Antithetic Variates Research Articles

On the use of antithetic variables to improve over the ranked set sampling estimator of the population mean

In the ranked set sampling algorithm a sample of size n2 is available. The data can be ranked without measurements. A subsample of size n is created using the information given by the ranks. The population mean is estimated by the subsample mean. In this paper, we investigate other ways for creating the subsample. To this end we introduce new sampling algorithms using the idea of antithetic variables. We propose a class of random estimators for the population mean which covers the ranked set sampling and simple random sampling estimators as special cases. A general dominance result leading to a suffcient condition for a random estimator \(\hat \mu _1\) to dominate another random estimator \(\hat \mu _2\) is established. The theory is done in a completely nonparametric basis and without making any assumption about the distribution of the underlying population. Finally, the superiority of our proposed estimators over the ranked set sampling estimator is established and the obtained results are evaluated through examples and numerical studies.

Modelling the Evolution of Credit Spreads using the Cox Process within the HJM Framework: A CDS Option Pricing Model

In this paper a simulation approach for defaultable yield curves is developed within the Heath et al. (1992) framework. The default event is modelled using the Cox process where the stochastic intensity represents the credit spread. The forward credit spread volatility function is affected by the entire credit spread term structure. The paper provides the defaultable bond and CDS option price in a probability setting equipped with a subfiltration structure. The Euler-Maruyama stochastic integral approximation and the Monte Carlo method are applied to develop a numerical algorithm for pricing. Finally, the Antithetic Variables technique is used to reduce the variance of CDSO estimations.

Assessing the Relationship between Computational Speed and Precision

Simulation techniques are well suited to modelling diseases yet can be computationally intensive. This study explores the relationship between modelled effect size, statistical precision, and efficiency gains achieved using variance reduction and an executable programming language. A published simulation model designed to model a population with type 2 diabetes mellitus based on the UKPDS 68 outcomes equations was coded in both Visual Basic for Applications (VBA) and C++. Efficiency gains due to the programming language were evaluated, as was the impact of antithetic variates to reduce variance, using predicted QALYs over a 40-year time horizon. The use of C++ provided a 75- and 90-fold reduction in simulation run time when using mean and sampled input values, respectively. For a series of 50 one-way sensitivity analyses, this would yield a total run time of 2 minutes when using C++, compared with 155 minutes for VBA when using mean input values. The use of antithetic variates typically resulted in a 53% reduction in the number of simulation replications and run time required. When drawing all input values to the model from distributions, the use of C++ and variance reduction resulted in a 246-fold improvement in computation time compared with VBA - for which the evaluation of 50 scenarios would correspondingly require 3.8 hours (C++) and approximately 14.5 days (VBA). The choice of programming language used in an economic model, as well as the methods for improving precision of model output can have profound effects on computation time. When constructing complex models, more computationally efficient approaches such as C++ and variance reduction should be considered; concerns regarding model transparency using compiled languages are best addressed via thorough documentation and model validation.

Efficiently pricing barrier options in a Markov-switching framework

An efficient Monte Carlo simulation for the pricing of barrier options in a Markov-switching model is presented. Compared to a brute-force approach, relying on the simulation of discretized trajectories, the presented algorithm simulates the underlying stock price process only at state changes and at maturity. Given these pieces of information, option prices are evaluated using the probability of Brownian bridges not to fall below some threshold level. It is illustrated how two methods of variance reduction, control variates and antithetic variates, further improve the algorithm. In a small case study, the algorithm is applied to the pricing of options with the EuroStoxx 50 as underlying.

The impact of sampling methods on bias and variance in stochastic linear programs

Stochastic linear programs can be solved approximately by drawing a subset of all possible random scenarios and solving the problem based on this subset, an approach known as sample average approximation (SAA). The value of the objective function at the optimal solution obtained via SAA provides an estimate of the true optimal objective function value. This estimator is known to be optimistically biased; the expected optimal objective function value for the sampled problem is lower (for minimization problems) than the optimal objective function value for the true problem. We investigate how two alternative sampling methods, antithetic variates (AV) and Latin Hypercube (LH) sampling, affect both the bias and variance, and thus the mean squared error (MSE), of this estimator. For a simple example, we analytically express the reductions in bias and variance obtained by these two alternative sampling methods. For eight test problems from the literature, we computationally investigate the impact of these sampling methods on bias and variance. We find that both sampling methods are effective at reducing mean squared error, with Latin Hypercube sampling outperforming antithetic variates. For our analytic example and the eight test problems we derive or estimate the condition number as defined in Shapiro et al. (Math. Program. 94:1---19, 2002). We find that for ill-conditioned problems, bias plays a larger role in MSE, and AV and LH sampling methods are more likely to reduce bias.

