is compared with empirical data to test the fitness of the model. Stock prices by minimal measure and Esscher transform measure are computed under geometric Lévy processes. Finally, univariate Variance Gamma process model is extended to Sato's two factor model for multivariate option pricing. The focus of this thesis is to give a detailed analysis of different option pricing models using mathematical and statistical concepts and theories, accompanied with simulations and empirical data to test the fitness of models. Extensions to numerous popular models are also discussed