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Parallel Computation of Multivariate Normal Probabilities
"... We present methods for the computation of multivariate normal probabilities on parallel/ distributed systems. After a transformation of the initial integral, an approximation can be obtained using MonteCarlo or quasirandom methods. We propose a metaalgorithm for asynchronous sampling methods and d ..."
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Cited by 207 (9 self)
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We present methods for the computation of multivariate normal probabilities on parallel/ distributed systems. After a transformation of the initial integral, an approximation can be obtained using MonteCarlo or quasirandom methods. We propose a metaalgorithm for asynchronous sampling methods and derive efficient parallel algorithms for the computation of MVN distribution functions, including a method based on randomized Korobov and Richtmyer sequences. Timing results of the implementations using the MPI parallel environment are given. 1 Introduction The computation of the multivariate normal distribution function F (a; b) = j\Sigmaj \Gamma 1 2 (2) \Gamma n 2 Z b a e \Gamma 1 2 x \Sigma \Gamma1 x dx: (1) often leads to computationalintensive integration problems. Here \Sigma is an n \Theta n symmetric positive definite covariance matrix; furthermore one of the limits in each integration variable may be infinite. Genz [5] performs a sequence of transformations resu...
Methods for the Computation of Multivariate tProbabilities
 Computing Sciences and Statistics
, 2000
"... This paper compares methods for the numerical computation of multivariate tprobabilities for hyperrectangular integration regions. Methods based on acceptancerejection, sphericalradial transformations and separationofvariables transformations are considered. Tests using randomly chosen problems ..."
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Cited by 83 (11 self)
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This paper compares methods for the numerical computation of multivariate tprobabilities for hyperrectangular integration regions. Methods based on acceptancerejection, sphericalradial transformations and separationofvariables transformations are considered. Tests using randomly chosen problems show that the most efficient numerical methods use a transformation developed by Genz (1992) for multivariate normal probabilities. These methods allow moderately accurate multivariate tprobabilities to be quickly computed for problems with as many as twenty variables. Methods for the noncentral multivariate tdistribution are also described. Key Words: multivariate tdistribution, noncentral distribution, numerical integration, statistical computation. 1 Introduction A common problem in many statistics applications is the numerical computation of the multivariate t (MVT) distribution function (see Tong, 1990) defined by T(a; b; \Sigma; ) = \Gamma( +m 2 ) \Gamma( 2 ) p j\Sigma...
Connecting Discrete and Continuous PathDependent Options
, 1999
"... . This paper develops methods for relating the prices of discrete and continuoustime versions of pathdependent options sensitive to extremal values of the underlying asset, including lookback, barrier, and hindsight options. The relationships take the form of correction terms that can be interpre ..."
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Cited by 54 (5 self)
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. This paper develops methods for relating the prices of discrete and continuoustime versions of pathdependent options sensitive to extremal values of the underlying asset, including lookback, barrier, and hindsight options. The relationships take the form of correction terms that can be interpreted as shifting a barrier, a strike, or an extremal price. These correction terms enable us to use closedform solutions for continuous option prices to approximate their discrete counterparts. We also develop discretetime discretestate lattice methods for determining accurate prices of discrete and continuous pathdependent options. In several cases, the lattice methods use correction terms based on the connection between discrete and continuoustime prices which dramatically improve convergence to the accurate price. Key words: Barrier options, lookback options, continuity corrections, trinomial trees JEL classification: G13, C63, G12 Mathematics Subject Classification (1991): 90A09, 60J15, 65N06 1
Numerical Computation of Rectangular Bivariate And Trivariate normal and t probabilities
 STATISTICS AND COMPUTING
, 2004
"... Algorithms for the computation of bivariate and trivariate normal and t probabilities for rectangles are reviewed. The algorithms use numerical integration to approximate transformed probability distribution integrals. A generalization of Plackett's formula is derived for bivariate and trivaria ..."
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Cited by 54 (1 self)
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Algorithms for the computation of bivariate and trivariate normal and t probabilities for rectangles are reviewed. The algorithms use numerical integration to approximate transformed probability distribution integrals. A generalization of Plackett's formula is derived for bivariate and trivariate t probabilities. New methods are described for the numerical computation of bivariate and trivariate t probabilities. Test results are provided, along with recommendations for the most efficient algorithms for single and double precision computations.
Pricing and Hedging American Options: A Recursive Integration Method
 Review of Financial Studies
, 1996
"... In this paper, we present a new method for pricing and hedging American options along with an efficient implementation procedure. The proposed method is efficient and accurate in computing both option values and various option hedge parameters. We demonstrate the computational accuracy and efficienc ..."
