Abstract We describe a simple Importance Sampling strategy for Monte Carlo simulations based on a least-squares optimization procedure. With several numerical examples, we show that such Least-squares Importance Sampling (LSIS) …
Abstract We describe a simple Importance Sampling strategy for Monte Carlo simulations based on a least-squares optimization procedure. With several numerical examples, we show that such Least-squares Importance Sampling (LSIS) provides efficiency gains comparable to the state-of-the-art techniques, for problems that can be formulated in terms of the determination of the optimal mean of a multivariate Gaussian distribution. In addition, LSIS can be naturally applied to more general Importance Sampling densities and is particularly effective when the ability to adjust higher moments of the sampling distribution, or to deal with non-Gaussian or multi-modal densities, is critical to achieve variance reductions. Keywords: Monte Carlo methodsDerivatives pricingFinancial derivativesFinancial engineering Acknowledgements It is a pleasure to acknowledge Gabriele Cipriani, David Shorthouse, and Mark Stedman for stimulating discussions, and Paul Glasserman for an enlightening lecture that inspired this work. Useful suggestions by the anonymous referees are also gratefully acknowledged. The opinion and views expressed in this paper are uniquely those of the author, and do not necessarily represent those of Credit Suisse Group. Notes †In the present discussion we will treat the Z i 's as continuous variables, however all the results also apply if some or all of them can assume only a discrete set of values. For any such variable, the symbol ∫ dZ i is to be interpreted as a sum over the possible outcomes. Here and in the following, with a slight abuse of notation, we indicate by Z both the random variables and the dummy variables of integration. †In particular, the Monte Carlo integration becomes unfeasible if the variance of the estimator diverges, giving rise to the so-called sign-problem instability. Although this problem is the crux of Monte Carlo simulations in several branches of the physical sciences [see, e.g., Sorella, S. and Capriotti, L., Phys. Rev. B, 2000, 61, 2599], this issue does not usually affect financial contexts. ‡It is possible to show (Press et al. Citation2002) that, when G(Z) does not have a definite sign, the optimal sampling density has the similar form P opt = |G(Z)|P(Z)/V, although in this case the optimal variance is not zero. †The use of other discretization schemes does not alter the present discussion. †B. Arouna, Private communication.
Computational methods play an important role in modern finance. Through the theory of arbitrage-free pricing, the price of a derivative security can be expressed as the expected value of its …
Computational methods play an important role in modern finance. Through the theory of arbitrage-free pricing, the price of a derivative security can be expressed as the expected value of its payouts under a particular probability measure. The resulting integral becomes quite complicated if there are several state variables or if payouts are path-dependent. Simulation has proved to be a valuable tool for these calculations. This paper summarizes some of the recent applications and developments of the Monte Carlo method to security pricing problems.
We describe a simple Importance Sampling strategy for Monte Carlo simulations based on a least squares optimization procedure. With several numerical examples, we show that such Least Squares Importance Sampling …
We describe a simple Importance Sampling strategy for Monte Carlo simulations based on a least squares optimization procedure. With several numerical examples, we show that such Least Squares Importance Sampling (LSIS) provides efficiency gains comparable to the state of the art techniques, when the latter are known to perform well. However, in contrast to traditional approaches, LSIS is not limited to the determination of the optimal mean of a Gaussian sampling distribution. As a result, it outperforms other methods when the ability to adjust higher moments of the sampling distribution, or to deal with non-Gaussian or multi-modal densities, is critical to achieve variance reductions.
We describe a simple Importance Sampling strategy for Monte Carlo simulations based on a least squares optimization procedure. With several numerical examples, we show that such Least Squares Importance Sampling …
We describe a simple Importance Sampling strategy for Monte Carlo simulations based on a least squares optimization procedure. With several numerical examples, we show that such Least Squares Importance Sampling (LSIS) provides efficiency gains comparable to the state of the art techniques, when the latter are known to perform well. However, in contrast to traditional approaches, LSIS is not limited to the determination of the optimal mean of a Gaussian sampling distribution. As a result, it outperforms other methods when the ability to adjust higher moments of the sampling distribution, or to deal with non-Gaussian or multi-modal densities, is critical to achieve variance reductions.
We describe a simple Importance Sampling strategy for Monte Carlo simulations based on a least squares optimization procedure. With several numerical examples, we show that such Least Squares Importance Sampling …
We describe a simple Importance Sampling strategy for Monte Carlo simulations based on a least squares optimization procedure. With several numerical examples, we show that such Least Squares Importance Sampling (LSIS) provides efficiency gains comparable to the state of the art techniques, when the latter are known to perform well. However, in contrast to traditional approaches, LSIS is not limited to the determination of the optimal mean of a Gaussian sampling distribution. As a result, it outperforms other methods when the ability to adjust higher moments of the sampling distribution, or to deal with non-Gaussian or multi-modal densities, is critical to achieve variance reductions.
