Risk neutral pricing and financial mathematics : a primer

338 Pages·2015·5.56 MB·English

by  Knopf

RISK NEUTRAL PRICING AND
FINANCIAL MATHEMATICS
RISK NEUTRAL
PRICING AND
FINANCIAL
MATHEMATICS
A Primer
Peter M. Knopf
John L. Teall
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Dedication
to thelovely andsupportive ladies in our lives,
Anne and Arli,
andthe many others who shall remainnameless here.
About the Authors
Peter Knopf obtained his Ph.D. from Cornell New York University, Cornell University, Pace
University and subsequently taught at Texas University, Dublin City University, University
A&M University and Rutgers University. He of Melbourne as well as other institutions
is currently Professor of Mathematics at Pace in the United States, Europe and Asia. His
University. He has numerous research publica- primary areas of research and publication have
tions in both pure and applied mathematics. been related to corporate finance and financial
His recent research interests have been in the institutions. He is the author/co-author of
areas of difference equations and stochastic 5 books. Dr. Teall obtained his Ph.D. from
delayequationmodelsforpricingsecurities. the Stern School of Business at New York
JohnL.Teallisavisitingprofessoroffinance University and is a former member of the
at the LUISS Business School, LUISS Guido American Stock Exchange. Dr. Teall has con-
Carli University. He was Jackson Tai ’72 sulted with numerous financial institutions
Professor of Practice at Rensselaer Polytechnic including Deutsche Bank, Goldman Sachs,
Institute, and also served on the faculties of NationalWestminsterBankandCiticorp.
ix
Preface
It’s no easy task to write a book that makes groups of readers: ambitious financial mathe-
quantitative finance seem easy. Writing such a matics students with relatively modest mathe-
book with two bull-headed authors can be a matics backgrounds, say with one or two
real battle, at least at times. But, now that calculus courses along with a course or two in
we’re finished, or at least ready to begin pon- linear mathematics and another in statistics.
dering the book’s eventual second edition, we While a stronger math background would only
both are surprised and quite pleased with the help the reader, we have provided a brief
result. Our various pre-publication readers review (Chapter 1 and much of Chapter 2) for
claim to like the chapters, and seem to believe readers who are a bit rusty or have gaps in
that the book well suits our intended audience their undergraduate preparation. We believe
(we’ll get to this shortly). Competition, skepti- that readers with technical backgrounds such
cism, and stubbornness do seem to have a as in mathematics and engineering as well as
place ina joint writing endeavor. readers with nontechnical backgrounds such
Somewhere in the middle of an early draft of as studies in undergraduate finance and eco-
Chapter 7, a passage appeared that sparked the nomics will be able to follow and benefit from
followingexchangebetweenthecoauthorsofthis the presentations of this book. After all, the
book,alteredsomewhattosoftenthelanguage: book is still focused on finance, and not even
most mathematicians will have seen at least
some of the mathematics used here and in the
Thisparagraphdoesn’tmeananything!
That’sbecauseyou’redense. financial industry more generally. We also
(exasperated)“You’resuchapinhead.WhydidI believe that the large number of solved end-of-
everagreetowritethisbook?” chapter exercises and online materials will
boost reader comfort with the material and
Actually, we knew from the start exactly deepenthe learningexperience.
why we needed to write this book. We’ve seen In order to keep the price of the textbook
a nice assortment of books in financial mathe- down, certain materials such as additional
matics, written by mathematicians, that are readings, proofs and verifications have been
nicely suited to readers with solid technical placed on the authors’ website at http://www.
