scholarly journals AN ADAPTIVE METHOD FOR EVALUATING MULTIDIMENSIONAL CONTINGENT CLAIMS: PART II

2003 ◽  
Vol 06 (04) ◽  
pp. 327-353 ◽  
Author(s):  
LARS O. DAHL
Keyword(s):  
Monte Carlo ◽  
Option Pricing ◽  
Variance Reduction ◽  
Adaptive Method ◽  
Unit Cube ◽  
Random Numbers ◽  
Quasi Monte Carlo ◽  
Low Discrepancy Sequences ◽  
Pricing Problems ◽  
Simulation Results

This is part two of a work on adaptive integration methods aimed at multidimensional option pricing problems in finance. It presents simulation results of an adaptive method developed in the companion article [3] for the evaluation of multidimensional integrals over the unit cube. The article focuses on a rather general test problem constructed to give insights in the success of the adaptive method for option pricing problems. We establish a connection between the decline rate of the ordered eigenvalues of the pricing problem and the efficiency of the adaptive method relative to the non-adaptive. This gives criteria for when the adaptive method can be expected to outperform the non-adaptive for other pricing problems. In addition to evaluating the method for different problem parameters, we present simulation results after adding various techniques to enhance the adaptive method itself. This includes using variance reduction techniques for each sub-problem resulting from the partitioning of the integration domain. All simulations are done with both pseudo-random numbers and quasi-random numbers (low discrepancy sequences), resulting in Monte Carlo (MC) and quasi-Monte Carlo (QMC) estimators and the ability to compare them in the given setting. The results show that the adaptive method can give performance gains in the order of magnitudes for many configurations, but it should not be used incautious, since this ability depends heavily on the problem at hand.

scholarly journals AN ADAPTIVE METHOD FOR EVALUATING MULTIDIMENSIONAL CONTINGENT CLAIMS: PART I

2003 ◽  
Vol 06 (03) ◽  
pp. 301-316 ◽  
Author(s):  
LARS O. DAHL
Keyword(s):  
Option Pricing ◽  
Variance Reduction ◽  
Adaptive Method ◽  
Companion Paper ◽  
Unit Cube ◽  
Convergence Properties ◽  
Local Variance ◽  
Reduction Techniques ◽  
Pricing Problems ◽  
Multidimensional Integrals

The paper presents an adaptive method for the evaluation of multidimensional integrals over the unit cube. The measure used to partition the domain is suited for integrands which are monotonic in each dimension individually, and is therefore suitable for problems stemming from finance where this is often the case. We use a QMC method for each sub-problem resulting from the partitioning of the domain. The article is part one of a work on this topic, and presents the method together with various local variance reduction techniques. The material is presented with an alignment to option pricing problems. In the companion paper we present an option pricing problem and simulation results on different setups of this. We compare the convergence properties of the adaptive method with the convergence properties of the QMC method used directly on the problem. We find that the adaptive method in many configurations outperform the conventional QMC method, and we develop criteria on the problem for when the adaptive method can be expected to outperform the conventional.

Monte Carlo and quasi-Monte Carlo methods

Acta Numerica ◽  
1998 ◽  
Vol 7 ◽  
pp. 1-49 ◽  
Author(s):  
Russel E. Caflisch
Keyword(s):  
Monte Carlo ◽  
Convergence Rate ◽  
Monte Carlo Methods ◽  
Variance Reduction ◽  
Rarefied Gas ◽  
Fluid Dynamic ◽  
Random Sequence ◽  
Convergence Theory ◽  
Quasi Monte Carlo ◽  
Low Discrepancy Sequences

Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N−1/2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasi-random (also called low-discrepancy) sequences, which are a deterministic alternative to random or pseudo-random sequences. The points in a quasi-random sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasi-Monte Carlo, has a convergence rate of approximately O((logN)kN−1). For quasi-Monte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less than the fluid dynamic length scale. Computational examples are presented throughout the text to illustrate the theory. A number of open problems are described.

