Computing images of Galois representations attached to elliptic curves

Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let G_E(ell) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ell-torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G_E(ell) up to local conjugacy for all primes ell by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G_E(ell) up to one of at most two isomorphic conjugacy classes of subgroups of GL_2(Z/ell Z) that have the same semisimplification, each of which occurs for an elliptic curve isogenous to E. Under the GRH, their running times are polynomial in the bit-size n of an integral Weierstrass equation for E, and for our Monte Carlo algorithm, quasi-linear in n. We have applied our algorithms to the non-CM elliptic curves in Cremona's tables and the Stein--Watkins database, some 140 million curves of conductor up to 10^10, thereby obtaining a conjecturally complete list of 63 exceptional Galois images G_E(ell) that arise for E/Q without CM. Under this conjecture we determine a complete list of 160 exceptional Galois images G_E(ell) the arise for non-CM elliptic curves over quadratic fields with rational j-invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the j-invariant is irrational.Comment: minor edits, 47 pages, to appear in Forum of Mathematics, Sigm

Simulation of asset prices using Lévy processes

Includes bibliographical references (leaves 93-97).This dissertation focuses on a Lévy process driven framework for the pricing of financial instruments. The main focus of this dissertation is not, however, to price these instruments; the main focus is simulation based. Simulation is a key issue under Monte Carlo pricing and risk-neutral valuation- it is the first step towards pricing and therefore must be done accurately and with care. This dissertation looks at different kinds of Lévy processes and the various approaches one can take when simulating them

Optimal Variance Swaps Portfolios and Estimating Greeks for Variance-Gamma

In this dissertation, we investigate two problems: constructing optimal variance swaps portfolios and estimating Greeks for options with underlying assets following a Variance Gamma process. By modeling the dependent non-Gaussian residual in a linear regression model through a L'evy Mixture (LM) model and a Variance Gamma Correlated (VGC) model, and running some optimizations, we construct an optimal variance swap portfolio. By implementing gradient estimation techniques, we estimate the Greeks for a series of basket options called Mountain Range options. Constructing an optimal variance swap portfolio consists of two steps: evaluations and optimization. Each variance swap has two legs: a fixed leg (also called the variance strike) and a floating leg (also called the realized variance). The value of a variance swap is the discounted difference between the realized variance and the variance strike. For the latter, one can use an option surface calibration to evaluate. For the former, the procedure is complicated due to the non-negligible residuals from a linear regression model. Through LM and VGC, we can estimate the realized variance on different sample paths and obtain the payoff of a variance swap numerically. Based on these numerical results, we can apply the optimization method to construct an optimal portfolio. In the second part of this dissertation, we consider gradient estimation for Mountain Range options including Everest options, Atlas options, Altiplano/Annapurna options and Himalayan options. Assuming the underlying assets follow a Variance-Gamma (VG) process, we derive estimators for sensitivities such as Greeks through Monte Carlo simulation. We implement and compare using numerical experiments several gradient estimation approaches: finite difference methods (forward difference), infinitesimal perturbation analysis (IPA), and likelihood ratio (LR) method using either the density function or the characteristic function

The Variance-Gamma Distribution: A Review

The variance-gamma (VG) distributions form a four-parameter family which includes as special and limiting cases the normal, gamma and Laplace distributions. Some of the numerous applications include financial modelling and distributional approximation on Wiener space. In this review, we provide an up-to-date account of the basic distributional theory of the VG distribution. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review methods for parameter estimation and some applications of the VG distribution, including the aforementioned applications to financial modelling and distributional approximation on Wiener space.Comment: 31pages, 3 figure

Bayesian changepoint models motivated by cyber-security applications

Changepoint detection has an important role to play in the next generation of cyber security defenses. A cyber attack typically changes the behaviour of the target network. Therefore, to detect the presence of a network intrusion, it can be informative to monitor for changes in the high-volume data sources that are collected inside an enterprise computer network. However, most traditional changepoint detection methods are not adapted to characterise what cyber security analysts mean by a change, and consequently raise too many false alerts but also overlook weak signals that are suggestive of a real attack. This thesis will present three novel Bayesian changepoint models that address some challenges raised by cyber data: the first model combines evidence across a graph of time series to identify patterns of changepoints that are a priori more likely to correspond to an attack; the second model offers robustness to non-exchangeable data within segments so that normal dynamic phenomena observed in cyber data can be captured; and, the third model relaxes the standard assumption that changes are instantaneous, so that time intervals where cyber data may be subject to non-instantaneous changes can be identified.Open Acces

A universal median quasi-Monte Carlo integration

We study quasi-Monte Carlo (QMC) integration over the multi-dimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.Comment: Major revision, 32 pages, 4 figure