Pricing and Hedging Asian Basket Options with Quasi-Monte Carlo Simulations

In this article we consider the problem of pricing and hedging high-dimensional Asian basket options by Quasi-Monte Carlo simulation. We assume a Black-Scholes market with time-dependent volatilities and show how to compute the deltas by the aid of the Malliavin Calculus, extending the procedure employed by Montero and Kohatsu-Higa (2003). Efficient path-generation algorithms, such as Linear Transformation and Principal Component Analysis, exhibit a high computational cost in a market with time-dependent volatilities. We present a new and fast Cholesky algorithm for block matrices that makes the Linear Transformation even more convenient. Moreover, we propose a new-path generation technique based on a Kronecker Product Approximation. This construction returns the same accuracy of the Linear Transformation used for the computation of the deltas and the prices in the case of correlated asset returns while requiring a lower computational time. All these techniques can be easily employed for stochastic volatility models based on the mixture of multi-dimensional dynamics introduced by Brigo et al. (2004).Comment: 16 page

Generalization of coloring linear transformation

The paper is focused on the technique of linear transformation between correlated and uncorrelated Gaussian random vectors, which is more or less commonly used in the reliability analysis of structures. These linear transformations are frequently needed to transform uncorrelated random vectors into correlated vectors with a prescribed covariance matrix (coloring transformation), and also to perform an inverse (whitening) transformation, i.e. to decorrelate a random vector with a non-identity covariance matrix. Two well-known linear transformation techniques, namely Cholesky decomposition and eigendecomposition (also known as principal component analysis, or the orthogonal transformation of a covariance matrix), are shown to be special cases of the generalized linear transformation presented in the paper. The proposed generalized linear transformation is able to rotate the transformation randomly, which may be desired in order to remove unwanted directional bias. The conclusions presented herein may be useful for structural reliability analysis with correlated random variables or random fields

Generation of correlated Rayleigh fading channels for accurate simulationof promising wireless communication systems

In this paper, a generalized method is proposed for the accurate simulation of equal/ unequal power correlated Rayleigh fading channels to overcome the shortcomings of existing methods. Spatial and spectral correlations are also considered in this technique for different transmission conditions. It employs successive coloring for the inphase and quadrature components of successive signals using real correlation vector of successive signal envelopes rather than complex covariance matrix of the Gaussian signals which is utilized in conventional methods. Any number of fading signals with any desired correlations of successive envelope pairs in the interval [0, 1] can be generated with high accuracy. Moreover, factorization of the desired covariance matrix is avoided to overcome the shortcomings and high computational complexity of conventional methods. Extensive simulations of different representative scenarios demonstrate the effectiveness of the proposedtechnique. The simplicity and accuracy of this method will help the researchers to study and simulate the impact of fading correlation on the performance evaluation of various multi-antenna and multicarrier communication systems. Moreover, it enables the engineers for efficient design and deployment of new schemes for feasible wireless application

Random quantum correlations and density operator distributions

Consider the question: what statistical ensemble corresponds to minimal prior knowledge about a quantum system ? For the case where the system is in fact known to be in a pure state there is an obvious answer, corresponding to the unique unitarily-invariant measure on the Hilbert sphere. However, the problem is open for the general case where states are described by density operators. Here two approaches to the problem are investigated. The first approach assumes that the system is randomly correlated with a second system, where the ensemble of composite systems is described by a random pure state. Results for qubits randomly correlated with other systems are presented, including average entanglement entropies. It is shown that maximum correlation is guaranteed in the limit as one system becomes infinite-dimensional. The second approach relies on choosing a metric on the space of density operators, and generating a corresponding ensemble from the induced volume element. Comparisons between the approaches are made for qubits, for which the second approach (based on the Bures metric) yields the most symmetric, and hence the least informative, ensemble of density operators.Comment: 13 pages, no figures; a new page of additional notes at end draws attention to 3 new references and their relevanc

Self-similar prior and wavelet bases for hidden incompressible turbulent motion

This work is concerned with the ill-posed inverse problem of estimating turbulent flows from the observation of an image sequence. From a Bayesian perspective, a divergence-free isotropic fractional Brownian motion (fBm) is chosen as a prior model for instantaneous turbulent velocity fields. This self-similar prior characterizes accurately second-order statistics of velocity fields in incompressible isotropic turbulence. Nevertheless, the associated maximum a posteriori involves a fractional Laplacian operator which is delicate to implement in practice. To deal with this issue, we propose to decompose the divergent-free fBm on well-chosen wavelet bases. As a first alternative, we propose to design wavelets as whitening filters. We show that these filters are fractional Laplacian wavelets composed with the Leray projector. As a second alternative, we use a divergence-free wavelet basis, which takes implicitly into account the incompressibility constraint arising from physics. Although the latter decomposition involves correlated wavelet coefficients, we are able to handle this dependence in practice. Based on these two wavelet decompositions, we finally provide effective and efficient algorithms to approach the maximum a posteriori. An intensive numerical evaluation proves the relevance of the proposed wavelet-based self-similar priors.Comment: SIAM Journal on Imaging Sciences, 201

Enabling Multi-level Trust in Privacy Preserving Data Mining

Privacy Preserving Data Mining (PPDM) addresses the problem of developing accurate models about aggregated data without access to precise information in individual data record. A widely studied \emph{perturbation-based PPDM} approach introduces random perturbation to individual values to preserve privacy before data is published. Previous solutions of this approach are limited in their tacit assumption of single-level trust on data miners. In this work, we relax this assumption and expand the scope of perturbation-based PPDM to Multi-Level Trust (MLT-PPDM). In our setting, the more trusted a data miner is, the less perturbed copy of the data it can access. Under this setting, a malicious data miner may have access to differently perturbed copies of the same data through various means, and may combine these diverse copies to jointly infer additional information about the original data that the data owner does not intend to release. Preventing such \emph{diversity attacks} is the key challenge of providing MLT-PPDM services. We address this challenge by properly correlating perturbation across copies at different trust levels. We prove that our solution is robust against diversity attacks with respect to our privacy goal. That is, for data miners who have access to an arbitrary collection of the perturbed copies, our solution prevent them from jointly reconstructing the original data more accurately than the best effort using any individual copy in the collection. Our solution allows a data owner to generate perturbed copies of its data for arbitrary trust levels on-demand. This feature offers data owners maximum flexibility.Comment: 20 pages, 5 figures. Accepted for publication in IEEE Transactions on Knowledge and Data Engineerin