DOCSLIB.ORG

Variance Reduction Techniques of Importance Sampling Monte Carlo Methods for Pricing Options*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Variance Reduction Techniques of Importance Sampling Monte Carlo Methods for Pricing Options* Journal of Mathematical Finance, 2013, 3, 431-436 Published Online November 2013 (http://www.scirp.org/journal/jmf) http://dx.doi.org/10.4236/jmf.2013.34045 Variance Reduction Techniques of Importance Sampling * Monte Carlo Methods for Pricing Options Qiang Zhao1, Guo Liu1, Guiding Gu2 1School of Finance, Shanghai University of Finance and Economics, Shanghai, China 2Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, China Email: [email protected] Received July 10, 2013; revised September 7, 2013; accepted September 18, 2013 Copyright © 2013 Qiang Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper we discuss the importance sampling Monte Carlo methods for pricing options. The classical importance sampling method is used to eliminate the variance caused by the linear part of the logarithmic function of payoff. The variance caused by the quadratic part is reduced by stratified sampling. We eliminate both kinds of variances just by importance sampling. The corresponding space for the eigenvalues of the Hessian matrix of the logarithmic function of payoff is enlarged. Computational Simulation shows the high efficiency of the new method. Keywords: Monte Carlo Method; Importance Sampling; Variance Reduction; Option Pricing 1. Introduction with probability 1 , where is the standard devia- Monte Carlo simulation is a numerical method based on tion of V , n is the number of samples, is the the probability theory. Its application in finance becomes significance le vel and Z is the quantile of standard more and more popular as the demand for pricing and 2 hedging of various complex financial derivatives, which normal distribution under . play an important role in the field of investment, risk 2 management and corporate governance. The advantage It is clear that the convergence rate of Monte Carlo of Monte Carlo method is that its convergence rate is method is independent on the number of state variables. Monte 1 Carlo simulation is often the only way available for the 2 On. pricing of complex path-dependent options if the number of relevant underlying assets is greater than three. However, Monte Carlo simulation is constantly criticized hence, in order to reduce the error by a factor of 10 one for its slow convergence. Let V be a random variable has to generate 100 times as much as samples as well as and we want to calculate EV . We can generate computation time. For this reason, Monte Carlo simu- independently and identically distributed samples lation needs to be run on large parallel computers with a V n of V . Law of Large Numbers guarantees that high financial cost in terms of hardware and software i i1 as.. developments. The computational demands of simulation 1 n n have motivated substantial interest in the financial VVi1 i n industry in demands for increased efficiency. Another and Central Limit Theorem guarantees that asymp- way to improve the accuracy is to reduce the standard totically falls in the confidence interval deviation . Motivated by this thought, several tech- niques to reduce the variance of the Monte Carlo simu- VZVZnn, lation have been proposed, such as control variates, nn22antithetic variables, importance sampling and stratifi- *This work is supported by Research Innovation Foundation of Shang- cation(see Boyle, Broadie and Glasserman [1], and Gla- hai University of Finance and Economics under Grant No. CXJJ-2013- sserman [2]. These techniques aim to reduce the variance 323; Basic Academic Discipline Program, the 11th five year plan of 211 per Monte Carlo observation so that a given level of Project for Shanghai University of Finance and Economics. Open Access JMF 432 Q. ZHAO ET AL. accuracy can be obtained with a smaller number of two families based on the strategy adopted. The first one simulations. Control variates and antithetic variables are is proposed by Glasserman, Heidelberge and Shaha- the most widely used variance reduction techniques, buddin in a remarkable paper [14] (GHS for short), relies mainly because of the simplicity of their implementations, on a deterministic optimization procedure which can be and the fact that they can be accommodated in an exist- applied for a specific class of payoffs. Xu and Zhang [13] ing Monte Carlo calculator with a small effort. Examples improve the optimization algorithm of the importance of successful implementations of control variates for sampling by Newton Raphson algorithm based on direct pricing the derivatives include Hull and White [3], simulation. The second one is the so-called adaptive Kemna and Vorst [4], Turnbull and Wakeman [5], Ma Monte Carlo method, such as Vázquez-Abad and Du- and Xu [6]. fresne [9], Su and Fu [10], Arouna [11], that aims to Importance sampling has the capacity to exploit determine the optimal