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Variance Reduction For Monte Carlo Simulation Of European American Or Barrier Options In A Stochastic Volatility Environment
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Book Synopsis Variance Reduction for Monte Carlo Simulation of European, American Or Barrier Options in a Stochastic Volatility Environment by :
Download or read book Variance Reduction for Monte Carlo Simulation of European, American Or Barrier Options in a Stochastic Volatility Environment written by and published by . This book was released on 2002 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work we develop a methodology to reduce the variance when applying Monte Carlo simulation to the pricing of a European, American or Barrier option in a stochastic volatility environment. We begin by presenting some applicable concepts in the theory of stochastic differential equations. Secondly, we develop the model for the evolution of an asset price under constant volatility. We next present the replicating portfolio and equivalent martingale measure approaches to the pricing of a European style option. Modeling an asset price utilizing constant volatility has been shown to be an inefficient model[8,16]. One way to compensate for this inefficiency is the use of stochastic volatility models, which involves modeling the volatility as a function of a stochastic process[26]. A class of these models is presented and a discussion is given on how to price European options in this framework. After developing the methods of how to price, we begin our discussion on Monte Carlo simulation of European options in a stochastic volatility environment. We start by describing how to simulate Monte Carlo for a diffusion process modeled as a stochastic differential equation. The essential element to our variance reduction technique, which is known as importance sampling, is hereafter presented. Importance sampling requires a preliminary approximation to the expectation of interest, which we obtain by a fast mean-reversion expansion of the pricing partial differential equation[22,6]. A detailed discussion is given on this fast mean-reversion expansion technique, which was first presented in [10]. We shall compare utilizing this method of expansion with that developed in [11], which is know as small noise expansion, and demonstrate numerically the efficiency of the fast mean-reversion expansion, in particular in the presence of a skew. We next wish to apply our variance reduction technique to the pricing of an American and barrier option. A discussion is given on how to price.
Book Synopsis Variance Reduction for Monte Carlo Simulation of European, American Or Barrier Options in a Stochastic Volatility Environment by : Tracey Andrew Tullie
Download or read book Variance Reduction for Monte Carlo Simulation of European, American Or Barrier Options in a Stochastic Volatility Environment written by Tracey Andrew Tullie and published by . This book was released on 2002 with total page 115 pages. Available in PDF, EPUB and Kindle. Book excerpt: Keywords: importance sampling, variance reduction, volatility, fast mean-reverting asymptotics.
Book Synopsis Monte Carlo Methods in Derivative Modelling by : Kai Zhang
Download or read book Monte Carlo Methods in Derivative Modelling written by Kai Zhang and published by . This book was released on 2011 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Dissertation Abstracts International by :
Download or read book Dissertation Abstracts International written by and published by . This book was released on 2003 with total page 674 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Valuation of Exotic Barrier Options and American Options Using Monte Carlo Simulation by : Pokpong Chirayukool
Download or read book The Valuation of Exotic Barrier Options and American Options Using Monte Carlo Simulation written by Pokpong Chirayukool and published by . This book was released on 2011 with total page 438 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Valuation of Barrier Options Using Sequential Monte Carlo by : Pavel V. Shevchenko
Download or read book Valuation of Barrier Options Using Sequential Monte Carlo written by Pavel V. Shevchenko and published by . This book was released on 2015 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sequential Monte Carlo (SMC) methods have successfully been used in many applications in engineering, statistics and physics. However, these are seldom used in financial option pricing literature and practice. This paper presents SMC method for pricing barrier options with continuous and discrete monitoring of the barrier condition. Under the SMC method, simulated asset values rejected due to barrier condition are re-sampled from asset samples that do not breach the barrier condition improving the efficiency of the option price estimator; while under the standard Monte Carlo many simulated asset paths can be rejected by the barrier condition making it harder to estimate option price accurately. We compare SMC with the standard Monte Carlo method and demonstrate that the extra effort to implement SMC when compared with the standard Monte Carlo is very little while improvement in price estimate can be significant. Both methods result in unbiased estimators for the price converging to the true value as 1/ sqrt{M}$, where $M$ is the number of simulations (asset paths). However, the variance of SMCestimator is smaller and does not grow with the number of time steps when compared to the standard Monte Carlo. In this paper we demonstrate that SMC can successfully be used for pricing barrier options. SMC can also be used for pricing other exotic options and also for cases with many underlying assets and additional stochastic factors such as stochastic volatility; we provide general formulas and references.