Valuing American options by least-squares randomized quasi-Monte Carlo methods

2014 ◽  
Vol 01 (02) ◽  
pp. 1450016
Author(s):  
Xin-Yu Wu ◽  
Hai-Lin Zhou ◽  
Shou-Yang Wang
Keyword(s):  
Monte Carlo ◽  
Least Squares ◽  
Financial Engineering ◽  
Variance Reduction ◽  
American Options ◽  
Computation Time ◽  
Quasi Monte Carlo ◽  
Low Discrepancy Sequences ◽  
Low Efficiency ◽  
Least Squares Approach

Valuation of American options is a difficult and challenging problem encountered in financial engineering. Longstaff and Schwartz [Longstaff, FA and ES Schwartz (2001). Valuing American Options by Simulation: A Simple Least-squares Approach, Review of Financial Studies, 14(1), 113–147.] Proposed the least-squares Monte Carlo (LSM) method for valuing American options. As this approach is intuitive and easy to apply, it has received much attention in the finance literature. However, a drawback of the LSM method is the low efficiency. In order to overcome this problem, we propose the least-squares randomized quasi-Monte Carlo (LSRQM) methods which can be viewed as a use low-discrepancy sequences as a variance reduction technique in the LSM method for valuing American options in this paper. Numerical results demonstrate that our proposed LSRQM methods are more efficient than the LSM method in terms of the valuation accuracy, the computation time and the convergence rate.

Monte Carlo & Quasi-Monte Carlo approach in option pricing

2012 ◽  
Author(s):  
M. A. Maasar ◽  
N. A. M. Nordin ◽  
M. Anthonyrajah ◽  
W. M. W. Zainodin ◽  
A. M. Yamin
Keyword(s):  
Monte Carlo ◽  
Option Pricing ◽  
Monte Carlo Approach ◽  
Quasi Monte Carlo

scholarly journals p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering

Algorithms ◽  
2020 ◽  
Vol 13 (5) ◽  
pp. 110
Author(s):  
Philippe Blondeel ◽  
Pieterjan Robbe ◽  
Cédric Van hoorickx ◽  
Stijn François ◽  
Geert Lombaert ◽  
...  
Keyword(s):  
Finite Element ◽  
Monte Carlo ◽  
Variance Reduction ◽  
Civil Engineering ◽  
Galerkin Finite Element ◽  
Quasi Monte Carlo ◽  
Stability Of Slopes ◽  
Galerkin Finite Element Methods ◽  
Significant Cost Reduction ◽  
Lattice Rule

Civil engineering applications are often characterized by a large uncertainty on the material parameters. Discretization of the underlying equations is typically done by means of the Galerkin Finite Element method. The uncertain material parameter can be expressed as a random field represented by, for example, a Karhunen–Loève expansion. Computation of the stochastic responses, i.e., the expected value and variance of a chosen quantity of interest, remains very costly, even when state-of-the-art Multilevel Monte Carlo (MLMC) is used. A significant cost reduction can be achieved by using a recently developed multilevel method: p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). This method is based on the idea of variance reduction by employing a hierarchical discretization of the problem based on a p-refinement scheme. It is combined with a rank-1 Quasi-Monte Carlo (QMC) lattice rule, which yields faster convergence compared to the use of random Monte Carlo points. In this work, we developed algorithms for the p-MLQMC method for two dimensional problems. The p-MLQMC method is first benchmarked on an academic beam problem. Finally, we use our algorithm for the assessment of the stability of slopes, a problem that arises in geotechnical engineering, and typically suffers from large parameter uncertainty. For both considered problems, we observe a very significant reduction in the amount of computational work with respect to MLMC.

A quasi-Monte Carlo implementation of the ziggurat method

2018 ◽  
Vol 24 (2) ◽  
pp. 93-99
Author(s):  
Nguyet Nguyen ◽  
Linlin Xu ◽  
Giray Ökten
Keyword(s):  
Monte Carlo ◽  
Random Variable ◽  
Numerical Results ◽  
Quasi Monte Carlo ◽  
Low Discrepancy Sequences ◽  
Gamma Distributions ◽  
Variable Generation ◽  
Monte Carlo Implementation

Abstract The ziggurat method is a fast random variable generation method introduced by Marsaglia and Tsang in a series of papers. We discuss how the ziggurat method can be implemented for low-discrepancy sequences, and present algorithms and numerical results when the method is used to generate samples from the normal and gamma distributions.

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