Variance reduction in stochastic homogenization: proof of concept, using antithetic variables

We show that we can reduce the variance in a simple problem of stochastic homogenization using the classical technique of antithetic variables. The setting, and the presentation, are deliberately kept elementary. We point out the main issues, show some illustrative results, and demonstrate, both theoretically and numerically, the efficiency of the approach on simple cases.

K-Antithetic Variates in Monte Carlo Simulation

Standard Monte Carlo simulation needs prohibitive time to achieve reasonable estimations. for untractable integrals (i.e. multidimensional integrals and/or intergals with complex integrand forms). Several statistical technique, called variance reduction methods, are used to reduce the simulation time. In this note, we propose a generalization of the well known antithetic variate method. Principally we propose a K−antithetic variate estimator (KAVE) based on the generation of K correlated uniform variates. Some numerical examples are presented to show the improvenment of our proposition.

Coupling from the past with randomized quasi-Monte Carlo

The coupling-from-the-past (CFTP) algorithm of Propp and Wilson permits one to sample exactly from the stationary distribution of an ergodic Markov chain. By using it n times independently, we obtain an independent sample from that distribution. A more representative sample can be obtained by creating negative dependence between these n replicates; other authors have already proposed to do this via antithetic variates, Latin hypercube sampling, and randomized quasi-Monte Carlo (RQMC). We study a new, often more effective, way of combining CFTP with RQMC, based on the array-RQMC algorithm. We provide numerical illustrations for Markov chains with both finite and continuous state spaces, and compare with the RQMC combinations proposed earlier.

Participating life insurance policies: an accurate and efficient parallel software for COTS clusters

In this paper we discuss the development of a parallel software for the numerical simulation of Participating Life Insurance Policies in distributed environments. The main computational kernels in the mathematical models for the solution of the problem are multidimensional integrals and stochastic differential equations. The former is solved by means of Monte Carlo method combined with the Antithetic Variates variance reduction technique, while differential equations are approximated via a fully implicit, positivity-preserving, Euler method. The parallelization strategy we adopted relies on the parallelization of Monte Carlo algorithm. We implemented and tested the software on a PC Linux cluster.

A farewell to the use of antithetic variates in Monte Carlo simulation

Antithetic variates (AV) is one the oldest and most popular variance reduction techniques (VRTs), commonly using complementary random numbers. The AV variance reduction is generally justified by the negative correlation it produces in paired simulation estimates. A new and simpler interpretation of the AV role is presented, showing AV as solely a procedure for input sample means compensation, without any further contribution from the complementary idea. The proposed interpretation is based on the descriptive sampling framework, viewing input samples as composed of a set of values and their sequencing. Simulation experiments and third-party results give support to this interpretation. However, when newer simulation sampling methods, like Latin Hypercube Sampling, Descriptive Sampling, Moment Matching and Quasi-Monte Carlo are adopted, all of them based on a controlled selection of the input sample values, AV turns irrelevant. Other VRTs are also affected by the ideas presented here.

Antithetic variates

AbstractAntithetic variates are used to reduce the experimental error in Monte Carlo experiments. Monte Carlo experiments use random numbers as the fundamental experimental units. The basic idea in antithetic variates is to introduce experimental units that are negatively correlated with other experimental units. The variance of the mean of two such units is the mean of their individual variances plus their negative covariance. Copyright © 2009 John Wiley & Sons, Inc.This article is categorized under: Algorithms and Computational Methods > Random Number Generation

Posouzení vybraných možností zefektivnění simulace Monte Carlo při opčním oceňování

In general, there exist many ways to detect the fair value of financial derivatives. However, each of them is suitable for different purposes. For example, when the payoff function is not very simple or the underlying process is too complex, the approach of Monte Carlo simulation can be useful. Unfortunately, the plain Monte Carlo simulation needs a very high number of independent paths to get reliable results. It is the reason why an improvement of the plain approach should be applied to decrease the number of paths required in order to get reliable results. In this paper we study more closely several such approaches and examine their potential of increasing the efficiency. To be more exact, we apply the antithetic variates method and stratified sampling approaches, including their combinations in order to get the fair price of a plain vanilla call. We consider three distinct underlying processes: geometric Brownian motion, variance gamma model and normal inverse Gaussian model. We also verify the confidence interval for the option price. We did not find any improvements of examined methods for complex processes considering the definition via two or more independent random numbers. However, if the required accuracy is very high, it might be useful to apply the stratification to the distribution function of the complex process.