drift through stochastic optimi- detailed knowledge about a model (often in the form of zation techniques that typically involve an iterative algo- asymptotic approximations) to produce potential variance rithm. reduction. Unfortunately, importance sampling technique Most closely related to our work is Glasserman, has not been widely used as other variance reduction Heidelberge and Shahabuddin [14], who applied impor- techniques in pricing financial derivatives until recently. tance sampling combined with stratified sampling to This is mainly because there is no general way to dramatically reduce variance in derivative pricing. In this implement importance sampling. If the transformation of paper, we propose a new importance sampling method by probability measure is chosen improperly, this method modifying the drift term and the quadratic term of the does not work. Importance sampling attempts to reduce simulated process simultaneously. In the previous variance by changing the probability measure from literature, the variance for the linear part is eliminated by which paths are generated. Our goal is to obtain a more importance sampling and those for the quadratic part is convenient representation of the expected value. The idea reduced by stratification. However, we eliminate both behind the importance sampling is to reduce the statis- kinds of variances just by importance sampling. The tical uncertainty of Monte Carlo calculation by focusing corresponding space for the eigenvalues of the Hessian on the most important region of the space from which the matrix of the log function of payoff is enlarged. Illu- random samples are drawn. Such regions depend both on strations of the use of the method with European options the random process simulated, and the structure of the and Asian options are given, which show the high effi- security priced. Just as mentioned by Glasserman [2], an ciency of the method. The method proposed in the paper effective importance sampling density should weight more points to the region where the product of their can be extended to the pricing of other financial deriva- probability and their payoff is large. For example, for a ives directly. deep out-of-the-money call option, most of the time the payoff from simulation is 0 , so simulating more paths 2. Importance Sampling Method with positive payoffs should reduce the variance in the Importance sampling attempts to reduce variance by estimation. changing the probability measure from which samples An early example of impo rtance sampling applied to are generated. To make this idea concrete, consider the security pricing is Reider [7], where the variance was problem of estimating reduced substantially by increasing the drift in simulation VEGX Gxfxxd, (1) for deep out-of-the-money European call options. Gla- f sserman, Heidelberger and Shahabuddin [8] applied where X is a random vector of n with probability importance sampling to reduce substantial variance by density f , and E f denotes the expectation under combining stratification in the stochastic volatility model. the original measure with density function f , G is a Other recent work on importance sampling methods in function from n to , genera lly denoting the finance has been done for Monte Carlo simulations payoffs of financial derivatives. The ordinary Monte driven by high-dimensional Gaussian vectors, such as Carlo estimator is Boyle, Broadie and Glasserman [1], Vázquez-Abad and M ˆ 1 Dufresne [9], Su and Fu [10], Arouna [11], Capriotti [12], VGXfi , (2) M i1 Xu and Zhang [13]. In this framework, Importance Sampling is applied by modifying the drift term of the with X12,,,XX M independent draws from f . Let n simulated process to construct a new measure in which g be any other probability density on satisfying more weight is given to important outcomes thereby f x>0 gx >0, (3) increasing sampling efficiency. The different methods proposed in the literature mainly differ in the way where for all x n . Then we can obtain by c hanging mea- such a change of drift is found, and can be divided into sure Open Access JMF Q. ZHAO ET AL. 433 fx fX 2 fX VG x gxxEGXdg . (4) min gfEGX . (8) gx gX gX Thus, V can be interpreted as an expectation with Unfortunately, there is no general way to find the respect to the density g . optimal g for an ordinary function f . So, we turn to If X12,,,XX M are now independent draws from find the relatively optimal g to make the variance as g , the importance sampling estimator associated with small as possible. g is 3. New Im portance Sampling Si mulation M fX ˆ 1 i VGXgi . (5) In this section, we propose a new importance sampling M i1 gX i simulation following the idea of GHS. Consider the The weight problem of estimating the expectation of the payoff fXii gX wX i VEGX Gxfxxd, f Rn is the likelihood ratio evaluated at X i . where X is a n -dimensional vector of standard It follows from (1) and (4) that normal variable with probability density ˆˆ n 1 T EVff EV gg V, xx T 2 2 fx 2π e,x xx12 ,,,.
