SUMMARY A generalization of the sampling method introduced by Metropolis et al. (1953) is presented along with an exposition of the relevant theory, techniques of application and methods and difficulties of assessing the error in Monte Carlo estimates. Examples of the methods, including the generation of random orthogonal matrices and potential applications of the methods to numerical problems arising in statistics, are discussed. For numerical problems in a large number of dimensions, Monte Carlo methods are often more efficient than conventional numerical methods. However, implementation of the Monte Carlo methods requires sampling from high dimensional probability distributions and this may be very difficult and expensive in analysis and computer time. General methods for sampling from, or estimating expectations with respect to, such distributions are as follows. (i) If possible, factorize the distribution into the product of one-dimensional conditional distributions from which samples may be obtained. (ii) Use importance sampling, which may also be used for variance reduction. That is, in order to evaluate the integral J = X) p(x)dx = Ev(f), where p(x) is a probability density function, instead of obtaining independent samples XI, ..., Xv from p(x) and using the estimate J, = Zf(xi)/N, we instead obtain the sample from a distribution with density q(x) and use the estimate J2 = Y{f(xj)p(x1)}/{q(xj)N}. This may be advantageous if it is easier to sample from q(x) thanp(x), but it is a difficult method to use in a large number of dimensions, since the values of the weights w(xi) = p(x1)/q(xj) for reasonable values of N may all be extremely small, or a few may be extremely large. In estimating the probability of an event A, however, these difficulties may not be as serious since the only values of w(x) which are important are those for which x -A. Since the methods proposed by Trotter & Tukey (1956) for the estimation of conditional expectations require the use of importance sampling, the same difficulties may be encountered in their use. (iii) Use a simulation technique; that is, if it is difficult to sample directly from p(x) or if p(x) is unknown, sample from some distribution q(y) and obtain the sample x values as some function of the corresponding y values. If we want samples from the conditional dis

https://www.jstor.org/stable/pdfplus/2334940.pdf

Monte Carlo Sampling Methods Using Markov Chains and Their Applications

From the Publisher:
This is the revised and greatly expanded Second Edition of the hugely popular Numerical Recipes: The Art of Scientific Computing. The product of a unique collaboration among four leading scientists in academic research and industry, Numerical Recipes is a complete text and reference book on scientific computing. In a self-contained manner it proceeds from mathematical and theoretical considerations to actual practical computer routines. With over 100 new routines (now well over 300 in all), plus upgraded versions of many of the original routines, this book is more than ever the most practical, comprehensive handbook of scientific computing available today. The book retains the informal, easy-to-read style that made the first edition so popular, with many new topics presented at the same accessible level. In addition, some sections of more advanced material have been introduced, set off in small type from the main body of the text. Numerical Recipes is an ideal textbook for scientists and engineers and an indispensable reference for anyone who works in scientific computing. Highlights of the new material include a new chapter on integral equations and inverse methods; multigrid methods for solving partial differential equations; improved random number routines; wavelet transforms; the statistical bootstrap method; a new chapter on "less-numerical" algorithms including compression coding and arbitrary precision arithmetic; band diagonal linear systems; linear algebra on sparse matrices; Cholesky and QR decomposition; calculation of numerical derivatives; Pade approximants, and rational Chebyshev approximation; new special functions; Monte Carlo integration in high-dimensional spaces; globally convergent methods for sets of nonlinear equations; an expanded chapter on fast Fourier methods; spectral analysis on unevenly sampled data; Savitzky-Golay smoothing filters; and two-dimensional Kolmogorov-Smirnoff tests. All this is in addition to material on such basic top

Numerical Recipes in C: The Art of Scientific Computing

A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions

One option-pricing problem that has hitherto been unsolved is the pricing of a European call on an asset that has a stochastic volatility. This paper examines this problem. The option price is determined in series form for the case in which the stochastic volatility is independent of the stock price. Numerical solutions are also produced for the case in which the volatility is correlated with the stock price. It is found that the Black-Scholes price frequently overprices options and that the degree of overpricing increases with the time to maturity. ONE OPTION-PRICING PROBLEM that has hitherto remained unsolved is the pricing of a European call on a stock that has a stochastic volatility. From the work of Merton [12], Garman [6], and Cox, Ingersoll, and Ross [3], the differential equation that the option must satisfy is known. The solution of this differential equation is independent of risk preferences if (a) the volatility is a traded asset or (b) the volatility is uncorrelated with aggregate consumption. If either of these conditions holds, the risk-neutral valuation arguments of Cox and Ross [4] can be used in a straightfoward way. This paper produces a solution in series form for the situation in which the stock price is instantaneously uncorrelated with the volatility. We do not assume that the volatility is a traded asset. Also, a constant correlation between the instantaneous rate of change of the volatility and the rate of change of aggregate consumption can be accommodated. The option price is lower than the BlackScholes (B-S) [1] price when the option is close to being at the money and higher when it is deep in or deep out of the money. The exercise prices for which overpricing by B-S takes place are within about ten percent of the security price. This is the range of exercise prices over which most option trading takes place, so we may, in general, expect the B-S price to overprice options. This effect is exaggerated as the time to maturity increases. One of the most surprising implications of this is that, if the B-S equation is used to determine the implied volatility of a near-the-money option, the longer the time to maturity the lower the implied volatility. Numerical solutions for the case in which the volatility is correlated with the stock price are also examined. The stochastic volatility problem has been examined by Merton [13], Geske [7], Johnson [10], Johnson and Shanno [11], Eisenberg [5], Wiggins [16], and

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1540-6261.1987.tb02568.x

The Pricing of Options on Assets with Stochastic Volatilities

Several Markov chain methods are available for sampling from a posterior distribution. Two important examples are the Gibbs sampler and the Metropolis algorithm. In addition, several strategies are available for constructing hybrid algorithms. This paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues. On the theoretical side, results from the theory of general state space Markov chains can be used to obtain convergence rates, laws of large numbers and central limit theorems for estimates obtained from Markov chain methods. These theoretical results can be used to guide the construction of more efficient algorithms. For the practical use of Markov chain methods, standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation.

/pdf/markov-chains-for-exploring-posterior-distributions-19mikfcnja.pdf

Markov Chains for Exploring Posterior Distributions

1 The general nature of Monte Carlo methods.- 2 Short resume of statistical terms.- 3 Random, pseudorandom, and quasirandom numbers.- 4 Direct simulation.- 5 General principles of the Monte Carlo method.- 6 Conditional Monte Carlo.- 7 Solution of linear operator equations.- 8 Radiation shielding and reactor criticality.- 9 Problems in statistical mechanics.- 10 Long polymer molecules.- 11 Percolation processes.- 12 Multivariable problems.- References.

D. C. Handscomb

Papers

Monte Carlo methods