Book Synopsis Student Solutions Manual to accompany Simulation and the Monte Carlo Method, Student Solutions Manual by : Dirk P. Kroese
Download or read book Student Solutions Manual to accompany Simulation and the Monte Carlo Method, Student Solutions Manual written by Dirk P. Kroese and published by John Wiley & Sons. This book was released on 2012-01-20 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: This accessible new edition explores the major topics in Monte Carlo simulation Simulation and the Monte Carlo Method, Second Edition reflects the latest developments in the field and presents a fully updated and comprehensive account of the major topics that have emerged in Monte Carlo simulation since the publication of the classic First Edition over twenty-five years ago. While maintaining its accessible and intuitive approach, this revised edition features a wealth of up-to-date information that facilitates a deeper understanding of problem solving across a wide array of subject areas, such as engineering, statistics, computer science, mathematics, and the physical and life sciences. The book begins with a modernized introduction that addresses the basic concepts of probability, Markov processes, and convex optimization. Subsequent chapters discuss the dramatic changes that have occurred in the field of the Monte Carlo method, with coverage of many modern topics including: Markov Chain Monte Carlo Variance reduction techniques such as the transform likelihood ratio method and the screening method The score function method for sensitivity analysis The stochastic approximation method and the stochastic counter-part method for Monte Carlo optimization The cross-entropy method to rare events estimation and combinatorial optimization Application of Monte Carlo techniques for counting problems, with an emphasis on the parametric minimum cross-entropy method An extensive range of exercises is provided at the end of each chapter, with more difficult sections and exercises marked accordingly for advanced readers. A generous sampling of applied examples is positioned throughout the book, emphasizing various areas of application, and a detailed appendix presents an introduction to exponential families, a discussion of the computational complexity of stochastic programming problems, and sample MATLAB® programs. Requiring only a basic, introductory knowledge of probability and statistics, Simulation and the Monte Carlo Method, Second Edition is an excellent text for upper-undergraduate and beginning graduate courses in simulation and Monte Carlo techniques. The book also serves as a valuable reference for professionals who would like to achieve a more formal understanding of the Monte Carlo method.
Book Synopsis Contributions to the Theory of Monte Carlo and Quasi-Monte Carlo Methods by : Giray Okten
Download or read book Contributions to the Theory of Monte Carlo and Quasi-Monte Carlo Methods written by Giray Okten and published by Universal-Publishers. This book was released on 1999 with total page 91 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quasi-Monte Carlo methods, which are often described as deterministic versions of Monte Carlo methods, were introduced in the 1950s by number theoreticians. They improve several deficiencies of Monte Carlo methods; such as providing estimates with deterministic bounds and avoiding the paradoxical difficulty of generating random numbers in a computer. However, they have their own drawbacks. First, although they provide faster convergence than Monte Carlo methods asymptotically, the advantage may not be practical to obtain in "high" dimensional problems. Second, there is not a practical way to measure the error of a quasi-Monte Carlo simulation. Finally, unlike Monte Carlo methods, there is a scarcity of error reduction techniques for these methods. In this dissertation, we attempt to provide remedies for the disadvantages of quasi-Monte Carlo methods mentioned above. In the first part of the dissertation, a hybrid-Monte Carlo sequence designed to obtain error reduction in high dimensions is studied. Probabilistic results on the discrepancy of this sequence as well as results obtained by applying the sequence to problems from numerical integration and mathematical finance are presented. In the second part of the dissertation, a new hybrid-Monte Carlo method is introduced, in an attempt to obtain a practical statistical error analysis using low-discrepancy sequences. It is applied to problems from mathematical finance and particle transport theory to compare its effectiveness with the conventional methods. In the last part of the dissertation, a generalized quasi-Monte Carlo integration rule is introduced. A Koksma-Hlawka type inequality for the rule is proved, using a new concept for the variation of a function. As a consequence of the rule, error reduction techniques and in particular an "importance sampling" type statement are derived. Problems from different disciplines are used as practical tests for our methods. The numerical results obtained in favor of the methods suggest the practical advantages that can be realized by their use in a wide variety of applications.
