Paolo Foschi

A comparative study of algorithms for solving seemingly unrelated regressions models

The computational efficiency of various algorithms for solving seemingly unrelated regressions (SUR) models is investigated. Some of the algorithms adapt known methods; others are new. The first transforms the SUR model to an ordinary linear model and uses the QR decomposition to solve it. Three others employ the generalized QR decomposition to solve the SUR model formulated as a generalized linear least-squares problem. Strategies to exploit the structure of the matrices involved are developed. The algorithms are reconsidered for solving the SUR model after it has been transformed to one of smaller dimensions.

Parametrix Approximation of Diffusion Transition Densities

A new analytical approximation tool, derived from the classical PDE theory, is introduced in order to build approximate transition densities of diffusions. The tool is useful for approximate pricing and hedging of financial derivatives and for maximum likelihood and method of moments estimates of diffusion parameters. The approximation is uniform with respect to time and space variables. Moreover, easily computable error bounds are available in any dimension.

Approximations for Asian options in local volatility models

We develop approximate formulae expressed in terms of elementary functions for the density, the price and the Greeks of path dependent options of Asian style, in a general local volatility model. An algorithm for computing higher order approximations is provided. The proof is based on a heat kernel expansion method in the framework of hypoelliptic, not uniformly parabolic, partial differential equations.

Estimation of VAR Models: Computational Aspects

The Vector Autoregressive (VAR) model with zero coefficient restrictions canbe formulated as a Seemingly Unrelated Regression Equation (SURE) model. Boththe response vectors and the coefficient matrix of the regression equationscomprise columns from a Toeplitz matrix. Efficient numerical and computationalmethods which exploit the Toeplitz and Kronecker product structure of thematrices are proposed. The methods are also adapted to provide numericallystable algorithms for the estimation of VAR(p) models with Granger-causedvariables.

Seemingly unrelated regression model with unequal size observations: computational aspects

The computational solution of the seemingly unrelated regression model with unequal size observations is considered. Two algorithms to solve the model when treated as a generalized linear least-squares problem are proposed. The algorithms have as a basic tool the generalized QR decomposition (GQRD) and efficiently exploit the block-sparse structure of the matrices. One of the algorithms reduces the computational burden of the estimation procedure by not computing explicitly the RQ factorization of the GQRD. The maximum likelihood estimation of the model when the covariance matrix is unknown is also considered.

ANALYSIS OF AN UNCERTAIN VOLATILITY MODEL

We examine, from both analytical and numerical viewpoints, the uncertain volatility model by Hobson-Rogers in the framework of degenerate parabolic PDEs of Kolmogorov type.

Algorithms for Computing the QR Decomposition of a Set of Matrices with Common Columns

Abstract The QR decomposition of a set of matrices which have common columns is investigated. The triangular factors of the QR decompositions are represented as nodes of a weighted directed graph. An edge between two nodes exists if and only if the columns of one of the matrices is a subset of the columns of the other. The weight of an edge denotes the computational complexity of deriving the triangular factor of the destination node from that of the source node. The problem is equivalent to constructing the graph and finding the minimum cost for visiting all the nodes. An algorithm which computes the QR decompositions by deriving the minimum spanning tree of the graph is proposed. Theoretical measures of complexity are derived and numerical results from the implementation of this and alternative heuristic algorithms are given.

Estimating seemingly unrelated regression models with vector autoregressive disturbances

Abstract The numerical solution of seemingly unrelated regression (SUR) models with vector autoregressive disturbances is considered. Initially, an orthogonal transformation is applied to reduce the model to one with smaller dimensions. The transformed model is expressed as a reduced-size SUR model with stochastic constraints. The generalized QR decomposition is used as the main computational tool to solve this model. An iterative estimation algorithm is proposed when the variance–covariance matrix of the disturbances and the matrix of autoregressive coefficients are unknown. Strategies to compute the orthogonal factorizations of the non-dense-structured matrices which arise in the estimation procedure are presented. Experimental results demonstrate the computational efficiency of the proposed algorithm.

Black-Scholes formulae for Asian options in local volatility models

We develop approximate formulae expressed in terms of elementary functions for the density, the price and the Greeks of path dependent options of Asian style, in a general local volatility model. An algorithm for computing higher order approximations is provided. The proof is based on a heat kernel expansion method in the framework of hypoelliptic, not uniformly parabolic, partial differential equations.

A computationally efficient method for solving sur models with orthogonal regressors

Abstract A computationally efficient method to estimate seemingly unrelated regression equations models with orthogonal regressors is presented. The method considers the estimation problem as a generalized linear least squares problem (GLLSP). The basic tool for solving the GLLSP is the generalized QR decomposition of the block-diagonal exogenous matrix and Cholesky factor C ⊗ I T of the covariance matrix of the disturbances. Exploiting the orthogonality property of the regressors the estimation problem is reduced into smaller and independent GLLSPs. The solution of each of the smaller GLLSPs is obtained by a single-column modification of C . This reduces significantly the computational burden of the standard estimation procedure, especially when the iterative feasible estimator of the model is needed. The covariance matrix of the estimators is also derived.