Monte Carlo and quasi-Monte Carlo methods

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.

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Pricing Lookback Options under Normal Inverse Gaussian Model by Variance Reduction and Randomized Quasi-Monte Carlo Methods

2014 Seventh International Joint Conference on Computational Sciences and Optimization ◽

10.1109/cso.2014.89 ◽

2014 ◽

Author(s):

Yongzeng Lai ◽

Jilin Zhang

Keyword(s):

Monte Carlo ◽

Monte Carlo Methods ◽

Variance Reduction ◽

Gaussian Model ◽

Inverse Gaussian ◽

Quasi Monte Carlo ◽

Lookback Options ◽

Normal Inverse Gaussian

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AN ADAPTIVE METHOD FOR EVALUATING MULTIDIMENSIONAL CONTINGENT CLAIMS: PART II

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024903001943 ◽

2003 ◽

Vol 06 (04) ◽

pp. 327-353 ◽

Cited By ~ 2

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.

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

Journal of Financial Engineering ◽

10.1142/s2345768614500160 ◽

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.

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