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    The aim of this book is to present a lively and palatable introduction to financialoption valuation for undergraduate students in mathematics, statistics and relatedareas. Prerequisites have been kept to a minimum. The reader is assumed to have abasic competence in calculus up to the level reached by a typical first year mathematicsprogramme. No background in probability, statistics or numerical analysisis required, although some previous exposure to material in these areas would undoubtedlymake the text easier to assimilate on first reading.
    The contents are presented in the form of short chapters, each of which couldreasonably be covered in a one hour teaching session. The book grew out of a finalyear undergraduate class called The Mathematics of Financial Derivatives that Ihave taught, in collaboration with Professor Xuerong Mao, at the University ofStrathclyde. The class is aimed at students taking honours degrees in Mathematicsor Statistics, or joint honours degrees in various combinations of Mathematics,Statistics, Economics, Business, Accounting, Computer Science and Physics. Inmy view, such a class has two great selling points.
    • From a student perspective, the topic is generally perceived as modern, sexy and likelyto impress potential employers.
    • From the perspective of a university teacher, the topic provides a focus for ideas frommathematical modelling, analysis, stochastics and numerical analysis.
    There are many excellent books on option valuation. However, in preparingnotes for a lecture course, I formed the opinion that there is a niche for a single,self-contained, introductory text that gives equal weight to
    • applied mathematics,
    • stochastics, and
    • computational algorithms.
    The classic applied mathematics view is provided by Wilmott, Howison andDewynne’s text (Wilmott et al., 1995). My aim has been to write a book at a similarlevel with a less ambitious scope (only option valuation is considered), lessemphasis on partial differential equations, and more attention paid to stochasticmodelling and simulation.
    Key features of this book are as follows.
    (i) Detailed derivation and discussion of the basic lognormal asset price model.
    (ii) Roughly equal weight given to binomial, finite difference and Monte Carlo methods.
    In particular, variance reduction techniques for Monte Carlo are treated in some detail.
    (iii) Heavy use of computational examples and figures as a means of illustration.
    (iv) Stand-alone MATLAB codes, with full listings and comprehensive descriptions, thatimplement the main algorithms. The core text can be read independently of the codes.
    Readers who are familiar with other programming languages or problem-solving environmentsshould have little difficulty in translating these examples.
    In a nutshell, this is the book that I wish had been available when I started toprepare lectures for the Strathclyde class.
    When designing a text like this, an immediate issue is the level at which stochasticcalculus is to be treated. One of the tenets of this book is thatrigorous, measure-theoretic, stochastic analysis, although beautiful, is hard and it isunrealistic to ask an undergraduate class to pick up such material on the fly. MonteCarlo-style simulation, on the other hand, is a relatively simple concept, and wellchosencomputational experiments provide an excellent way to back up heuristicarguments.
    Hence, the approach here is to treat stochastic calculus on a nonrigorous leveland give plenty of supporting computational examples. I rely heavily on the CentralLimit Theorem as a basis for heuristic arguments. This involves a deliberatecompromise – convergence in distribution must be swapped for a stronger type ofconvergence if these arguments are to be made rigorous – but I feel that erring onthe side of accessibility is reasonable, given the aims of this text.
    In fact, in deriving the Black–Scholes partial differential equation, I do not makeexplicit reference to Itˆo’s Lemma. I decided that a heuristic derivation of Itˆo’sLemma in a general setting followed by a single application of the lemma in onesimple case makes less pedagogical sense than a direct ‘in situ’ heuristic treatment,a decision inspired by Almgren’s expository article (Almgren, 2002). I hope thatat least some undergraduate readers will be sufficiently motivated to follow up onthe references and become exposed to the real thing.
    You can get a feeling for the contents of the book by skimming through theoutline bullet points that appear at the start of each chapter. Many of the laterchapters can be read independently of each other, or, of course, omitted.
    Exercises are given at the end of each chapter. It is my experience that activeproblem solving is the best learning tool, so I strongly encourage students to makeuse of them. I have used a starring system: one star for questions whose solutionis relatively easy/short, rising to three stars for the hardest/longest questions. Briefsolutions to the odd-numbered exercises are available from the book website givenbelow. This leaves the even-numbered questions as a teaching resource. Certainquestions are central to the text. I have tried to ensure that these come up in theodd-numbered list, in order to aid independent study.
    A short, introductory treatment like this can only scratch the surface. Hence,each chapter concludes with a Notes and references section, which gives my own,necessarily biased, hints about important omissions. References can be followedup via the References section at the end of the book.