Monte Carlo Methodologies and Applications for Pricing and Risk Management CONTENTS Authors Introduction Bruno Dupire of Nikko Financial Products GENERALITIES Options: A Monte Carlo Approach Phelim P. Boyle of University …
Monte Carlo Methodologies and Applications for Pricing and Risk Management CONTENTS Authors Introduction Bruno Dupire of Nikko Financial Products GENERALITIES Options: A Monte Carlo Approach Phelim P. Boyle of University of Waterloo Monte Carlo Methods for Security Pricing Phelim P. Boyle of University of Waterloo, Mark Broadie and Paul Glasserman of Columbia University Monte Carlo Toolkit Bruno Dupire of Nikko Financial Products PRICING Dimension Reduction and Other Ways of Speeding Monte Carlo Simulation Bruno Dupire of Nikko Financial Products and Antoine Savine of General Re Financial Products Average Intelligence Edmond Levy of HSBC MIDLAND and Stuart Turnbull of Queens University, Canada Beyond Average Intelligence Michael Curran of RiskCare Ltd Strata Gems Michael Curran of RiskCare Ltd Recovering Identity Michael Curran of RiskCare Ltd Greeks in Monte Carlo Michael Curran of RiskCare Ltd Quicker on the Curves Les Clewlow of University of Warwick and Andrew Carverhill of the University of Science and Technology, Hong Kong Exact Exotics Leif Andersen and Rupert Brotherton-Ratcliffe of General Re Financial Products Monte Carlo Simulation of Options on Joint Minima and Maxima Leif Andersen of General Re Financial Products Model Calibration in the Monte Carlo Framework Rapha?l Douady of Ecole Normale Sup?rieure, CMLA AMERICAN-STYLE Valuing American Options in a Path-Simulation Model James A. Tilley of Morgan Stanley Numerical Valuation of High-Dimensional Multivariate American Securities J?r?me Barraquand of Salomon Brothers International and Didier Martineau of Long-Term Captial Management Monte Carlo Methods for Pricing High-Dimensional American Options: An Overview Mark Broadie and Paul Glasserman of Columbia University FIXED INCOME Pricing Interst Rate Exotics by Monte Carlo Simulation Les Clewlow of Lacima Consultants Ltd, Warwick Business School and Chris Strickland of the University of Technology, Sydney, Australia Efficient and Flexible Bond Option Valuation in the Heath, Jarrow and Morton Framework Andrew Carverhill of Hong Kong University of Science and Technology and Kin Pang of Morgan Stanley Dean Witter & Co Term Structure Dynamics and Mortgage Valuation Oren Cheyette of BARRA Inc VAR Calculating Value-at-Risk with Monte Carlo Simulation Evan Picoult of Citibank Beyond VAR and Stress Testing Julian Shaw of NatWest Markets Using Non-Normal Monte Carlo Simulation to Compute Value-at-Risk Gerald D. Quinlan of TrueRisk Inc Scrambled Nets for Value-at-Risk Calculations Art Owen of Stanford University and Domingo Tavella of Align Risk Analysis DETERMINISTIC METHODS Quasi-Monte Carlo Methods in Numberical Finance Corwin Joy of Positron Energy Consulting, Phelim P. Boyle and Ken Seng Tan of University of Waterloo New Methodologies for Valuiing Derivatives Spassimir H. Paskov of Barclays Capital Valuation of Mortgage-Backed Securities Using Brownian Bridges to Reduce Effective Dimension Russel E. Caflisch of UCLA, William Morokoff of Goldman Sachs and Art Owen of Stanford University Smoothness and Dimension Reduction in Quasi-Monte Carlo Methods Bradley Moskowitz of Bettis Laboratory and Russel E. Caflisch of UCLA Beating Monte Carlo Anargyros Papageorgiou and Joseph Traub of Columbia University Monte Carlo Motoring Rupert Brotherton-Ratcliffe of General RE Financial Products Laudable Lattices Craig Stetson of Arizona Public Service, Steve Marshall and David Loebell of Chase Manhattan Inelegant Efficiency John Barrett and Gerald Moore of Imperial College and Paul Wilmott of Imperial College and University of Oxford Glossary of Monte Carlo terms
In this paper we consider the problem of computing tail probabilities of the distribution of a random sum of positive random variables. We assume that the individual claim variables follow …
In this paper we consider the problem of computing tail probabilities of the distribution of a random sum of positive random variables. We assume that the individual claim variables follow a reproducible natural exponential family (NEF) distribution, and that the random number has a NEF counting distribution with a cubic variance function. This specific modeling is supported by data of the aggregated claim distribution of an insurance company. Large tail probabilities are important as they reflect the risk of large losses, however, analytic or numerical expressions are not available. We propose several simulation algorithms which are based on an asymptotic analysis of the distribution of the counting variable and on the reproducibility property of the claim distribution. The aggregated sum is simulated efficiently by importance sampling using an exponential change of measure. We conclude by numerical experiments of these algorithms, based on real car insurance claim data.
In this paper we consider the problem of computing tail probabilities of the distribution of a random sum of positive random variables. We assume that the individual variables follow a …
In this paper we consider the problem of computing tail probabilities of the distribution of a random sum of positive random variables. We assume that the individual variables follow a reproducible natural exponential family (NEF) distribution, and that the random number has a NEF counting distribution with a cubic variance function. This specific modelling is supported by data of the aggregated claim distribution of an insurance company. Large tail probabilities are important as they reflect the risk of large losses, however, analytic or numerical expressions are not available. We propose several simulation algorithms which are based on an asymptotic analysis of the distribution of the counting variable and on the reproducibility property of the claim distribution. The aggregated sum is simulated efficiently by importancesampling using an exponential cahnge of measure. We conclude by numerical experiments of these algorithms.
In this paper we consider the problem of computing tail probabilities of the distribution of a random sum of positive random variables. We assume that the individual variables follow a …
In this paper we consider the problem of computing tail probabilities of the distribution of a random sum of positive random variables. We assume that the individual variables follow a reproducible natural exponential family (NEF) distribution, and that the random number has a NEF counting distribution with a cubic variance function. This specific modelling is supported by data of the aggregated claim distribution of an insurance company. Large tail probabilities are important as they reflect the risk of large losses, however, analytic or numerical expressions are not available. We propose several simulation algorithms which are based on an asymptotic analysis of the distribution of the counting variable and on the reproducibility property of the claim distribution. The aggregated sum is simulated efficiently by importancesampling using an exponential cahnge of measure. We conclude by numerical experiments of these algorithms.
An option is a contract which gives the owner (buyer) of the option the right, but not obligation, to buy or sell the underlying financial instrument at a predetermined price …
An option is a contract which gives the owner (buyer) of the option the right, but not obligation, to buy or sell the underlying financial instrument at a predetermined price on a specified future date or multiple dates. They are part of a broader class of financial instruments known as derivatives. Because an option is a contract that gives the owner the right and not the obligation to buy/sell, it is intuitive that its price should be greater than 0. That is true; however, it is harder to determine exactly what the price should be. Therefore, in the last 50 years, multiple algorithms were developed, which, under certain assumptions, determine the correct option price. Monte Carlo methods are one of those algorithms, which use probability theory and pseudorandom numbers to valuate options.
The aim of this thesis is to present Monte Carlo methods and their use in the valuation of options. We will describe financial markets and characteristics of derivatives. We will look at how pseudorandom numbers are generated and how to use them with Monte Carlo methods. The theory behind Monte Carlo methods will be shown, together with the assessment of their accuracy. Algorithms will be implemented in Matlab programming language.
This paper firstly introduces the method of option pricing using Monte Carlo,then,proposes one kind of Quasi-Monte Carlo Simulation,which uses Halton sequences to improve Monte Carlo Simulation.This paper also introduces generated …
This paper firstly introduces the method of option pricing using Monte Carlo,then,proposes one kind of Quasi-Monte Carlo Simulation,which uses Halton sequences to improve Monte Carlo Simulation.This paper also introduces generated rule of Halton Low Discrepancy Sequences and Moro algorithm.Finally,the performances of three kinds of Quasi-Monte Carlo method are compared.