Load more

Recommended publications
  • Lecture 15: Approximate Inference: Monte Carlo Methods
    10-708: Probabilistic Graphical Models 10-708, Spring 2017 Lecture 15: Approximate Inference: Monte Carlo methods Lecturer: Eric P. Xing Name: Yuan Liu(yuanl4), Yikang Li(yikangl) 1 Introduction We have already learned some exact inference methods in the previous lectures, such as the elimination algo- rithm, message-passing algorithm and the junction tree algorithms. However, there are many probabilistic models of practical interest for which exact inference is intractable. One important class of inference algo- rithms is based on deterministic approximation schemes and includes methods such as variational methods. This lecture will include an alternative very general and widely used framework for approximate inference based on stochastic simulation through sampling, also known as the Monte Carlo methods. The purpose of the Monte Carlo method is to find the expectation of some function f(x) with respect to a probability distribution p(x). Here, the components of x might comprise of discrete or continuous variables, or some combination of the two. Z hfi = f(x)p(x)dx The general idea behind sampling methods is to obtain a set of examples x(t)(where t = 1; :::; N) drawn from the distribution p(x). This allows the expectation (1) to be approximated by a finite sum N 1 X f^ = f(x(t)) N t=1 If the samples x(t) is i.i.d, it is clear that hf^i = hfi, and the variance of the estimator is h(f − hfi)2i=N. Monte Carlo methods work by drawing samples from the desired distribution, which can be accomplished even when it it not possible to write out the pdf.
    [Show full text]
  • Estimating Standard Errors for Importance Sampling Estimators with Multiple Markov Chains
    Estimating standard errors for importance sampling estimators with multiple Markov chains Vivekananda Roy1, Aixin Tan2, and James M. Flegal3 1Department of Statistics, Iowa State University 2Department of Statistics and Actuarial Science, University of Iowa 3Department of Statistics, University of California, Riverside Aug 10, 2016 Abstract The naive importance sampling estimator, based on samples from a single importance density, can be numerically unstable. Instead, we consider generalized importance sampling estimators where samples from more than one probability distribution are combined. We study this problem in the Markov chain Monte Carlo context, where independent samples are replaced with Markov chain samples. If the chains converge to their respective target distributions at a polynomial rate, then under two finite moment conditions, we show a central limit theorem holds for the generalized estimators. Further, we develop an easy to implement method to calculate valid asymptotic standard errors based on batch means. We also provide a batch means estimator for calculating asymptotically valid standard errors of Geyer’s (1994) reverse logistic estimator. We illustrate the method via three examples. In particular, the generalized importance sampling estimator is used for Bayesian spatial modeling of binary data and to perform empirical Bayes variable selection where the batch means estimator enables standard error calculations in high-dimensional settings. arXiv:1509.06310v2 [math.ST] 11 Aug 2016 Key words and phrases: Bayes factors, Markov chain Monte Carlo, polynomial ergodicity, ratios of normalizing constants, reverse logistic estimator. 1 Introduction Let π(x) = ν(x)=m be a probability density function (pdf) on X with respect to a measure µ(·). R Suppose f : X ! R is a π integrable function and we want to estimate Eπf := X f(x)π(x)µ(dx).