backgrounds in engineering, the physical jteall.com/books.html (click the “Resources”
sciences and math. There are a smaller number link), and are also available online at the
of financial math books geared towards under- Elsevier textbook site (see overleaf for
graduates and MBA students who really aren’t address). These readings, proofs and verifica-
interested in the stochastic processes underly- tions offer a wealth of insight into many of
ing models for pricing derivatives and fixed the processes used to model financial instru-
income instruments. Your coauthors sought to ments and to the mathematics underlying val-
fill a rapidly growing gap between these two uation and hedging. For example, in the
xi
xii
PREFACE
additional readings for Chapter 6, we derive Writing a textbook is never a solo effort, or
the solution to the Black-Scholes differential even an effort undertaken by only two coau-
equation. Of similar importance to Chapter 8, thors. As have most authors, we benefitted
the text website contains a derivation of the immensely from the help from many people,
Vasicek single factor model for pricing bonds. including friends, family, students, colleagues
These website readings are presented to and practitioners. For example, we owe special
improve the balance between rigor and peda- thankstosomeofourfavoritestudents,includ-
gogy in order for the student to gain as much ing Xiao (Sean) Tang, T.J. Wu, Alban Leung,
intuition and understanding of the most Haoyu Li, Victor Shen, and Matthew Spector.
important derivations in quantitative finance A number of our colleagues were also most
without losing the ability to apply the impor- helpful, including Professors Steven Kalikow
tantresults. and Maury Bramson for valuable input con-
Introduction to Risk-Neutral Pricing and cerning stochastic processes and Professor
Financial Mathematics: A Primer seeks to intro- Matthew Hyatt for programming and use of
duce financial mathematics to students in software tools. We are particularly grateful to
quantitative finance, financial engineering, Hong Tu Yan for his careful reading of the text
actuarial science and computational finance. and competent coauthoring of supplemental
The central theme of the book is risk-neutral materials. We thank Ryan Cummings for her
(martingale) pricing, though it does venture delightful cartoon appearing on the text com-
into a number of other areas. The text endea- panionwebsite,andYingSueHuangandImon
vors to provide a foundation in financial math- Palit for their thoughtful comments. And then
ematics for use in an introductory financial there’sChrisStone,alwaystherewhenweneed
engineering, financial modeling, or financial help, and always going above and beyond the
mathematics course, primarily to students call of duty, such as for her 11th hour sacrifice
whose math backgrounds do not extend much of vacation time to ensure that we met our
beyond two semesters of calculus, linear math manuscriptsubmissiondeadline.Ofcourse,we
and statistics. Students with stronger mathe- are alsogratefulto Scott Bentley,Melissa Read,
matics preparations may find this book some- Susan Ikeda, and Mckenna Bailey at Academic
what easier to follow, but most will not find Press/Elsevier. And finally, we enjoyed the
significant amounts of material repeating what unfailing support and encouragement from
they have seen elsewhere. The book will Arli Epton and the unique culinary skills of
present and apply topics such as stochastic AnneTeall.EmilyTeall’sunboundedinterestin
processes, equivalent martingales, Radon(cid:1) everythingrelatedtofinanceboostedourmorale
Nikodym derivatives, and stochastic calculus, as well. Unfortunately, a number of errors will
butitwillstartfromtheperspectivethatreader surely persist evenafter the book goes intopro-
mathematical training does not extend beyond duction.Weapologizesincerelyfortheseerrors,
basiccalculus,linearmathandstatistics. andweblameourparentsforthem.
Additionalreadings,proofsandverificationscanbefoundonthecompanionwebsite.
Goonlinetoaccessitat:
http://booksite.elsevier.com/9780128015346
C H A P T E R
1
Preliminaries and Review
1.1 FINANCIAL MODELS
A model can be characterized as an artificial structure describing the relationships among vari-
ables or factors. Practically all of the methodology in this book is geared toward the development
andimplementationoffinancialmodelstosolvefinancialproblems.Forexample,valuationmodels
provide a foundation for investment decision-making and models describing stochastic processes
provideanimportanttooltoaccountforriskindecision-making.
The use of models is important in finance because “real world” conditions that underlie finan-
cial decisions are frequently extraordinarily complicated. Financial decision-makers frequently use
existing models or construct new ones that relate to the types of decisions they wish to make.