In the following paper we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed …
In the following paper we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed to approximate expectations w.r.t a sequence of related probability measures. These approaches have been used, successfully, for a wide class of applications in engineering, statistics, physics and operations research. SMC methods are highly suited to many option pricing problems and sensitivity/Greek calculations due to the nature of the sequential simulation. However, it is seldom the case that such ideas are explicitly used in the option pricing literature. This article provides an up-to date review of SMC methods, which are appropriate for option pricing. In addition, it is illustrated how a number of existing approaches for option pricing can be enhanced via SMC. Specifically, when pricing the arithmetic Asian option w.r.t a complex stochastic volatility model, it is shown that SMC methods provide additional strategies to improve estimation.
In this article, we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed to …
In this article, we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed to approximate expectations w.r.t a sequence of related probability measures. These approaches have been used successfully for a wide class of applications in engineering, statistics, physics, and operations research. SMC methods are highly suited to many option pricing problems and sensitivity/Greek calculations due to the nature of the sequential simulation. However, it is seldom the case that such ideas are explicitly used in the option pricing literature. This article provides an up-to-date review of SMC methods, which are appropriate for option pricing. In addition, it is illustrated how a number of existing approaches for option pricing can be enhanced via SMC. Specifically, when pricing the arithmetic Asian option w.r.t a complex stochastic volatility model, it is shown that SMC methods provide additional strategies to improve estimation.
Value-at-risk (VaR) and conditional value-at-risk (CVaR) are two widely used risk measures of large losses and are employed in the financial industry for risk management purposes. In practice, loss distributions …
Value-at-risk (VaR) and conditional value-at-risk (CVaR) are two widely used risk measures of large losses and are employed in the financial industry for risk management purposes. In practice, loss distributions typically do not have closed-form expressions, but they can often be simulated (i.e., random observations of the loss distribution may be obtained by running a computer program). Therefore, Monte Carlo methods that design simulation experiments and utilize simulated observations are often employed in estimation, sensitivity analysis, and optimization of VaRs and CVaRs. In this article, we review some of the recent developments in these methods, provide a unified framework to understand them, and discuss their applications in financial risk management.
The purpose of this paper is to compare the use of Quasi-Monte Carlo methods, especially the use of recent developed (t; m; s)-nets, versus classical Monte Carlo method for valuing …
The purpose of this paper is to compare the use of Quasi-Monte Carlo methods, especially the use of recent developed (t; m; s)-nets, versus classical Monte Carlo method for valuing _nancial derivatives. Some research has indicate that under certain condition Quasi-Monte Carlo is superior than the traditional Monte Carlo in terms of rate of convergence and accuracy. In particular, theoretic results hinted that the so-called (t; m; s)-net suppose to be the most powerful one among all the Quasi-Monte Carlo methods when the problem is smooth. However, the application of (t; m; s)-net was not included in the exist-ing simulation literatures. In this paper I will introduce the algorithms of generate the most common Quasi-Monte Carlo sequences, then im- plement these sequences in several path-dependent options. Our in- vestigation showed that Quasi-Monte Carlo methods outperform the traditional Monte Carlo.
Article Free Access Share on Variance reduction of Monte Carlo and randomized quasi-Monte Carlo estimators for stochastic volatility models in finance Authors: Hatem Ben Ameur École des Hautes Études Commerciales, …
Article Free Access Share on Variance reduction of Monte Carlo and randomized quasi-Monte Carlo estimators for stochastic volatility models in finance Authors: Hatem Ben Ameur École des Hautes Études Commerciales, 3000, chemin de la Côte-Ste-Catherine, Montréal, H3T 2A7, Canada École des Hautes Études Commerciales, 3000, chemin de la Côte-Ste-Catherine, Montréal, H3T 2A7, CanadaView Profile , Pierre L'Ecuyer Département d'Informatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada Département d'Informatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, CanadaView Profile , Christiane Lemieux Département d'Informatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada Département d'Informatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, CanadaView Profile Authors Info & Claims WSC '99: Proceedings of the 31st conference on Winter simulation: Simulation---a bridge to the future - Volume 1December 1999 Pages 336–343https://doi.org/10.1145/324138.324237Published:01 December 1999Publication History 7citation763DownloadsMetricsTotal Citations7Total Downloads763Last 12 Months32Last 6 weeks6 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF
A backward induction procedure for pricing arithmetic Asian options in Lévy models is realized in the dual space. Each step of the procedure is the composition of multiplication operators by …
A backward induction procedure for pricing arithmetic Asian options in Lévy models is realized in the dual space. Each step of the procedure is the composition of multiplication operators by explicitly given functions and the convolution operator ${\mathcal H}_{\Gamma}$ which belongs to a class of natural generalizations of the Hilbert transform ${\mathcal H}$. The kernel of ${\mathcal H}_\Gamma$ being $(1/2\pi )^{-1}{\Gamma}(i(\eta-\xi))$, we call ${\mathcal H}_\Gamma$ the Gamma transform. An efficient realization of the procedure (double spiral method) is based on calculations of the functions on two parallel lines, using the fast convolution.
We outline the basic ideas and techniques underpinning the simulation of stochastic differential equations. In particular we focus on strong simulation and its context. We also provide illustratory examples and …
We outline the basic ideas and techniques underpinning the simulation of stochastic differential equations. In particular we focus on strong simulation and its context. We also provide illustratory examples and sample matlab algorithms for the reader to use and follow. Our target audience is advanced undergraduate and graduate students interested in learning about simulating stochastic differential equations. We try to address the FAQs we have encountered.
Based on new exact formulations of American option problems on bounded domains, error estimates are established for finite element approximations of American option prices under admissible regularity. Some numerical results …
Based on new exact formulations of American option problems on bounded domains, error estimates are established for finite element approximations of American option prices under admissible regularity. Some numerical results are also presented.
Monte Carlo simulation has become an essential tool for pricing and risk estimation in financial applications. It allows finance professionals to incorporate uncertainty in financial models, and to consider additional …
Monte Carlo simulation has become an essential tool for pricing and risk estimation in financial applications. It allows finance professionals to incorporate uncertainty in financial models, and to consider additional layers of complexity that are difficult to incorporate in analytical models. The main idea of Monte Carlo simulation is to represent the uncertainty in market variables through scenarios, and to evaluate parameters of interest that depend on these market variables in complex ways. The advantage of such an approach is that it can easily capture the dynamics of underlying processes and the otherwise complex effects of interactions among market variables. A substantial amount of research in recent years has been dedicated to making scenario generation more accurate and efficient, and a number of sophisticated computational techniques are now available to the financial modeler.
The paper develops an efficient Monte Carlo method to price discretely monitored Parisian options based on a control variate approach. The paper also modifies the Parisian option design by assuming …
The paper develops an efficient Monte Carlo method to price discretely monitored Parisian options based on a control variate approach. The paper also modifies the Parisian option design by assuming the option is exercised when the barrier condition is met rather than at maturity. We obtain formulas for this new design when the underlying is continuously monitored and develop an efficient Monte Carlo method for the discrete case. Our method can also be used for the case of multiple barriers. We use numerical examples to illustrate the approach and reveal important features of the different types of options considered. Some performance-based executive stock options include different tranches of discretely monitored Parisian options and we illustrate this with a practical example.