Book Synopsis Lectures on Monte Carlo Methods by : Neal Noah Madras
Download or read book Lectures on Monte Carlo Methods written by Neal Noah Madras and published by American Mathematical Soc.. This book was released on 2002 with total page 113 pages. Available in PDF, EPUB and Kindle. Book excerpt: Monte Carlo methods form an experimental branch of mathematics that employs simulations driven by random number generators. These methods are often used when others fail, since they are much less sensitive to the ``curse of dimensionality'', which plagues deterministic methods in problems with a large number of variables. Monte Carlo methods are used in many fields: mathematics, statistics, physics, chemistry, finance, computer science, and biology, for instance. This book is an introduction to Monte Carlo methods for anyone who would like to use these methods to study various kinds of mathematical models that arise in diverse areas of application. The book is based on lectures in a graduate course given by the author. It examines theoretical properties of Monte Carlo methods as well as practical issues concerning their computer implementation and statistical analysis. The only formal prerequisite is an undergraduate course in probability. The book is intended to be accessible to students from a wide range of scientific backgrounds. Rather than being a detailed treatise, it covers the key topics of Monte Carlo methods to the depth necessary for a researcher to design, implement, and analyze a full Monte Carlo study of a mathematical or scientific problem. The ideas are illustrated with diverse running examples. There are exercises sprinkled throughout the text. The topics covered include computer generation of random variables, techniques and examples for variance reduction of Monte Carlo estimates, Markov chain Monte Carlo, and statistical analysis of Monte Carlo output.
Book Synopsis First Passage Time Density Approach to Pricing Barrier Options and Monte Carlo Simulation of the HJM Interest Rate Model by : Zhenyu Duanmu
Download or read book First Passage Time Density Approach to Pricing Barrier Options and Monte Carlo Simulation of the HJM Interest Rate Model written by Zhenyu Duanmu and published by . This book was released on 1994 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Simulation and the Monte Carlo Method by : Reuven Y. Rubinstein
Download or read book Simulation and the Monte Carlo Method written by Reuven Y. Rubinstein and published by Wiley. This book was released on 2016-11-21 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Valuation of American Options by : David Animante
Download or read book Valuation of American Options written by David Animante and published by . This book was released on 2016 with total page 55 pages. Available in PDF, EPUB and Kindle. Book excerpt: The use of American style equity options as hedging instrument has gained currency in recent times. This phenomenon devolves from the ever-expanding need by individuals, corporations and governments to hedge away their financial risks and the clarion call for derivative securities that give the holder increased flexibility in exercise. Nevertheless, pricing American options is complex and there exists no analytic solution to the problem except a profusion of approximation and finite difference techniques. Indeed, many researchers have shown that these methods cannot handle multifactor situations where the underlying asset follows a jump-diffusion process and where the derivative security depends on multiple sources of uncertainty such as stochastic volatility, stochastic interest rate among others. Monte-Carlo simulation techniques therefore developed out of the search for a pricing formula that has the capacity to accommodate all forms of uncertainty and at the same time able to produce speedy and accurate results. Some scholars at first rejected these techniques as yielding inaccurate results but in recent times, many researchers have demonstrated the efficacy of Monte-Carlo simulation in option pricing. The aim of this study is to assess the effectiveness of Monte-Carlo simulation methods in comparison with other option pricing techniques. To achieve this objective, the research builds an algorithm to compute Call and Put prices based on a wide range of input parameters. It also develops a model where volatility or interest rate is stochastic and a deterministic function of time. The results indicate that Monte-Carlo simulation techniques produce option values and exercise boundaries that are very similar to the Binomial, Barone-Adesi and Whaley as well as the Explicit Finite Difference methods. The results also show that the stochastic volatility and stochastic interest rate models yield slightly different but more accurate results. Consequently, the study recommends simulation techniques that incorporate multiple sources of uncertainty simultaneously for fast, efficient and more accurate option pricing.