    [Show full text]
  • Distilling Importance Sampling
    Distilling Importance Sampling Dennis Prangle1 1School of Mathematics, University of Bristol, United Kingdom Abstract in almost any setting, including in the presence of strong posterior dependence or discrete random variables. How- ever it only achieves a representative weighted sample at a Many complicated Bayesian posteriors are difficult feasible cost if the proposal is a reasonable approximation to approximate by either sampling or optimisation to the target distribution. methods. Therefore we propose a novel approach combining features of both. We use a flexible para- An alternative to Monte Carlo is to use optimisation to meterised family of densities, such as a normal- find the best approximation to the posterior from a family of ising flow. Given a density from this family approx- distributions. Typically this is done in the framework of vari- imating the posterior, we use importance sampling ational inference (VI). VI is computationally efficient but to produce a weighted sample from a more accur- has the drawback that it often produces poor approximations ate posterior approximation. This sample is then to the posterior distribution e.g. through over-concentration used in optimisation to update the parameters of [Turner et al., 2008, Yao et al., 2018]. the approximate density, which we view as dis- A recent improvement in VI is due to the development of tilling the importance sampling results. We iterate a range of flexible and computationally tractable distribu- these steps and gradually improve the quality of the tional families using normalising flows [Dinh et al., 2016, posterior approximation. We illustrate our method Papamakarios et al., 2019a]. These transform a simple base in two challenging examples: a queueing model random distribution to a complex distribution, using a se- and a stochastic differential equation model.
    [Show full text]
  • Importance Sampling & Sequential Importance Sampling
    Importance Sampling & Sequential Importance Sampling Arnaud Doucet Departments of Statistics & Computer Science University of British Columbia A.D. () 1 / 40 Each distribution πn (dx1:n) = πn (x1:n) dx1:n is known up to a normalizing constant, i.e. γn (x1:n) πn (x1:n) = Zn We want to estimate expectations of test functions ϕ : En R n ! Eπn (ϕn) = ϕn (x1:n) πn (dx1:n) Z and/or the normalizing constants Zn. We want to do this sequentially; i.e. …rst π1 and/or Z1 at time 1 then π2 and/or Z2 at time 2 and so on. Generic Problem Consider a sequence of probability distributions πn n 1 de…ned on a f g sequence of (measurable) spaces (En, n) n 1 where E1 = E, f F g 1 = and En = En 1 E, n = n 1 . F F F F F A.D. () 2 / 40 We want to estimate expectations of test functions ϕ : En R n ! Eπn (ϕn) = ϕn (x1:n) πn (dx1:n) Z and/or the normalizing constants Zn. We want to do this sequentially; i.e. …rst π1 and/or Z1 at time 1 then π2 and/or Z2 at time 2 and so on. Generic Problem Consider a sequence of probability distributions πn n 1 de…ned on a f g sequence of (measurable) spaces (En, n) n 1 where E1 = E, f F g 1 = and En = En 1 E, n = n 1 . F F F F F Each distribution πn (dx1:n) = πn (x1:n) dx1:n is known up to a normalizing constant, i.e.
    [Show full text]
  • Blackimportance Sampling and Its Optimality for Stochastic
    Electronic Journal of Statistics ISSN: 1935-7524 Importance Sampling and its Optimality for Stochastic Simulation Models Yen-Chi Chen and Youngjun Choe University of Washington, Department of Statistics Seattle, WA 98195 e-mail: [email protected]∗ University of Washington, Department of Industrial and Systems Engineering Seattle, WA 98195 e-mail: [email protected]∗ Abstract: We consider the problem of estimating an expected outcome from a stochastic simulation model. Our goal is to develop a theoretical framework on importance sampling for such estimation. By investigating the variance of an importance sampling estimator, we propose a two-stage procedure that involves a regression stage and a sampling stage to construct the final estimator. We introduce a parametric and a nonparametric regres- sion estimator in the first stage and study how the allocation between the two stages affects the performance of the final estimator. We analyze the variance reduction rates and derive oracle properties of both methods. We evaluate the empirical performances of the methods using two numerical examples and a case study on wind turbine reliability evaluation. MSC 2010 subject classifications: Primary 62G20; secondary 62G86, 62H30. Keywords and phrases: nonparametric estimation, stochastic simulation model, oracle property, variance reduction, Monte Carlo. 1. Introduction The 2011 Fisher lecture (Wu, 2015) features the landscape change in engineer- ing, where computer simulation experiments are replacing physical experiments thanks to the advance of modeling and computing technologies. An insight from the lecture highlights that traditional principles for physical experiments do not necessarily apply to virtual experiments on a computer. The virtual environ- ment calls for new modeling and analysis frameworks distinguished from those arXiv:1710.00473v2 [stat.ME] 25 Sep 2019 developed under the constraint of physical environment.
