# Monte Carlo Simulation

Topics: Normal distribution, Variance, Monte Carlo methods in finance Pages: 753 (218886 words) Published: February 13, 2011
Preface

This is a book about Monte Carlo methods from the perspective of ﬁnancial engineering. Monte Carlo simulation has become an essential tool in the pricing of derivative securities and in risk management; these applications have, in turn, stimulated research into new Monte Carlo techniques and renewed interest in some old techniques. This is also a book about ﬁnancial engineering from the perspective of Monte Carlo methods. One of the best ways to develop an understanding of a model of, say, the term structure of interest rates is to implement a simulation of the model; and ﬁnding ways to improve the eﬃciency of a simulation motivates a deeper investigation into properties of a model. My intended audience is a mix of graduate students in ﬁnancial engineering, researchers interested in the application of Monte Carlo methods in ﬁnance, and practitioners implementing models in industry. This book has grown out of lecture notes I have used over several years at Columbia, for a semester at Princeton, and for a short course at Aarhus University. These classes have been attended by masters and doctoral students in engineering, the mathematical and physical sciences, and ﬁnance. The selection of topics has also been inﬂuenced by my experiences in developing and delivering professional training courses with Mark Broadie, often in collaboration with Leif Andersen and Phelim Boyle. The opportunity to discuss the use of Monte Carlo methods in the derivatives industry with practitioners and colleagues has helped shaped my thinking about the methods and their application. Students and practitioners come to the area of ﬁnancial engineering from diverse academic ﬁelds and with widely ranging levels of training in mathematics, statistics, ﬁnance, and computing. This presents a challenge in setting the appropriate level for discourse. The most important prerequisite for reading this book is familiarity with the mathematical tools routinely used to specify and analyze continuous-time models in ﬁnance. Prior exposure to the basic principles of option pricing is useful but less essential. The tools of mathematical ﬁnance include Itˆ calculus, stochastic diﬀerential equations, o and martingales. Perhaps the most advanced idea used in many places in

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this book is the concept of a change of measure. This idea is so central both to derivatives pricing and to Monte Carlo methods that there is simply no avoiding it. The prerequisites to understanding the statement of the Girsanov theorem should suﬃce for reading this book. Whereas the language of mathematical ﬁnance is essential to our topic, its technical subtleties are less so for purposes of computational work. My use of mathematical tools is often informal: I may assume that a local martingale is a martingale or that a stochastic diﬀerential equation has a solution, for example, without calling attention to these assumptions. Where convenient, I take derivatives without ﬁrst assuming diﬀerentiability and I take expectations without verifying integrability. My intent is to focus on the issues most important to Monte Carlo methods and to avoid diverting the discussion to spell out technical conditions. Where these conditions are not evident and where they are essential to understanding the scope of a technique, I discuss them explicitly. In addition, an appendix gives precise statements of the most important tools from stochastic calculus. This book divides roughly into three parts. The ﬁrst part, Chapters 1–3, develops fundamentals of Monte Carlo methods. Chapter 1 summarizes the theoretical foundations of derivatives pricing and Monte Carlo. It explains the principles by which a pricing problem can be formulated as an integration problem to which Monte Carlo is then applicable. Chapter 2 discusses random number generation and methods for sampling from nonuniform distributions, tools fundamental to every application of Monte Carlo. Chapter 3 provides an overview of some of the...