The least squares method for option pricing revisited

It is shown that the the popular least squares method of option pricing converges even under very general assumptions. This substantially increases the freedom of creating different implementations of the method, with varying levels of computational complexity and flexible approach to regression. It is also argued that in many practical applications even modest non-linear extensions of standard regression may produce satisfactory results. This claim is illustrated with examples

Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis

We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale (Dynamical systems, New York, 1973, pp. 531-544). The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. The method distinguishes the hierarchy between singular set, Pareto critical set and stable Pareto critical set, and can handle the problem of superposition of local Pareto fronts, occurring in the general nonconvex case. Furthermore, a quadratic convergence result in a suitable set-wise sense is proven and tested in a number of numerical examples.Comment: 29 pages, 12 figure

Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States

We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, led to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in $O(n\log n)$ complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D $n\times n\times n$ Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D ``density fitting`` scheme. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excited states, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is related to the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for finite lattice-structured systems, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a $L\times L\times L$ lattice manifests the linear in $L$ computational work, $O(L)$, instead of the usual $O(L^3 \log L)$ scaling by the Ewald-type approaches

Efficient Bayesian estimation of a multivariate stochastic volatility model with cross leverage and heavy-tailed errors

The efficient Bayesian estimation method using Markov chain Monte Carlo is proposed for a multivariate stochastic volatility model that is a natural extension of the univariate stochastic volatility model with leverage and heavy-tailed errors, where we further incorporate cross leverage effects among stock returns. Our method is based on a multi-move sampler which samples a block of latent volatility vectors and is described first in the literature for a multivariate stochastic volatility model with cross leverage and heavy-tailed errors. Its high sampling efficiency is shown using numerical examples in comparison with a single-move sampler which samples one latent volatility vector at a time given other latent vectors and parameters. The empirical studies are given using five dimensional stock return indices in Tokyo Stock Exchange.