Modelsproposingdecisionsthatoughttobemadearecallednormativemodels.1
The purpose of models is to simulate or behave like real financial situations. When construct-
ing financial models, analysts exclude the “real world” conditions that seem to have little or no
effect on the outcomes of their decisions, concentrating on those factors that are most relevant to
their situations. In some instances, analysts may have to make unrealistic assumptions in order
to simplify their models and make them easier to analyze. After simple models have been con-
structed with what may be unrealistic assumptions, they can be modified to match more closely
“real world” situations. A good financial model is one that accounts for the major factors that
will affect the financial decision (a good model is complete and accurate), is simple enough
for its use to be practical (inexpensive to construct and easy to understand), and can be used to
predict actual outcomes. A model is not likely to be useful if it is not able to project an outcome
with an acceptable degree of accuracy. Completeness and simplicity may directly conflict with
one another. The financial analyst must determine the appropriate trade-off between complete-
ness and simplicity in the model hewishes toconstruct.
In finance, mathematical models are usually the easiest to develop, manipulate, and modify.
Thesemodelsareusuallyadaptabletocomputersandelectronicspreadsheets.Mathematicalmodels
are obviously most useful for those comfortable with math; the primary purpose of this book is to
provideafoundationforimprovingthequantitativepreparationofthelessmathematicallyoriented
analyst. Other models used in finance include those based on graphs and those involving simula-
tions.However,thesemodelsareoftenbasedonorcloselyrelatedtomathematicalmodels.
P.M.Knopf&J.L.Teall:RiskNeutralPricingandFinancialMathematics:APrimer.
DOI:http://dx.doi.org/10.1016/B978-0-12-801534-6.00001-1 1 ©2015ElsevierInc.Allrightsreserved.
2
1. PRELIMINARIESANDREVIEW
The concepts of market efficiency and arbitrage are essential to the development of many financial
models. Market efficiency is the condition in which security prices fully reflect all available
information. Such efficiency is more likely to exist when wealth-maximizing market participants
can instantaneously and costlessly execute transactions as information is revealed. Transactions
costs,irrationality,andpoorexecutionsystemsreduceefficiency.Arbitrage,initssimplestscenario,
isthesimultaneouspurchaseandsaleofthesameasset,ormoregenerally,thenearlysimultaneous
purchaseand saleof assets generatingnearlyidenticalcashflow structures.Ineither case, thearbi-
trageurseekstoproduceaprofitbypurchasingatapricethatislessthanthesellingprice.Proceeds
of the sales are used to finance purchases such that the portfolio of transactions is self-financing,
andthatovertime,noadditionalcapitalisdevotedtoorlostfromtheportfolio.Thus,theportfolio
is assured a non-negative profit at each time period. The arbitrage process is riskless if purchase
andsalepricesareknownatthetimestheyareinitiated.Arbitrageursfrequentlyseektoprofitfrom
marketinefficiencies.Theexistenceofarbitrageprofitsisinconsistentwithmarketefficiency.
1.2 FINANCIAL SECURITIES AND INSTRUMENTS
A security is a tradable claim on assets. Real assets contribute to the productive capacity of the
economy; securities are financial assets that represent claims on real assets or other securities.
Most securities are marketable to the general public, meaning that they can be sold or assigned
to other investors in the open marketplace. Some of the more common types of securities and
tradable instruments are briefly introduced in the following:
1. Debtsecurities:Denotecreditorshipofanindividual,firmorotherinstitution.Theytypically
involvepaymentsofafixedseriesofinterest(oftenknownascouponpayments)oramounts
towardsprincipalalongwithprincipalrepayment(oftenknownasfacevalue).Examplesinclude:
(cid:129) Bonds: Long-term debt securities issued by corporations, governments, or other institutions.
Bonds arenormally ofthe coupon variety (they make periodicinterestpayments onthe
principal) orpurediscount (they are zero coupon instruments that are sold ata discount from
face value, thebond’sfinal maturity value).
(cid:129) Treasury securities:Debt securities issued by theTreasury of theUnited Statesfederal
government. They are often considered to bepractically free ofdefault risk.