Regime-switching models have been heavily studied recently, as they have some clear advantages of over other non-constant volatility model to resolve the so-called smirk effect displayed when constant volatility models …
Regime-switching models have been heavily studied recently, as they have some clear advantages of over other non-constant volatility model to resolve the so-called smirk effect displayed when constant volatility models are used to price financial derivatives such as options. However, due to the increased model complexity, the associated computational effort usually increases as well, particularly when they are used to price American-style options. In this paper, a novel computational approach based on integral equations is presented. A distinctive feature of our approach, in comparison with other numerical approaches, is that the coupled partial differential equations (PDEs) in a PDE system have been decoupled in the Fourier space, resulting in a completely decoupled integral equation for each economical states, and thus has greatly reduced computational effort. Some examples with preliminary results for a two-state regime-switching model are used to demonstrate our approach.
Here we develop an approach for efficient pricing discrete-time American and Bermudan options which employs the fact that such options are equivalent to the European ones with a consumption, combined …
Here we develop an approach for efficient pricing discrete-time American and Bermudan options which employs the fact that such options are equivalent to the European ones with a consumption, combined with analysis of the market model over a small number of steps ahead. This approach allows constructing both upper and low bounds for the true price by Monte Carlo simulations. An adaptive choice of local low bounds and use of the kernel interpolation technique enhance efficiency of the whole procedure, which is supported by numerical experiments.
The pricing of American‐style options by simulation‐based methods is an important but difficult task primarily due to the feature of early exercise, particularly for high‐dimensional derivatives. In this paper, a …
The pricing of American‐style options by simulation‐based methods is an important but difficult task primarily due to the feature of early exercise, particularly for high‐dimensional derivatives. In this paper, a bundling method based on quasi‐Monte Carlo sequences is proposed to price high‐dimensional American‐style options. The proposed method substantially extends Tilley's bundling algorithm to higher‐dimensional situations. By using low‐discrepancy points, this approach partitions the state space and forms bundles. A dynamic programming algorithm is then applied to the bundles to estimate the continuation value of an American‐style option. A convergence proof of the algorithm is provided. A variety of examples with up to 15 dimensions are investigated numerically and the algorithm is able to produce computationally efficient results with good accuracy.
In this paper, we investigate the feasibility of using low-discrepancy sequences to allow complex derivatives, such as mortgage-backed securities (MBSs) and exotic options, to be calculated considerably faster than is …
In this paper, we investigate the feasibility of using low-discrepancy sequences to allow complex derivatives, such as mortgage-backed securities (MBSs) and exotic options, to be calculated considerably faster than is possible by using conventional Monte Carlo methods. In our experiments, we examine classical classes of low-discrepancy sequences, such as Halton, Sobol', and Faure sequences, as well as the very recent class called generalized Niederreiter sequences, in the light of the actual convergence rate of numerical integration with practical numbers of dimensions. Our results show that for the problems of pricing financial derivatives that we tested: (1) generalized Niederreiter sequences perform markedly better than both classical sequences and Monte Carlo methods; and (2) classical low-discrepancy sequences often perform worse than Monte Carlo methods. Finally, we discuss several important research issues from both practical and theoretical viewpoints.
Monte-Carlo simulations have been utilized greatly in the pricing of derivative securities. Over the years, several variance reduction techniques have been developed to curb the instability, as well as, increase …
Monte-Carlo simulations have been utilized greatly in the pricing of derivative securities. Over the years, several variance reduction techniques have been developed to curb the instability, as well as, increase the simulation efficiencies of the Monte-Carlo methods. Our approach in this research work will consider the use of antithetic variate techniques to estimate the fair prices of barrier options. Next, we use the quasi-Monte Carlo method, together with Sobol sequence to estimate the values of the same option. An extended version of the Black-Scholes model will serve as basis for the exact prices of these exotic options. The resulting simulated prices will be compared to the exact prices. The research concludes by showing some results which proves that when random numbers are generated via low discrepancy sequences in contrast to the normal pseudo-random numbers, a more efficient simulation method is ensued. This is further applicable in pricing complex derivatives without closed formsolutions.
Using number-theoretic methods, we investigate low-discrepancy sequences and weighted-sum estimators which outperform standard low-discrepancy techniques for pricing multi-asset European options on up to 5 underlying factors. The sequences used are …
Using number-theoretic methods, we investigate low-discrepancy sequences and weighted-sum estimators which outperform standard low-discrepancy techniques for pricing multi-asset European options on up to 5 underlying factors. The sequences used are simpler to implement than most low-discrepancy sequences, and computation time is considerably faster.
This work provides a simulation-based approach of assessing the risk and uncertainty involved in estimating the expected earnings of an organization. The procedure involves using Monte Carlo Simulation (MCS) in …
This work provides a simulation-based approach of assessing the risk and uncertainty involved in estimating the expected earnings of an organization. The procedure involves using Monte Carlo Simulation (MCS) in creating various possible outcomes and scenarios. The MCS is found to be more effective than single point estimates or guesswork. Hence, it is an efficient and useful tool in risk management analysis. The analysis of the output of the simulation reveals that the expected earnings is a little bit lower than the most likely forecasted value of N30m but there is 37% chance that the expected earnings might drop below or rise above the estimated value by margin of N10.9m and the wide range of possible outcomes make the venture to be very risky as uncertainties in unit sales, unit price or variable cost can push the earnings to assume any value within the wide range. It is also observed that a large increase in the unit sales and a moderate increase in the unit price will increase the expected revenue which will in turn increase the earnings. The regression analysis gives almost the same result as MCS.
An efficient conditioning technique, the so-called Brownian Bridge simulation, has previously been applied to eliminate pricing bias that arises in applications of the standard discrete-time Monte Carlo method to evaluate …
An efficient conditioning technique, the so-called Brownian Bridge simulation, has previously been applied to eliminate pricing bias that arises in applications of the standard discrete-time Monte Carlo method to evaluate options written on the continuous-time extrema of an underlying asset. It is based on the simple and easy to implement analytic formulas for the distribution of one-dimensional Brownian Bridge extremes. This paper extends the technique to the valuation of multi-asset options with knock-out barriers imposed for all or some of the underlying assets. We derive formula for the unbiased option price estimator based on the joint distribution of the multi-dimensional Brownian Bridge dependent extrema. As analytic formulas are not available for the joint distribution in general, we develop upper and lower biased option price estimators based on the distribution of independent extrema and the Fréchet lower and upper bounds for the unknown distribution. All estimators are simple and easy to implement. They can always be used to bind the true value by a confidence interval. Numerical tests indicate that our biased estimators converge rapidly to the true option value as the number of time steps for the asset path simulation increases in comparison to the estimator based on the standard discrete-time method. The convergence rate depends on the correlation and barrier structures of the underlying assets.