Variance Reduction for Multivariate Monte Carlo Simulation

As derivative securities become more complex, solution by Monte Carlo simulation is an increasingly necessary tool. But Monte Carlo methods are computationally demanding, and the size of the simulation sample required to achieve reasonable accuracy rapidly escalates beyond what is feasible given current technology when multiple stochastic factors are involved. Variance reduction techniques help considerably. One of the simplest is use of antithetic variables, which imposes on the simulated data the true constraint that the distribution from which the sample is to be drawn is symmetric. Moment matching goes further, by constraining the mean and variance of the sample to match the desired values. In this article, Wang goes further still, to show how to force simulated multivariate vectors to have the right correlations. In the first step a multivariate sample of independent variables is simulated and its sample correlation matrix is calculated. This will be close to the identity matrix but not exactly equal, due to sampling noise. Cholesky factorization of the sample correlation matrix is then used to transform the initial sample of slightly correlated factors into one whose elements are perfectly uncorrelated in the sample. A second Cholesky factorization of the target correlation matrix that is desired for the variables transforms them into a sample with exactly the correct correlations. As Wang demonstrates, this produces a sharp increase in performance for multivariate Monte Carlo problems, even when the simulation sample is constructed from nonrandom low-discrepancy sequences rather than by stochastic simulation. <b>TOPICS:</b>Derivatives, simulations, factor-based models

Reducing the Variance of Likelihood Ratio Greeks in Monte Carlo

We investigate the use of Antithetic Variables, Control Variates and Importance Sampling to reduce the statistical errors of option sensitivities calculated with the Likelihood Ratio Method in Monte Carlo. We show how Antithetic Variables solve the well-known problem of the divergence of the variance of Delta for short maturities and small volatilities. With numerical examples within a Gaussian Copula framework, we show how simple Control Variates and Importance Sampling strategies provide computational savings up to several orders of magnitude.

Variance Reduction for Radial Line Sampling Coverage Estimators

Abstract It may be convenient to use radial line sampling for coverage estimates in small irregularly shaped areas. Radial line samples emanate from a fixed point located in the interior of the area and would be run according to a randomly selected azimuth. Thus, radial line sampling is analogous to techniques of local stereology, which considers quantitative aspects of spatial structures in the neighborhood of a central point. Coverage estimates from radial line sampling can be based on coverage observed on several sample lines. Several variance reduction techniques commonly used in Monte Carlo integration are proposed to reduce the variance of coverage estimates based on radial line sampling: importance sampling, control variates, and antithetic variates. The success of these techniques in irregularly shaped areas depends on the assumption that total line length is correlated with length of coverage. Radial line sampling as proposed here differs somewhat from ordinary line sampling because coverage estimates are based on squared line lengths.

Variance reduction techniques: Experimental comparison and analysis for single systems

We provide a thorough analysis of the effectiveness of different Variance Reduction Techniques (VRTs). We consider both stand-alone and combined applications of two input techniques, Antithetic Variates (AV) and Latin Hypercube Sampling (LHS), and two output techniques, Control Variates (CV) and Poststratified Sampling (PS). Previous research in the area mainly focuses on asymptotic variance reduction. In this experimental study, we measure the performance of VRTs under finite simulation run lengths and analyze their effects. Our findings show that the asymptotic variance reduction results do not readily apply to finite-length simulations. We consider three different types of systems (M/M/1, serial production line and (s, S) inventory control systems) and compare the VRTs under various experimental conditions. We observe that a variance reduction cannot be guaranteed for every instance a VRT is applied. Our results also indicate that the output VRTs (CV, PS) are better than input VRTs (AV, LHS) on the average for the single systems considered in this study. More interestingly, the less-sophisticated techniques (AV, CV) often perform better than the relatively more-complex techniques (LHS, PS). A comprehensive bibliography is also provided.