    [Show full text]
  • Importance Sampling: Intrinsic Dimension And
    Submitted to Statistical Science Importance Sampling: Intrinsic Dimension and Computational Cost S. Agapiou∗, O. Papaspiliopoulos† D. Sanz-Alonso‡ and A. M. Stuart§ Abstract. The basic idea of importance sampling is to use independent samples from a proposal measure in order to approximate expectations with respect to a target measure. It is key to understand how many samples are required in order to guarantee accurate approximations. Intuitively, some notion of distance between the target and the proposal should determine the computational cost of the method. A major challenge is to quantify this distance in terms of parameters or statistics that are pertinent for the practitioner. The subject has attracted substantial interest from within a variety of communities. The objective of this paper is to overview and unify the resulting literature by creating an overarching framework. A general theory is presented, with a focus on the use of importance sampling in Bayesian inverse problems and filtering. 1. INTRODUCTION 1.1 Our Purpose Our purpose in this paper is to overview various ways of measuring the com- putational cost of importance sampling, to link them to one another through transparent mathematical reasoning, and to create cohesion in the vast pub- lished literature on this subject. In addressing these issues we will study impor- tance sampling in a general abstract setting, and then in the particular cases of arXiv:1511.06196v3 [stat.CO] 14 Jan 2017 Bayesian inversion and filtering. These two application settings are particularly important as there are many pressing scientific, technological and societal prob- lems which can be formulated via inversion or filtering.
    [Show full text]
  • Importance Sampling
    Deep Learning Srihari Importance Sampling Sargur N. Srihari [email protected] 1 Deep Learning Srihari Topics in Monte Carlo Methods 1. Sampling and Monte Carlo Methods 2. Importance Sampling 3. Markov Chain Monte Carlo Methods 4. Gibbs Sampling 5. Mixing between separated modes 2 Deep Learning Srihari Importance Sampling: Choice of p(x) • Sum of integrand to be computed: Discrete case Continuous case s = p(x)f(x) = E ⎡f(x)⎤ s = p(x)f(x)dx = E ⎡f(x)⎤ ∑ p ⎣ ⎦ ∫ p ⎣ ⎦ x • An important step is deciding which part of the integrand has the role of the probability p(x) – from which we sample x(1),..x(n) • And which part has role of f (x) whose expected value (under the probability distribution) is estimated as 1 n sˆ f( (i)) n = ∑ x n i=1 Deep Learning Decomposition of Integrand Srihari • In the equation s = ∑ p(x)f(x) s = ∫ p(x)f(x)dx x • There is no unique decomposition because. p(x) f (x) can be rewritten as p(x)f(x) p(x)f(x) = q(x) q( ) x – where we now sample from q and average p(x)f(x) q( ) x s = p(x)f(x)dx – In many cases the problem ∫ is specified naturally as expectation of f (x) given distribution p(x) • But it may not be optimal in no of samples required Deep LearningDifficulty of sampling from p(x) Srihari • Principal reason for sampling p(x) is evaluating expectation of some f (x) E[f ] = f(x)p(x)dx ∫ • Given samples x(i), i=1,..,n, from p(x), the finite sum approximation is 1 n fˆ = f (x (i) ) n∑ i=1 • But drawing samples p(x) may be impractical 5 Deep Learning Srihari Using a proposal distribution Rejection sampling uses
    [Show full text]
  • Uniformly Efficient Importance Sampling for the Tail Distribution Of