This paper presents an overview of techniques for improving the efficiency of option pricing simulations, including quasi-Monte Carlo methods, variance reduction, and methods for dealing with discretization error.
This paper presents an overview of techniques for improving the efficiency of option pricing simulations, including quasi-Monte Carlo methods, variance reduction, and methods for dealing with discretization error.
In this paper we present a simple yet generic method for fast and robust Monte-Carlo calculation of sensitivities of Collateralized Debt Obligations (CDOs). The method is product independent and only …
In this paper we present a simple yet generic method for fast and robust Monte-Carlo calculation of sensitivities of Collateralized Debt Obligations (CDOs). The method is product independent and only relies on four pricings against modified models. From a modeling perspective the method is also fairly general as it only relies on the availability of a conditional cumulative distribution function for the default time. In our presentation we concentrate on conditional independent loss models as given in [Li2000].
In this article we give a short overview on sensitivity calculation using Monte-Carlo simulation and an introduction to the proxy simulation scheme method. We shortly discuss the localization technique and …
In this article we give a short overview on sensitivity calculation using Monte-Carlo simulation and an introduction to the proxy simulation scheme method. We shortly discuss the localization technique and the implementation.
For the numerical calculation of partial derivatives (aka.~sensitivites or greeks) from a Monte-Carlo simulation there are essentially two possible approaches: The pathwise method and the likelihood ratio method. Both methods …
For the numerical calculation of partial derivatives (aka.~sensitivites or greeks) from a Monte-Carlo simulation there are essentially two possible approaches: The pathwise method and the likelihood ratio method. Both methods have their shortcomings: While the pathwise method works very well for smooth payouts it fails for discontinuous payouts. On the other hand, the likelihood ratio gives much better results on discontinuous payouts, but falls short of the pathwise method if smooth payouts are considered.
In this paper, we present a generic method for the Monte-Carlo pricing of (generalized) auto-callable products (aka. trigger products), i.e., products for which the payout function features a discontinuity with …
In this paper, we present a generic method for the Monte-Carlo pricing of (generalized) auto-callable products (aka. trigger products), i.e., products for which the payout function features a discontinuity with a (possibly) stochastic location (the trigger) and value (the payout). The Monte-Carlo pricing of products with discontinuous payout is known to come with a high Monte-Carlo error. The numerical calculation of sensitivities (i.e., partial derivatives) of such prices by finite differences gives very noisy results, since the Monte-Carlo approximation (being a finite sum of discontinuous functions) is not smooth. Additionally, the Monte-Carlo error of the finite-difference approximation explodes as the shift size tends to zero. Our method combines a product specific modification of the underlying numerical scheme, which is to some extent similar to an importance sampling and/or partial proxy simulation scheme and a reformulation of the payoff function into an equivalent smooth payout. From the financial product we merely require that hitting of the stochastic trigger will result in an conditionally analytic value. Many complex derivatives can be written in this form. A class of products where this property is usually encountered are the so called auto-callables, where a trigger hit results in cancellation of all future payments except for one redemption payment, which can be valued analytically, conditionally on the trigger hit. From the model we require that its numerical implementation allows for a calculation of the transition probability of survival (i.e., non-trigger hit). Many models allows this, e.g., Euler schemes of Itô processes, where the trigger is a model primitive. The method presented is effective across a large range of cases where other methods fail, e.g. small finite difference shift sizes or short time to trigger reset (approaching maturity); this means that a practitioner can use this method and be confident that it will work consistently. The method itself can be viewed as a generalization of the method proposed by Glasserman and Staum (2001), both with respect to the type (and shape) of the boundaries, as well as, with respect to the class of products considered. In addition we explicitly consider the calculation of sensitivities.
In this paper we discuss the valuation and sensitivities of financial products with early exercise rights (e.g., Bermudan options) using a Monte-Carlo simulation. The usual way to value early exercise …
In this paper we discuss the valuation and sensitivities of financial products with early exercise rights (e.g., Bermudan options) using a Monte-Carlo simulation. The usual way to value early exercise rights is the backward algorithm. As we will point out, the Monte-Carlo version of the backward algorithm is given by an unconditional expectation of a random variable whose paths are discontinuous functions of the initial data. This results in noisy sensitivities, when sensitivities are calculated from finite differences of valuations.
In this paper we present a generic method for the Monte-Carlo pricing of (generalized) auto-callable products (aka. trigger products), i.e., products for which the payout function features a discontinuity with …
In this paper we present a generic method for the Monte-Carlo pricing of (generalized) auto-callable products (aka. trigger products), i.e., products for which the payout function features a discontinuity with a (possibly) stochastic location (the trigger) and value (the payout).
We consider a generic framework for generating likelihood ratio weighted Monte Carlo simulation paths, where we use one simulation scheme K° (proxy scheme) to generate realizations and then reinterpret them …
We consider a generic framework for generating likelihood ratio weighted Monte Carlo simulation paths, where we use one simulation scheme K° (proxy scheme) to generate realizations and then reinterpret them as realizations of another scheme K* (target scheme) by adjusting measure (via likelihood ratio) to match the distribution of K° such that E( f(K*) | F_t ) = E( f(K°) w | F_t ). This is done numerically in every time step, on every path. This makes the approach independent of the product (the function f) and even of the model, it only depends on the numerical scheme. The approach is essentially a numerical version of the likelihood ratio method [Broadie & Glasserman, 1996] and Malliavin's Calculus [Fournie et al., 1999; Malliavin, 1997] reconsidered on the level of the discrete numerical simulation scheme. Since the numerical scheme represents a time discrete stochastic process sampled on a discrete probability space the essence of the method may be motivated without a deeper mathematical understanding of the time continuous theory (e.g. Malliavin's Calculus). The framework is completely generic and may be used for high accuracy drift approximations and the robust calculation of partial derivatives of expectations w.r.t. model parameters (i.e. sensitivities, aka. Greeks) by applying finite differences by reevaluating the expectation with a model with shifted parameters. We present numerical results using a Monte-Carlo simulation of the LIBOR Market Model for benchmarking.