Use of Control Variates in a Large-Scale Aggregated Combat Model

In early fall 2004, the Air Force Studies and Analyses Agency (AFSAA) forwarded a proposed research topic, endorsed by the Air Force Chief of Staff, to the Department of Operational Sciences at the Air Force Institute of Technology. This topic requested an examination of potential variance reduction techniques (VRTs) for use with the THUNDER combat model. THUNDER is a large-scale, highly aggregated, discrete-event simulation of campaign-level military operations used to examine issues involving the utility and effectiveness of air and space power in a theater-level, joint warfare context. A new, state of the art combined multiple recursive generator was incorporated into THUNDER to synchronize the random inputs for common random numbers (CRN) and antithetic variates (AV). This work discusses the application of CRN, AV, and control variates (CV), within THUNDER, and the resulting variance reductions obtained for an example scenario.

Simulation optimization of logistics systems through the use of variance reduction techniques and criterion models

Simulation is a useful tool for modelling logistics systems. However, simulation itself is not an optimization tool. Therefore attempts have been made to combine simulation and optimization. Optimization of a logistics system through the use of simulation is difficult for several reasons. Because of the size and complexity of logistics systems, it is often necessary to consider the trade-offs between multiple conflicting performance measures for the system. One major drawback associated with commercially available tools is that users are not able to consider these trade-offs easily. To overcome this drawback, a simulation model can be developed to employ multi-objective decision analysis techniques such as criterion models which can then be optimized. This article illustrates how criterion models can be interfaced with simulation models of logistics systems. In addition, this article includes the programming and implementation of the variance reduction techniques of common random numbers and antithetic variates.

Variance Reduction for Sector Sampling

Abstract Variance reduction techniques used in Monte Carlo integration including control variates and importance sampling can use estimated or actual shapes of vegetated land areas to reduce the variance of estimators from sector sampling. The estimated shapes of forested areas could come from maps, aerial photos, or similar sources. Antithetic variates for variance reduction in Monte Carlo integration can be applied to sector sampling without using any estimated shape or map. Sector sampling selects as samples all trees or other vegetation of interest located in randomly chosen sectors that have a vertex at a common point located in the interior of a forested area. Each sector is associated with an angle of fixed magnitude emanating from the interior point. Sector orientation is based on a randomly selected azimuth originating at the interior point that is the vertex for each sector. This technique is well adapted to application on relatively small areas that have irregularly shaped boundaries. Sector sampling can be demonstrated to be unbiased. Unbiasedness holds with angle reduction for adjustment of sample size and with the variance reduction techniques that are proposed. The method is not restricted to application with trees because sector sampling methods could be applied just as well to other vegetation types or any other objects located in the land area of interest.

Universal parameter optimisation in games based on SPSA

Most game programs have a large number of parameters that are crucial for their performance. While tuning these parameters by hand is rather difficult, efficient and easy to use generic automatic parameter optimisation algorithms are known only for special problems such as the adjustment of the parameters of an evaluation function. The SPSA algorithm (Simultaneous Perturbation Stochastic Approximation) is a generic stochastic gradient method for optimising an objective function when an analytic expression of the gradient is not available, a frequent case in game programs. Further, SPSA in its canonical form is very easy to implement. As such, it is an attractive choice for parameter optimisation in game programs, both due to its generality and simplicity. The goal of this paper is twofold: (i) to introduce SPSA for the game programming community by putting it into a game-programming perspective, and (ii) to propose and discuss several methods that can be used to enhance the performance of SPSA. These methods include using common random numbers and antithetic variables, a combination of SPSA with RPROP, and the reuse of samples of previous performance evaluations. SPSA with the proposed enhancements was tested in some large-scale experiments on tuning the parameters of an opponent model, a policy and an evaluation function in our poker program, MCRAISE. Whilst SPSA with no enhancements failed to make progress using the allocated resources, SPSA with the enhancements proved to be competitive with other methods, including TD-learning; increasing the average payoff per game by as large as 0.19 times the size of the amount of the small bet. From the experimental study, we conclude that the use of an appropriately enhanced variant of SPSA for the optimisation of game program parameters is a viable approach, especially if no good alternative exist for the types of parameters considered.