    MATHEMATICS OF OPERATIONS RESEARCH Vol. 33, No. 1, February 2008, pp. 36–50 informs ® issn 0364-765X eissn 1526-5471 08 3301 0036 doi 10.1287/moor.1070.0276 © 2008 INFORMS Uniformly Efficient Importance Sampling for the Tail Distribution of Sums of Random Variables Paul Glasserman Columbia University, New York, New York 10027 [email protected], http://www2.gsb.columbia.edu/faculty/pglasserman/ .org/. Sandeep Juneja ms Tata Institute of Fundamental Research, Mumbai, India 400005 author(s). or [email protected], http://www.tcs.tifr.res.in/~sandeepj/ .inf the Successful efficient rare-event simulation typically involves using importance sampling tailored to a specific rare event. However, in applications one may be interested in simultaneous estimation of many probabilities or even an entire distribution. to nals In this paper, we address this issue in a simple but fundamental setting. Specifically, we consider the problem of efficient estimation of the probabilities PSn ≥ na for large n, for all a lying in an interval , where Sn denotes the sum of n tesy independent, identically distributed light-tailed random variables. Importance sampling based on exponential twisting is known to produce asymptotically efficient estimates when reduces to a single point. We show, however, that this procedure fails http://jour cour to be asymptotically efficient throughout when contains more than one point. We analyze the best performance that can at a be achieved using a discrete mixture of exponentially twisted distributions, and then present a method using a continuous le mixture. We show that a continuous mixture of exponentially twisted probabilities and a discrete mixture with a sufficiently as large number of components produce asymptotically efficient estimates for all a ∈ simultaneously.
    [Show full text]
  • Advances in Importance Sampling Arxiv:2102.05407V2 [Stat.CO]
    Advances in Importance Sampling V´ıctorElvira? and Luca Martinoy ? School of Mathematics, University of Edinburgh (United Kingdom) y Universidad Rey Juan Carlos de Madrid (Spain) Abstract Importance sampling (IS) is a Monte Carlo technique for the approximation of intractable distributions and integrals with respect to them. The origin of IS dates from the early 1950s. In the last decades, the rise of the Bayesian paradigm and the increase of the available computational resources have propelled the interest in this theoretically sound methodology. In this paper, we first describe the basic IS algorithm and then revisit the recent advances in this methodology. We pay particular attention to two sophisticated lines. First, we focus on multiple IS (MIS), the case where more than one proposal is available. Second, we describe adaptive IS (AIS), the generic methodology for adapting one or more proposals. Keywords: Monte Carlo methods, computational statistics, importance sampling. 1 Problem Statement In many problems of science and engineering intractable integrals must be approximated. Let us denote an integral of interest Z I(f) = E [f(x)] = f(x)π(x)dx; (1) πe e dx dx 1 where f : R ! R, and πe(x) is a distribution of the r.v. X 2 R . Note that although Eq. (1) involves a distribution, more generic integrals could be targeted with the techniques described below. The integrals of this form appear often in the Bayesian framework, where a set of observations are available in y 2 Rdy , and the goal is in inferring some hidden parameters and/or latent arXiv:2102.05407v2 [stat.CO] 23 Apr 2021 variables x 2 Rdy that are connected to the observations through a probabilistic model [61].