Abstract Monte Carlo simulation methods are widely used in several scientific disciplines. They are flexible and easy to apply, but their corresponding error is sometimes deemed too large. This error …
Abstract Monte Carlo simulation methods are widely used in several scientific disciplines. They are flexible and easy to apply, but their corresponding error is sometimes deemed too large. This error can be reduced by increasing the sample size, but a more effective approach is to apply cleverly designed variance reduction techniques . Control variates is an example of such a technique. It aims at reducing the Monte Carlo estimator's variance using a correcting factor that depends on the distance between a control variate and its expectation. This article explains how to use this technique and why it works.
The calculation of option Greeks is vital for risk management. Traditional pathwise and finite-difference methods work poorly for higher-order Greeks and options with discontinuous payoff functions. The Quasi-Monte Carlo-based conditional …
The calculation of option Greeks is vital for risk management. Traditional pathwise and finite-difference methods work poorly for higher-order Greeks and options with discontinuous payoff functions. The Quasi-Monte Carlo-based conditional pathwise method (QMC-CPW) for options Greeks allows the payoff function of options to be effectively smoothed, allowing for increased efficiency when calculating sensitivities. Another way to increase efficiency is to utilize the increased computational speed by applying GPUs to highly parallelisable finance problems such as calculating Greeks. We pair QMC-CPW with simulation on the GPU using the CUDA platform. We estimate the delta, vega and gamma Greeks of three exotic options: arithmetic Asian, binary Asian, and look-back. The increased computational speed through usage of the GPU is also shown with achieved speedups over sequential CPU implementations of more than 200x for our most accurate method.
Monte Carlo simulations based on quasi-random Sobol sequences and pseudo-random numbers are compared with and without a stratified sampling variance reduction procedure (VR). We find that Sobol sequences are superior …
Monte Carlo simulations based on quasi-random Sobol sequences and pseudo-random numbers are compared with and without a stratified sampling variance reduction procedure (VR). We find that Sobol sequences are superior to pseudo-random without VR and comparable with VR.
This paper introduces random versions of successive approximations and multigrid algorithms for computing approximate solutions to a class of finite and infinite horizon Markovian decision problems (MDPs). We prove that …
This paper introduces random versions of successive approximations and multigrid algorithms for computing approximate solutions to a class of finite and infinite horizon Markovian decision problems (MDPs). We prove that these algorithms succeed in breaking the curse of dimensionality for a subclass of MDPs known as discrete decision processes (DDPs).
The effectiveness of a deterministic algorithm recently proposed by Wozniakowski (1991) for the numerical estimation of high-dimensional integrals based on so-called optimal sampling points is studied by using it to …
The effectiveness of a deterministic algorithm recently proposed by Wozniakowski (1991) for the numerical estimation of high-dimensional integrals based on so-called optimal sampling points is studied by using it to compute virial coefficients for the hard-sphere-fluid. Although the algorithm is inherently more efficient than the familiar Monte Carlo method for sufficiently large samples, in practice this advantage diminishes rapidly with increasing dimensionality. For the viral coefficient problem, the authors find the new algorithm to be noticeably more efficient than crude Monte Carlo methods for dimensions less than 10 and of comparable efficiency for larger dimensionality up to 14.
Monte Carlo simulation is one alternative for analyzing options markets when the assumptions of simpler analytical models are violated. We introduce techniques for the sensitivity analysis of option pricing, which …
Monte Carlo simulation is one alternative for analyzing options markets when the assumptions of simpler analytical models are violated. We introduce techniques for the sensitivity analysis of option pricing, which can be efficiently carried out in the simulation. In particular, using these techniques, a single run of the simulation would often provide not only an estimate of the option value but also estimates of the sensitivities of the option value to various parameters of the model. Both European and American options are considered, starting with simple analytically tractable models to present the idea and proceeding to more complicated examples. We then propose an approach for the pricing of options with early exercise features by incorporating the gradient estimates in an iterative stochastic approximation algorithm. The procedure is illustrated in a simple example estimating the option value of an American call. Numerical results indicate that the additional computational effort required over that required to estimate a European option is relatively small.
The discrepancy of a sequence of pseudo-random numbers generated by the linear congruential method is estimated for parts of the period which are somewhat larger than the square root of …
The discrepancy of a sequence of pseudo-random numbers generated by the linear congruential method is estimated for parts of the period which are somewhat larger than the square root of the modulus. Applications to numerical integration are mentioned.
The likelihood ratio method for gradient estimation is briefly surveyed. Two applications settings are described, namely Monte Carlo optimization and statistical analysis of complex stochastic systems. Steady-state gradient estimation is …
The likelihood ratio method for gradient estimation is briefly surveyed. Two applications settings are described, namely Monte Carlo optimization and statistical analysis of complex stochastic systems. Steady-state gradient estimation is emphasized, and both regenerative and non-regenerative approaches are given. The paper also indicates how these methods apply to general discrete-event simulations; the idea is to view such systems as general state space Markov chains.
article Free AccessArtifacts Evaluated & ReusableArtifacts Available Share on Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators Author: Bennett L. Fox Computer Science Department, University of Montreal, P.O. …
article Free AccessArtifacts Evaluated & ReusableArtifacts Available Share on Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators Author: Bennett L. Fox Computer Science Department, University of Montreal, P.O. Box 6128, Station A, Montreal, Quebec, Canada H3C 3J7 Computer Science Department, University of Montreal, P.O. Box 6128, Station A, Montreal, Quebec, Canada H3C 3J7View Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 12Issue 4pp 362–376https://doi.org/10.1145/22721.356187Published:01 December 1986Publication History 117citation1,807DownloadsMetricsTotal Citations117Total Downloads1,807Last 12 Months78Last 6 weeks10 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF
We show that under the (sufficient) conditions usually given for infinitesimal perturbation analysis (IPA) to apply for derivative estimation, a finite-difference scheme with common random numbers (FDC) has the same …
We show that under the (sufficient) conditions usually given for infinitesimal perturbation analysis (IPA) to apply for derivative estimation, a finite-difference scheme with common random numbers (FDC) has the same order of convergence, namely O(n −1/2 ), provided that the size of the finite-difference interval converges to zero fast enough. This holds for both one- and two-sided FDC. This also holds for different variants of IPA, such as some versions of smoothed perturbation analysis (SPA), which is based on conditional expectation. Finally, this also holds for the estimation of steady-state performance measures by truncated-horizon estimators, under some ergodicity assumptions. Our developments do not involve monotonicity, but are based on continuity and smoothness. We give examples and numerical illustrations which show that the actual difference in mean square error (MSE) between IPA and FDC is typically negligible. We also obtain the order of convergence of that difference, which is faster than the convergence of the MSE to zero.