2. Equitysecurities (stock): Denote ownership in abusiness or corporation. They typically permit
for dividend payments if the firm’s debt obligations have been satisfied.
3. Derivative securities: Have payofffunctions derived from the values of other securities,rates, or
indices.Some of themorecommon derivative securities are:
(cid:129) Options: Securitiesthat grant their owners rights to buy(call)or sell (put) anunderlying
assetor security at aspecific price (exercise price) on orbefore its expirationdate.
(cid:129) Forward and futures contracts: Instruments that oblige their participants to either purchaseor
sell a given asset orsecurity at aspecified price (settlement price) onthe future settlement
date of that contract. A long positionobligates theinvestor to purchasethe given asset on
the settlementdateof the contract and ashort position obligates the investor to sell the
given asset on thesettlement date of thecontract.
(cid:129) Swaps:Provide for the exchange of cash flows associated with one asset, rate, or indexfor
the cash flows associated with anotherasset, rate, or index.
RISKNEUTRALPRICINGANDFINANCIALMATHEMATICS:APRIMER
3
1.3REVIEWOFMATRICESANDMATRIXARITHMETIC
4. Commodities:Contracts,includingfuturesandoptionsonphysicalcommoditiessuchasoil,
metals,corn,etc.Commoditiesaretradedinspotmarkets,wheretheexchangeofassetsand
moneyoccursatthetimeofthetransactionorinforwardandfuturesmarkets.
5. Currencies: Exchange rates denote the number ofunits ofone currency that mustbegiven up
for one unit of asecond currency. Exchange transactionscan occur ineitherspot or forward
markets. As with commodities, in the spot market,the exchange of one currency for another
occurswhenthe agreement is made. In a forward market transaction,the actual exchange of
one currency for another actuallyoccurs ata date later than that of the agreement. Spot and
forwardcontract participants take one positionin each of two currencies:
(cid:129) Long: An investorhas a “long”position inthat currency that he will accept atthe later date.
(cid:129) Short: An investor hasa “short” positionin that currency that he must deliver inthe
transaction.
6. Indices: Contracts pegged to measures of market performance suchas the Dow Jones
IndustrialsAverage or the S&P 500 Index.Theseare frequentlyfuturescontracts onportfolios
structured to perform exactly as the indices for whichthey are named. Index traders also
trade options on these futures contracts.
This list of security types is far from comprehensive; it only reflects some of those instruments
that will be emphasized in this book. In addition, most of the instrument types will have many
differentvariations.
1.3 REVIEW OF MATRICES AND MATRIX ARITHMETIC
A matrix is simply an ordered rectangular array of numbers. A matrix is an entity that
enables one to represent a series of numbers as a single object, thereby providing for
convenient systematic methods for completing large numbers of repetitive computations. Such
objects are essential for the management of large data structures. Rules of matrix arithmetic
and other matrix operations are often similar to rules of ordinary arithmetic and other opera-
tions, but they are not always identical. In this text, matrices will usually be denoted with
bold uppercase letters. When the matrix has only one row or one column, bold lowercase let-
ters will be used for identification. The following are examples of matrices:
2 3 2 3
4 2 6 " # 1
6 7 2 23 6 7
A5643 7 475B5 c564575d5½4(cid:2)
3=4 21=2
8 25 9 7
The dimensions of a matrix are given by the ordered pair m3n, where m is the number of
rows and n isthe number of columns in the matrix. The matrix is said to be of order m3n where,
by convention, the number of rows is listed first. Thus, A is 333, B is 232, c is 331, and d is
131. Each number in a matrix is referred to as an element. The symbol a denotes the element
i,j
in Row i and Column j of Matrix A, b denotes the element in Row i and Column j of Matrix B,
i,j
andso on. Thus, a is 25 andc 55.
3,2 2,1
There are specific terms denoting various types of matrices. Each of these particular types of
matrices has useful applications and unique properties for working with. For example, a vector is
RISKNEUTRALPRICINGANDFINANCIALMATHEMATICS:APRIMER