Papers by Giorgio Pauletto
The solution of large and sparse models presents in many ways a suitable structure for implementa... more The solution of large and sparse models presents in many ways a suitable structure for implementation on parallel computers. However, efficient use of these computing devices requires that the code be specifi-cally structured to exploit the particular type of parallel computer used. This article discusses the implementa-tion of data parallel processing algorithms as well as performance results based on the solution of a macro-econometric model on a Connection Machine 2.
Abstract. Monte Carlo (MC) methods have proved to be flexible, robust and very useful techniques ... more Abstract. Monte Carlo (MC) methods have proved to be flexible, robust and very useful techniques in computational finance. Several studies have investigated ways to achieve greater efficiency of such methods for serial computers. In this paper, we concentrate on the parallelization potentials of the MC methods. While MC is generally thought to be “embarrassingly parallel”, the results eventually depend on the quality of the underlying parallel pseudo-random number generators. There are several methods for obtaining pseudo-random numbers on a parallel computer and we briefly present some alternatives. Then, we turn to an application of security pricing where we empirically investigate the pros and cons of the different generators. This also allows us to assess the potentials of parallel MC in the computational finance framework. 1
: Macroeconometric models with rational expectations constitute a challenge when they have to be ... more : Macroeconometric models with rational expectations constitute a challenge when they have to be solved repeatedly a large number of times, as it is the case in stochastic simulation or in sensitivity analysis. This paper presents an efficient implementation for solving such models on an IBM SP1. The results are illustrated with the model MULTIMOD from the International Monetary Fund. Key words: Parallel computing, solution of rational expectation models, Jacobi and Gauss-Seidel algorithms. 1 Introduction Solution techniques for large nonlinear rational expectations models have been discussed in [8, Fair and Taylor 1983], [14, Holly and Zarrop 1983], [13, Hall and Henry 1988], [9, Fisher and Hughes Hallett 1988] and [10, Fisher 1992] among others. Even though the simulation of such models implies the solution of fairly large systems of equations, one may consider this as a solved problem, given the performance of the computers available at present time. This is probably true if we run...
The simulation of large macroeconometric models containing forward-looking variables can become i... more The simulation of large macroeconometric models containing forward-looking variables can become impractical when using exact Newton methods. The difficulties generally arise from the use of direct methods for the solution of the linear system in the Newton step. In such cases, nonstationary iterative methods, also called Krylov methods, provide an interesting alternative. In this paper we apply such methods to simulate a real world econometric model. Our numerical experiments confirm the interesting features of these techniques: low computational complexity and storage requirements. We also discuss a block preconditioner suitable for the particular class of models solved.
This paper investigates computational and implementation issues for the valuation of options on t... more This paper investigates computational and implementation issues for the valuation of options on three underlying assets, focusing on the use of the finite difference methods. We demonstrate that implicit methods, which have good convergence and stability properties, can now be implemented efficiently due to the recent development of techniques that allow the efficient solution of large and sparse linear systems. In the trivariate option valuation problem, we use nonstationary iterative methods (also called Krylov methods) for the solution of the large and sparse linear systems arising while using implicit methods. Krylov methods are investigated both in serial and in parallel implementations. Computational results show that the parallel implementation is particularly efficient if a fine grid space is needed.
Research Papers in Economics, 2000
Monte Carlo (MC) methods have proved flexible, robust and very useful techniques in computational... more Monte Carlo (MC) methods have proved flexible, robust and very useful techniques in computational finance. Several studies have investigated ways to achieve greater efficiency for such methods for serial computers.In this paper, we concentrate on the parallelization potentials of the MC methods. While MC is generally thought to be `embarrassingly parallel', the results eventually depend on the quality of the underlying parallel pseudo-random number generators. There are several methods for obtaining pseudo-random numbers on a parallel computer and we briefly present some alternatives. Then, we turn to an application of security pricing where we empirically investigate the pros and cons of the different generators. This also allows us to assess the advantages or inconveniences of parallel MC versus its serial version in the computational finance framework.
Computing in Economics and Finance, 2000
This paper presents a dynamic model of the joint labor/leisure and consumption/saving decision ov... more This paper presents a dynamic model of the joint labor/leisure and consumption/saving decision over the life cycle. Such a dynamic model provides a framework for considering the important policy experiments related to the reforms in Social Security. We address the role of labor supply in a life cyle utility maximization model formally, building upon recent work by Low (1998), and extending the classical optimal lifetime consumption problem under uncertainty first formalized in Phelps (1962) and later in Hakansson (1970). We begin by solving the finite horizon consumption/saving problem analytically and numerically and compare the two solutions. We also simulate this benchmark model. Once the labor choice is considered, the stochastic dynamic programming utility maximization problem of the individual is solved numerically, since analytical solutions are infeasible when the individual is maximizing utility over consumption and leisure, given non-linear marginal utility. We show how su...