In this paper, we investigate the feasibility of using low-discrepancy sequences to allow complex derivatives, such as mortgage-backed securities (MBSs) and exotic options, to be calculated considerably faster than is …
In this paper, we investigate the feasibility of using low-discrepancy sequences to allow complex derivatives, such as mortgage-backed securities (MBSs) and exotic options, to be calculated considerably faster than is possible by using conventional Monte Carlo methods. In our experiments, we examine classical classes of low-discrepancy sequences, such as Halton, Sobol', and Faure sequences, as well as the very recent class called generalized Niederreiter sequences, in the light of the actual convergence rate of numerical integration with practical numbers of dimensions. Our results show that for the problems of pricing financial derivatives that we tested: (1) generalized Niederreiter sequences perform markedly better than both classical sequences and Monte Carlo methods; and (2) classical low-discrepancy sequences often perform worse than Monte Carlo methods. Finally, we discuss several important research issues from both practical and theoretical viewpoints.
This paper introduces and illustrates a new version of the Monte Carlo method that has attractive properties for the numerical valuation of derivatives. The traditional Monte Carlo method has proven …
This paper introduces and illustrates a new version of the Monte Carlo method that has attractive properties for the numerical valuation of derivatives. The traditional Monte Carlo method has proven to be a powerful and flexible tool for many types of derivatives calculations. Under the conventional approach pseudo-random numbers are used to evaluate the expression of interest. Unfortunately, the use of pseudo-random numbers yields an error bound that is probabilistic which can be a disadvantage. Another drawback of the standard approach is that many simulations may be required to obtain a high level of accuracy. There are several ways to improve the convergence of the standard method. This paper suggests a new approach which promises to be very useful for applications in finance. Quasi-Monte Carlo methods use sequences that are deterministic instead of random. These sequences improve convergence and give rise to deterministic error bounds. The method is explained and illustrated with several examples. These examples include complex derivatives such as basket options, Asian options, and energy swaps.
Preface 1. Monte Carlo methods and Quasi-Monte Carlo methods 2. Quasi-Monte Carlo methods for numerical integration 3. Low-discrepancy point sets and sequences 4. Nets and (t,s)-sequences 5. Lattice rules for …
Preface 1. Monte Carlo methods and Quasi-Monte Carlo methods 2. Quasi-Monte Carlo methods for numerical integration 3. Low-discrepancy point sets and sequences 4. Nets and (t,s)-sequences 5. Lattice rules for numerical integration 6. Quasi- Monte Carlo methods for optimization 7. Random numbers and pseudorandom numbers 8. Nonlinear congruential pseudorandom numbers 9. Shift-Register pseudorandom numbers 10. Pseudorandom vector generation Appendix A. Finite fields and linear recurring sequences Appendix B. Continued fractions Bibliography Index.
Introduction. In this paper we shall give an account of some methods developed for the numerical evaluation of multidimensional integrals.These methods are based on the theory of Diophantine approximation.They are …
Introduction. In this paper we shall give an account of some methods developed for the numerical evaluation of multidimensional integrals.These methods are based on the theory of Diophantine approximation.They are suitable for some problems for which the Monte Carlo method is commonly used and, like the Monte Carlo method, are well fitted for use with an electronic digital computer.We shall show, however, that they are superior to the Monte Carlo method provided that the integrand satisfies certain conditions.We shall also show that they are superior, for integrals in space of several dimensions, to formulas typified by those of Gauss and Simpson; they may be superior even to certain new integration formulas specially constructed for the evaluation of multiple integrals (see for example Hammer [2], who gives a bibliography, and Miller [5], [6], [7]).The method of antithetic variâtes which is described by Hammersley and others [3], [4] may be used to obtain better estimates than the Monte Carlo method but the author thinks that the method described in the present paper is simpler to apply and gives better results.Various authors have suggested methods which are particular cases of those described in this paper but without the underlying theory.See for example Davis and Rabinowitz [1].In this section we shall give a short account of the behavior of the error in the Monte Carlo method and the direct-product Gauss-type methods so that we can compare these with the errors of the new methods.We shall not give an account of the method of antithetic variâtes.Suppose that we wish to estimate the integral 1 = / / " ' / f(Xl >Xl> " ' ' x*) dxi d%2 • • ■ dxk. Jo Jo JoWe shall denote the vector (xi, x2, • ■ ■ , xk) by x.Numerical methods for the evaluation of I involve the calculation of /(x) at a number N of points x¿.The most desirable of such methods for use on an electronic computer are those which require the evaluation of/(x) at the smallest number of points x, in order to obtain an estimate with a sufficiently small error.The Monte Carlo method gives as an estimate for I the sum where the points x¿ are chosen at random in the range of integration.The error of such an estimate has standard deviation 0{N~m) provided that the function /(x) satisfies certain conditions.It is sufficient that the function be bounded.A Gauss-type formula for a one-dimensional integral takes the form r>a+h v \ g(y) dy = h^Ci g(a + «¿Ä) + R.
We compare empirically accuracy and speed of low-discrepancy sequence generators of Sobol' and Faure. These generators are useful for multidimensional integration and global optimization. We discuss our implementation of the …
We compare empirically accuracy and speed of low-discrepancy sequence generators of Sobol' and Faure. These generators are useful for multidimensional integration and global optimization. We discuss our implementation of the Sobol' generator.
The proper justification of the normal practice of Monte Carlo integration must be based not on the randomness of the procedure, which is spurious, but on equidistribution properties of the …
The proper justification of the normal practice of Monte Carlo integration must be based not on the randomness of the procedure, which is spurious, but on equidistribution properties of the sets of points at which the integrand values are computed. Besides the discrepancy, which it is proposed to call henceforth extreme discrepancy, another concept, that of mean square discrepancy, can be regarded as a measure of the lack of equidistribution of a sequence of points in a multidimensional cube. Determinate upper bounds can be obtained, in terms of either discrepancy, for the absolute value of the error in the computation of the integral. There exist sequences of points yielding, for sufficiently smooth functions, errors of a much smaller order of magnitude than that which is claimed by the Monte Carlo method. In the case of two dimensions, sequences with optimum properties can be generated with the help of Fibonacci numbers. The previous arguments do not apply to domains of integration which cannot be reduced to multidimensional intervals. Difficult questions arising in this connection still await an answer.
In this paper, we discuss some research issues related to the general topic of optimizing a stochastic system via simulation. In particular, we devote extensive attention to finite-difference estimators of …
In this paper, we discuss some research issues related to the general topic of optimizing a stochastic system via simulation. In particular, we devote extensive attention to finite-difference estimators of objective function gradients and present a number of new limit theorems. We also discuss a new family of orthogonal function approximations to the global behavior of the objective function. We show that if the objective function is sufficiently smooth, the convergence rate can be made arbitrarily close to n-1/2 in the number of observations required. The paper concludes with a brief discussion of how these ideas can be integrated into an optimization algorithm.