    Scattered at the end of each chapter are a few quotes, designed to enlighten andentertain. Some of these reinforce the ideas in the text and others cast doubt onthem. Mathematical option valuation is a strange business of sophisticated analysisbased on simple models that have obvious flaws and perhaps do not merit suchdetailed scrutiny. When preparing lecture notes, I have found that authoritative,pithy quotes are a particularly powerful means to highlight some of this tension.I have an uneasy feeling that some Strathclyde students spent more time perusingthe quotes than the main text, so I have aimed to make the quotes at least forma reasonable mini-summary of the contents. Most quotes relate directly to theirchapter, but a few general ones have been dispersed throughout the book on thegrounds that they were too good to leave out.
    A website for this book has been created at www.maths.strath.ac.uk/∼aas96106/option book.html. It includes the following.
    • The MATLAB codes listed in the book.
    • Outline solutions to the odd-numbered exercises.
    • Links to the websites mentioned in the book.
    • Colour versions of some of the figures.
    • A list of corrections.
    • Some extra quotes that did not make it into the book.
    I am grateful to several people who have influenced this book. Nick Highamcast a critical eye over an early draft and made many helpful suggestions. VickyHenderson checked parts of the text and patiently answered a number of questions.
    Petter Wiberg gave me access to his MATLAB files for processing stockmarket data. Xuerong Mao, through animated discussions and research collaboration,has enriched my understanding of stochastics and its role in mathematicalfinance. Additionally, five anonymous reviewers provided unbiased feedback. Inparticular, one reviewer who was not in favour of the nonrigorous approach tostochastic analysis in this book was nevertheless generous enough to provide detailedcomments that allowed me to improve the final product. Finally, three years’worth of Strathclyde honours students have helped to shape my views on how topresent this material to a wide audience.
    MATLAB programs
    I firmly believe that the best way to check your understanding of a computationalalgorithm is to examine, and interactively experiment with, a real program. Forthis reason, I have included a Program of the Chapter at the end of every chapter,followed by two programming exercises. Each program illustrates a key topic.
    They can be downloaded from the website previously mentioned.The programs are written in MATLAB.1 I chose this environment for a numberof reasons.
    • It offers excellent random number generation and graphical output facilities.
    • It has powerful, built-in, high-level commands for matrix computations and statistics.
    • It runs on a variety of platforms.
    It is widely available in mathematics and computer science departments and is oftenused as the basis for scientific computing or numerical analysis courses. Students maypurchase individual copies at a modest price.
    I wrote the programs with accuracy and clarity in mind, rather than efficiencyor elegance. I have made quite heavy use of MATLAB’s vectorization facilities,where possible working with arrays directly and eschewing unnecessary forloops. This tends to make the codes shorter, snappier and less daunting than alternativesthat operate on individual array components. Meaningful comments havebeen inserted into the codes and a ‘walkthrough’ commentary is appended in eachcase. Those walkthroughs provide MATLAB information on a just-in-time basis.
    For a comprehensive guide to MATLAB, see (Higham and Higham, 2000).
    I have not made use of any of the toolboxes that are available, at extra cost, toMATLAB users. This is because (a) the emphasis in the book is on understandingthe underlying models and algorithms, not on the use of black-box packages,and (b) only a small percentage of MATLAB users will have access to toolboxes.However, those who wish to perform serious option valuation computations inMATLAB are advised to investigate the toolboxes, especially those for Finance,Statistics, Optimization and PDEs.
    Readers with some experience of scientific computing in languages such asJava, C or FORTRAN should find it relatively easy to understand the codes. Thosewith no computing background may need to put in more effort, but should find theprocess rewarding.
    1 MATLAB is a registered trademark of The MathWorks, Inc.
    MATLAB is a commercial software product produced by The Mathworks,whose homepage is at www.mathworks.com/.
    Let me re-emphasize that these programs are entirely stand-alone; the book canbe read without reference to them. However, I believe that they form a major element– if you understand the programs, you understand a big chunk of the materialin this book.
    Disclaimer of warranty
    We make no warranties, express or implied, that the programs contained in thisvolume are free of error, or are consistent with any particular standard of merchantability,or that they will meet your requirements for any particular application.
    They should not be relied on for solving a problem whose incorrect solutioncould result in injury to a person or loss of property. If you do use the programs insuch a manner, it is at your own risk. The author and publisher disclaim all liabilityfor direct or consequential damages resulting from your use of the programs.