    [Show full text]
  • Sampling and Monte Carlo Integration
    Sampling and Monte Carlo Integration Michael Gutmann Probabilistic Modelling and Reasoning (INFR11134) School of Informatics, University of Edinburgh Spring semester 2018 Recap Learning and inference often involves intractable integrals, e.g. I Marginalisation Z p(x) = p(x, y)dy y I Expectations Z E [g(x) | yo] = g(x)p(x|yo)dx for some function g. I For unobserved variables, likelihood and gradient of the log lik Z L(θ) = p(D; θ) = p(u, D; θdu), u ∇θ`(θ) = Ep(u|D;θ) [∇θ log p(u, D; θ)] Notation: Ep(x) is sometimes used to indicate that the expectation is taken with respect to p(x). Michael Gutmann Sampling and Monte Carlo Integration 2 / 41 Recap Learning and inference often involves intractable integrals, e.g. I For unnormalised models with intractable partition functions p˜(D; θ) L(θ) = R x p˜(x; θ)dx ∇θ`(θ) ∝ m(D; θ) − Ep(x;θ) [m(x; θ)] I Combined case of unnormalised models with intractable partition functions and unobserved variables. I Evaluation of intractable integrals can sometimes be avoided by using other learning criteria (e.g. score matching). I Here: methods to approximate integrals like those above using sampling. Michael Gutmann Sampling and Monte Carlo Integration 3 / 41 Program 1. Monte Carlo integration 2. Sampling Michael Gutmann Sampling and Monte Carlo Integration 4 / 41 Program 1. Monte Carlo integration Approximating expectations by averages Importance sampling 2. Sampling Michael Gutmann Sampling and Monte Carlo Integration 5 / 41 Averages with iid samples I Tutorial 7: For Gaussians, the sample average is an estimate (MLE) of the mean (expectation) E[x] 1 n x¯ = X x ≈ [x] n i E i=1 I Gaussianity not needed: assume xi are iid observations of x ∼ p(x).
    [Show full text]
  • Monte Carlo Integration
    A Monte Carlo Integration HE techniques developed in this dissertation are all Monte Carlo methods. Monte Carlo T methods are numerical techniques which rely on random sampling to approximate their results. Monte Carlo integration applies this process to the numerical estimation of integrals. In this appendix we review the fundamental concepts of Monte Carlo integration upon which our methods are based. From this discussion we will see why Monte Carlo methods are a particularly attractive choice for the multidimensional integration problems common in computer graphics. Good references for Monte Carlo integration in the context of computer graphics include Pharr and Humphreys[2004], Dutré et al.[2006], and Veach[1997]. The term “Monte Carlo methods” originated at the Los Alamos National Laboratory in the late 1940s during the development of the atomic bomb [Metropolis and Ulam, 1949]. Not surprisingly, the development of these methods also corresponds with the invention of the first electronic computers, which greatly accelerated the computation of repetitive numerical tasks. Metropolis[1987] provides a detailed account of the origins of the Monte Carlo method. Las Vegas algorithms are another class of method which rely on randomization to compute their results. However, in contrast to Las Vegas algorithms, which always produce the exact or correct solution, the accuracy of Monte Carlo methods can only be analyzed from a statistical viewpoint. Because of this, we first review some basic principles from probability theory before formally describing Monte Carlo integration. 149 150 A.1 Probability Background In order to define Monte Carlo integration, we start by reviewing some basic ideas from probability.
    [Show full text]
  • 9 Importance Sampling 3 9.1 Basic Importance Sampling
    Contents 9 Importance sampling 3 9.1 Basic importance sampling . 4 9.2 Self-normalized importance sampling . 8 9.3 Importance sampling diagnostics . 11 9.4 Example: PERT . 13 9.5 Importance sampling versus acceptance-rejection . 17 9.6 Exponential tilting . 18 9.7 Modes and Hessians . 19 9.8 General variables and stochastic processes . 21 9.9 Example: exit probabilities . 23 9.10 Control variates in importance sampling . 25 9.11 Mixture importance sampling . 27 9.12 Multiple importance sampling . 31 9.13 Positivisation . 33 9.14 What-if simulations . 35 End notes . 37 Exercises . 40 1 2 Contents © Art Owen 2009{2013,2018 do not distribute or post electronically without author's permission 9 Importance sampling In many applications we want to compute µ = E(f(X)) where f(x) is nearly zero outside a region A for which P(X 2 A) is small. The set A may have small volume, or it may be in the tail of the X distribution. A plain Monte Carlo sample from the distribution of X could fail to have even one point inside the region A. Problems of this type arise in high energy physics, Bayesian inference, rare event simulation for finance and insurance, and rendering in computer graphics among other areas. It is clear intuitively that we must get some samples from the interesting or important region. We do this by sampling from a distribution that over- weights the important region, hence the name importance sampling. Having oversampled the important region, we have to adjust our estimate somehow to account for having sampled from this other distribution.
    [Show full text]