By a filtered Monte Carlo estimator we mean one whose constituent parts—summands or integral increments—are conditioned on an increasing family of σ-fields. Unbiased estimators of this type are suggested by …
By a filtered Monte Carlo estimator we mean one whose constituent parts—summands or integral increments—are conditioned on an increasing family of σ-fields. Unbiased estimators of this type are suggested by compensator identities. Replacing a point-process integrator with its intensity gives rise to one class of examples; exploiting Levy's formula gives rise to another. We establish variance inequalities complementing compensator identities. Among estimators that are (Stieltjes) stochastic integrals, we show that filtering reduces variance if the integrand and the increments of the integrator have conditional positive correlation. We also provide more primitive hypotheses that ensure this condition, making use of stochastic monotonicity properties. Our most detailed conditions apply in a Markov setting where monotone, up-down, and convex generators play a central role. We give examples. As an application of our results, we compare certain estimators that do and do not exploit the property that Poisson arrivals see time averages.
A decision-theoretic framework is proposed for evaluating the efficiency of simulation estimators. The framework includes the cost of obtaining the estimate as well as the cost of acting based on …
A decision-theoretic framework is proposed for evaluating the efficiency of simulation estimators. The framework includes the cost of obtaining the estimate as well as the cost of acting based on the estimate. The cost of obtaining the estimate and the estimate itself are represented as realizations of jointly distributed stochastic processes. In this context, the efficiency of a simulation estimator based on a given computational budget is defined as the reciprocal of the risk (the overall expected cost). This framework is appealing philosophically, but it is often difficult to apply in practice (e.g., to compare the efficiency of two different estimators) because only rarely can the efficiency associated with a given computational budget be calculated. However, a useful practical framework emerges in a large sample context when we consider the limiting behavior as the computational budget increases. A limit theorem established for this model supports and extends a fairly well known efficiency principle, proposed by J. M. Hammersley and D. C. Handscomb: “The efficiency of a Monte Carlo process may be taken as inversely proportional to the product of the sampling variance and the amount of labour expended in obtaining this estimate.”
High-dimensional integrals are usually solved with Monte Carlo algorithms although theory suggests that low-discrepancy algorithms are sometimes superior. We report on numerical testing which compares low-discrepancy and Monte Carlo algorithms …
High-dimensional integrals are usually solved with Monte Carlo algorithms although theory suggests that low-discrepancy algorithms are sometimes superior. We report on numerical testing which compares low-discrepancy and Monte Carlo algorithms on the evaluation of financial derivatives. The testing is performed on a Collateralized Mortgage Obligation (CMO) which is formulated as the computation of ten integrals of dimension up to 360. We tested two low-discrepancy algorithms (Sobol and Halton) and two randomized algorithms (classical Monte Carlo and Monte Carlo combined with antithetic variables). We conclude that for this CMO the Sobol algorithm is always superior to the other algorithms. We believe that it will be advantageous to use the Sobol algorithm for many other types of financial derivatives. Our conclusion regarding the superiority of the Sobol algorithm also holds when a rather small number of sample points are used, an important case in practice. We have built a software system called FINDER for computing high-dimensional integrals. FINDER runs on a heterogeneous network of workstations under PVM 3.2 (Parallel Virtual Machine). Since workstations are ubiquitous, this is a cost-effect way to do large computations fast. The measured speedup is at least .9N for $N$ workstations, $N$ less than or equal to 25. The software can also be used to compute high-dimensional integrals on a single workstation. A routine for generating Sobol points may be found, for example, in Numerical Recipes in C by Press et al. However, we incorporated major improvements in FINDER and we stress that the results reported in this paper were obtained using FINDER. One of the improvements was developing the table of primitive polynomials and initial direction numbers for dimensions up to 360.
Much of the recent work dealing with quasi-random methods has been aimed at establishing the best possible asymptotic rates of convergence to zero of the error resulting when a finite-dimensional …
Much of the recent work dealing with quasi-random methods has been aimed at establishing the best possible asymptotic rates of convergence to zero of the error resulting when a finite-dimensional integral is replaced by a finite sum of integrand values. In contrast with this perspective to concentrate on asymptotic convergence rates, this paper emphasizes quasi-random methods that are effective for all sample sizes. Throughout the paper, the problem of estimating finite-dimensional integrals is used to illustrate the major ideas, although much of what is done applies equally to the problem of solving certain Fredholm integral equations. Some new techniques, based on error-reducing transformations of the integrand, are described that have been shown to be useful both in estimating high-dimensional integrals and in solving integral equations. These techniques illustrate the utility of carrying over to the quasi-Monte Carlo method certain devices that have proven to be very valuable in statistical (pseudorandom) Monte Carlo applications.
Low-discrepancy sequences are used for numerical integration, in simulation, and in related applications. Techniques for producing such sequences have been proposed by, among others, Halton, Sobol´, Faure, and Niederreiter. Niederreiter's …
Low-discrepancy sequences are used for numerical integration, in simulation, and in related applications. Techniques for producing such sequences have been proposed by, among others, Halton, Sobol´, Faure, and Niederreiter. Niederreiter's sequences have the best theoretical asymptotic properties. The paper describes two ways to implement the latter sequences on a computer and discusses the results obtained in various practical tests on particular integrals.
The quasi-Monte Carlo method for financial valuation and other integration problems has error bounds of size O((log N)k N-1), or even O((log N)k N-3/2), which suggests significantly better performance than …
The quasi-Monte Carlo method for financial valuation and other integration problems has error bounds of size O((log N)k N-1), or even O((log N)k N-3/2), which suggests significantly better performance than the error size O(N-1/2) for standard Monte Carlo. But in high-dimensional problems, this benefit might not appear at feasible sample sizes. Substantial improvements from quasi-Monte Carlo integration have, however, been reported for problems such as the valuation of mortgage-backed securities, in dimensions as high as 360. The authors believe that this is due to a lower effective dimension of the integrand in those cases. This paper defines the effective dimension and shows in examples how the effective dimension may be reduced by using a Brownian bridge representation.
The discrepancy of a sequence of pseudo-random numbers generated by the linear congruential method, both homogeneous and inhomogeneous, is estimated for parts of the period that are somewhat larger than …
The discrepancy of a sequence of pseudo-random numbers generated by the linear congruential method, both homogeneous and inhomogeneous, is estimated for parts of the period that are somewhat larger than the square root of the modulus. The analogous problem for an arbitrary linear congruential generator modulo a prime is also considered, the result being particularly interesting for maximal period sequences. It is shown that the discrepancy estimates in this paper are best possible apart from logarithmic factors.