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Cited by 53 (3 self)
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In this paper, we present a new method for pricing and hedging American options along with an efficient implementation procedure. The proposed method is efficient and accurate in computing both option values and various option hedge parameters. We demonstrate the computational accuracy and efficiency of this numerical procedure in relation to other competing approaches. We also suggest how the method can be applied to the case of any American option for which a closedform solution exists for the corresponding European option. A variety of financial products such as fixedincome derivatives, mortgagebacked securities and corporate securities have earlyexercise or Americanstyle features that significantly influence their valuation and hedging. Considerable interest exists, therefore, in both academic and practitioner circles, in methods of valuation and hedging Americanstyle options that are conceptually sound, as well as efficient in their implementation. It has been recognized early in the ...
Methods for Approximating Integrals in Statistics with Special Emphasis on Bayesian Integration Problems
 Statistical Science
"... This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain method ..."
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Cited by 49 (5 self)
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This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain methods. Each method is discussed giving an outline of the basic supporting theory and particular features of the technique. Conclusions are drawn concerning the relative merits of the methods based on the discussion and their application to three examples. The following broad recommendations are made. Asymptotic methods should only be considered in contexts where the integrand has a dominant peak with approximate ellipsoidal symmetry. Importance sampling, and preferably adaptive importance sampling, based on a multivariate Student should be used instead of asymptotics methods in such a context. Multiple quadrature, and in particular subregion adaptive integration, are the algorithms of choice for...
Numerical Methods for Fitting and Simulating AutoregressiveToAnything Processes
 INFORMS Journal on Computing
, 1997
"... An ARTA (AutoRegressive to Anything) Process is a time series with arbitrary marginal distribution and autocorrelation structure specified through finite lag p. We develop an efficient numerical method for fitting ARTA processes and discuss its implementation in the software ARTAFACTS. We also prese ..."
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Cited by 37 (7 self)
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An ARTA (AutoRegressive to Anything) Process is a time series with arbitrary marginal distribution and autocorrelation structure specified through finite lag p. We develop an efficient numerical method for fitting ARTA processes and discuss its implementation in the software ARTAFACTS. We also present the software ARTAGEN that generates observations from ARTA processes for use as inputs to a computer simulation. We illustrate the use of the software with a realworld example. Subject classification: simulation Other Keywords: time series, input modeling, numerical integration Dependent, timeseries input processes occur naturally in the simulation of many service, communications and manufacturing systems. For example, the sizes of the demands on an inventory system in successive periods are often dependent because a large demand in one period implies that fewer items will be needed in the following period. Ware, Page and 1 Nelson [13] observed that the times between file accesses on ...
Tight error bounds for nonuniform signaling over AWGN Channels
 IEEE TRANS. INFORM. THEORY
, 2000
"... We consider a Bonferronitype lower bound due to Kounias on the probability of a finite union. The bound is expressed in terms of only the individual and pairwise event probabilities; however, it suffers from requiring an exponentially complex search for its direct implementation. We address this p ..."
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Cited by 24 (12 self)
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We consider a Bonferronitype lower bound due to Kounias on the probability of a finite union. The bound is expressed in terms of only the individual and pairwise event probabilities; however, it suffers from requiring an exponentially complex search for its direct implementation. We address this problem by presenting a practical algorithm for its evaluation. This bound is applied together with two other bounds, a recent lower bound (the KAT bound) and a greedy algorithm implementation of an upper bound due to Hunter, to examine the symbol error () and bit error ( ) probabilities of an uncoded communication system used in conjunction withary phaseshift keying (PSK)/quadrature amplitude (QAM) (PSK/QAM) modulations and maximum a posteriori (MAP) decoding over additive white Gaussian noise (AWGN) channels. It is shown that the bounds—which can be efficiently computed—provide an excellent estimate of the error probabilities over the entire range of the signaltonoise ratio (SNR). The new algorithmic bound and the greedy bound are particularly impressive as they agree with the simulation results even during very severe channel conditions.
Valuation of PerformanceDependent Options
, 2006
"... Performancedependent options are financial derivatives whose payoff depends on the performance of one asset in comparison to a set of benchmark assets. In this paper, we present a novel approach for the valuation of general performancedependent options. To this end, we use a multidimensional Black ..."
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Cited by 4 (4 self)
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Performancedependent options are financial derivatives whose payoff depends on the performance of one asset in comparison to a set of benchmark assets. In this paper, we present a novel approach for the valuation of general performancedependent options. To this end, we use a multidimensional BlackScholes model to describe the temporal development of the asset prices. The martingale approach then yields the fair price of such options as a multidimensional integral whose dimension is the number of stochastic processes used in the model. The integrand is typically discontinuous which makes accurate solutions difficult to achieve by numerical approaches, though. Using tools from computational geometry, we are able to derive a pricing formula which only involves the evaluation of several smooth multivariate normal distributions. This way, performancedependent options can efficiently be priced even for highdimensional problems as is shown by numerical results.