Book Synopsis Handbook in Monte Carlo Simulation by : Paolo Brandimarte
Download or read book Handbook in Monte Carlo Simulation written by Paolo Brandimarte and published by John Wiley & Sons. This book was released on 2014-06-20 with total page 620 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible treatment of Monte Carlo methods, techniques, and applications in the field of finance and economics Providing readers with an in-depth and comprehensive guide, the Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics presents a timely account of the applicationsof Monte Carlo methods in financial engineering and economics. Written by an international leading expert in thefield, the handbook illustrates the challenges confronting present-day financial practitioners and provides various applicationsof Monte Carlo techniques to answer these issues. The book is organized into five parts: introduction andmotivation; input analysis, modeling, and estimation; random variate and sample path generation; output analysisand variance reduction; and applications ranging from option pricing and risk management to optimization. The Handbook in Monte Carlo Simulation features: An introductory section for basic material on stochastic modeling and estimation aimed at readers who may need a summary or review of the essentials Carefully crafted examples in order to spot potential pitfalls and drawbacks of each approach An accessible treatment of advanced topics such as low-discrepancy sequences, stochastic optimization, dynamic programming, risk measures, and Markov chain Monte Carlo methods Numerous pieces of R code used to illustrate fundamental ideas in concrete terms and encourage experimentation The Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics is a complete reference for practitioners in the fields of finance, business, applied statistics, econometrics, and engineering, as well as a supplement for MBA and graduate-level courses on Monte Carlo methods and simulation.
Book Synopsis Adaptive Control Variates in Monte Carlo Simulation by : Sujin Kim
Download or read book Adaptive Control Variates in Monte Carlo Simulation written by Sujin Kim and published by . This book was released on 2006 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Sequential Monte Carlo Pricing of American-Style Options Under Stochastic Volatility Models by : Bhojnarine Rambharat
Download or read book Sequential Monte Carlo Pricing of American-Style Options Under Stochastic Volatility Models written by Bhojnarine Rambharat and published by . This book was released on 2013 with total page 43 pages. Available in PDF, EPUB and Kindle. Book excerpt: We introduce a new method to price American-style options on underlying investments governed by stochastic volatility (SV) models. The method does not require the volatility process to be observed. Instead, it exploits the fact that the optimal decision functions in the corresponding dynamic programming problem can be expressed as functions of conditional distributions of volatility, given observed data. By constructing statistics summarizing information about these conditional distributions, one can obtain high quality approximate solutions. Although the required conditional distributions are in general intractable, they can be arbitrarily precisely approximated using sequential Monte Carlo schemes. The drawback, as with many Monte Carlo schemes, is potentially heavy computational demand. We present two variants of the algorithm, one closely related to the well-known least-squares Monte Carlo algorithm of Longstaff and Schwartz (2001), and the other solving the same problem using a “brute force” gridding approach. We estimate an illustrative SV model using Markov chain Monte Carlo (MCMC) methods for three equities. We also demonstrate the use of our algorithm by estimating posterior distributions of the market price of volatility risk for each of the three equities.
Book Synopsis Hybrid Monte Carlo Methods in Machine Learning by : Wilson Tsakane Mongwe
Download or read book Hybrid Monte Carlo Methods in Machine Learning written by Wilson Tsakane Mongwe and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Dimension and Variance Reduction Monte-Carlo Method for Option Pricing Under Jump-Diffusion Models by : Duy-Minh Dang
Download or read book A Dimension and Variance Reduction Monte-Carlo Method for Option Pricing Under Jump-Diffusion Models written by Duy-Minh Dang and published by . This book was released on 2017 with total page 35 pages. Available in PDF, EPUB and Kindle. Book excerpt: We develop a highly efficient MC method for computing plain vanilla European option prices and hedging parameters under a very general jump-diffusion option pricing model which includes stochastic variance and multi-factor Gaussian interest short rate(s). The focus of our MC approach is variance reduction via dimension reduction. More specifically, the option price is expressed as an expectation of a unique solution to a conditional Partial Integro-Differential Equation (PIDE), which is then solved using a Fourier transform technique. Important features of our approach are (i) the analytical tractability of the conditional PIDE is fully determined by that of the Black-Scholes-Merton model augmented with the same jump component as in our model, and (ii) the variances associated with all the interest rate factors are completely removed when evaluating the expectation via iterated conditioning applied to only the Brownian motion associated with the variance factor. For certain cases when numerical methods are either needed or preferred, we propose a discrete fast Fourier transform method to numerically solve the conditional PIDE efficiently. Our method can also effectively compute hedging parameters. Numerical results show that the proposed method is highly efficient.