The problem of solution of large and sparse models presents in many points a suitable structure f... more The problem of solution of large and sparse models presents in many points a suitable structure for an implementation on parallel computers. However, an eecient use of these computing devices requires the code to be speciically structured in order to exploit the particular type of parallel computer used. The paper discusses the implementation of data parallel processing algorithms as well as performance results based on the solution of a macroeconometric model on a CM2 computer.
We compare alternative numerical methods for approximating solutions to continuous-state dynamic ... more We compare alternative numerical methods for approximating solutions to continuous-state dynamic programming (DP) problems. We distinguish two approaches: discrete approximation and parametric approximation. In the former, the continuous state space is discretized into a finite number of points N , and the resulting finite-state DP problem is solved numerically. In the latter, a function associated with the DP problem such as the value function, the policy function, or some other related function is approximated by a smooth function of K unknown parameters. Values of the parameters are chosen so that the parametric function approximates the true function as closely as possible. We focus on approximations that are linear in parameters, i.e. where the parametric approximation is a linear combination of K basis functions. We also focus on methods that approximate the value function V as the solution to the Bellman equation associated with the DP problem. In finite state DP problems...
This paper investigates computational and implementation issues for the valuation of options on t... more This paper investigates computational and implementation issues for the valuation of options on three underlying assets, focusing on the use of the finite difference methods. We demonstrate that implicit methods, which have good convergence and stability prooperties, can now be implemented efficiently due to the recent development of techniques that allow the efficient solution of large and sparse linear systems. In the trivariate option valuation problem, we use nonstationary iterative methods (also called Krylov methods) for the solution of the large and sparse linear systems arising while using implicit methods. Krylov methods are investigated both in serial and in parallel implementations. Computational results show that the parallel implementation is particularly efficient if a fine grid space is needed.
Monte Carlo (MC) methods have proved to be flexible, robust and very useful techniques in computa... more Monte Carlo (MC) methods have proved to be flexible, robust and very useful techniques in computational finance. Several studies have investigated ways to achieve greater efficiency of such methods for serial computers. In this paper, we concentrate on the parallelization potentials of the MC methods. While MC is generally thought to be "embarrassingly parallel", the results eventually depend on the quality of the underlying parallel pseudo-random number generators. There are several methods for obtaining pseudo-random numbers on a parallel computer and we briefly present some alternatives. Then, we turn to an application of security pricing where we empirically investigate the pros and cons of the different generators. This also allows us to assess the potentials of parallel MC in the computational finance framework.
Many numerical methods to price options have been suggested in the finance literature. This paper... more Many numerical methods to price options have been suggested in the finance literature. This paper aims at reviewing several numerical approaches in order to discuss their practical strenghts and/or weaknesses. The problem under investigation is a multivariate contingent claims model with three underlying assets. We compare several alternatives in the partial differential equation framework: explicit, ADI, and implicit methods, the Fourier grid method, and the Monte Carlo approach, which becomes the only amenable method when the dimension of the problem grows. The comparison criteria are computational complexity, robustness with respect to initial conditions and parameter settings, and potential for a parallel implementation.
Krylov subspace methods have proven to be powerful methods for solving sparse linear systems aris... more Krylov subspace methods have proven to be powerful methods for solving sparse linear systems arising in several engineering problems. More recently, these methods have been successfully applied in computational economics, for instance in the solution of forward-looking macroeconometric models (Gilli and Pauletto and Pauletto and Gilli), dynamic programming problems (Mrkaic) and pricing of financial options (Gilli, Kellezi and Pauletto). Since Krylov methods can suffer from slow convergence, one can modify the original linear system in order to improve convergence properties. This is known as preconditioning. In this paper, we investigate the effects of several preconditioning techniques in the framework of dynamic programming problems and financial option pricing. Very few theoretical results on preconditioning are known and experiments have to be conducted to recognize which classes of problems can be best solved using a given Krylov method and a given preconditioner.
This paper investigates computational and implementation issues for the valuation of options on t... more This paper investigates computational and implementation issues for the valuation of options on three underlying assets, focusing on the use of the finite difference methods. We demonstrate that implicit methods, which have good convergence and stability prooperties, can now be implemented efficiently due to the recent development of techniques that allow the efficient solution of large and sparse linear systems.
In this paper, different strategies to exploit the sparse structure in the solution techniques fo... more In this paper, different strategies to exploit the sparse structure in the solution techniques for macroeconometric models with forward-looking variables are discussed. First, the stacked model is decomposed into recursive submodels without destroying its original block pattern. Next, we concentrate on how to efficiently solve the sparse linear system in the Newton algorithm. In this frame, a multiple block diagonal LU factorization and a sparse Gaussian elimination are presented. The algorithms are compared by solving the country model for Japan in MULTIMOD.
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Papers by Giorgio Pauletto