Risk Analysis in Finance and Insurance, Second Edition

324 Pages·2011·1.81 MB·English

by  Alexander Melnikov

Finance and Investing
Second
Alexander Melnikov
Risk Analysis in Finance and Insurance Edition
Second Edition
R
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Risk Analysis in Finance and Insurance, Second Edition presents an
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accessible yet comprehensive introduction to the main concepts and methods

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that transform risk management into a quantitative science. Taking into account
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the interdisciplinary nature of risk analysis, the author discusses many important
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ideas from mathematics, finance, and actuarial science in a simplified manner. He
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explores the interconnections among these disciplines and encourages readers
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toward further study of the subject. This edition continues to study risks associated s
with financial and insurance contracts, using an approach that estimates the value i
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of future payments based on current financial, insurance, and other information.

i Risk Analysis
New to the Second Edition n
• Expanded section on the foundations of probability and stochastic analysis
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• Coverage of new topics, including financial markets with stochastic volatility,
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risk measures, risk-adjusted performance measures, and equity-linked n in Finance and
insurance a
• More worked examples and problems n
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Reorganized and expanded, this updated book illustrates how to use quantitative e
Insurance
methods of stochastic analysis in modern financial mathematics. These methods
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can be naturally extended and applied in actuarial science, thus leading to unified
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methods of risk analysis and management.
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Features
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• Introduces the main ideas, techniques, and stochastic models of financial n Second Edition
mathematics s
• Focuses on the foundations and key concepts of the modern methodology of u
quantitative financial analysis r
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• Explores the problems of managing insurance risks
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• Examines the multiple intrinsic connections between insurance risks and
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financial risks
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• Shows how stochastic analysis is a powerful tool for modeling financial and
insurance risk processes
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Risk Analysis
in Finance and
Insurance
Second Edition
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CHAPMAN & HALL/CRC

Financial Mathematics Series
Aims and scope:
The field of financial mathematics forms an ever-expanding slice of the financial sector. This series
aims to capture new developments and summarize what is known over the whole spectrum of this
field. It will include a broad range of textbooks, reference works and handbooks that are meant to
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Series Editors
M.A.H. Dempster Dilip B. Madan Rama Cont
Centre for Financial Research Robert H. Smith School Center for Financial
Department of Pure of Business Engineering
Mathematics and Statistics University of Maryland Columbia University
University of Cambridge New York
Published Titles
American-Style Derivatives; Valuation and Computation, Jerome Detemple
Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing,
 Pierre Henry-Labordère
Credit Risk: Models, Derivatives, and Management, Niklas Wagner
Engineering BGM, Alan Brace
Financial Modelling with Jump Processes, Rama Cont and Peter Tankov
Interest Rate Modeling: Theory and Practice, Lixin Wu
Introduction to Credit Risk Modeling, Second Edition, Christian Bluhm, Ludger Overbeck, and
 Christoph Wagner
Introduction to Stochastic Calculus Applied to Finance, Second Edition,
 Damien Lamberton and Bernard Lapeyre
Monte Carlo Methods and Models in Finance and Insurance, Ralf Korn, Elke Korn,
 and Gerald Kroisandt
Numerical Methods for Finance, John A. D. Appleby, David C. Edelman, and John J. H. Miller
Portfolio Optimization and Performance Analysis, Jean-Luc Prigent
Quantitative Fund Management, M. A. H. Dempster, Georg Pflug, and Gautam Mitra
Risk Analysis in Finance and Insurance, Second Edition, Alexander Melnikov
Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers
Stochastic Finance: A Numeraire Approach, Jan Vecer
Stochastic Financial Models, Douglas Kennedy
Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and Ludger Overbeck
Understanding Risk: The Theory and Practice of Financial Risk Management, David Murphy
Unravelling the Credit Crunch, David Murphy
Proposals for the series should be submitted to one of the series editors above or directly to:
CRC Press, Taylor & Francis Group
4th, Floor, Albert House
1-4 Singer Street
London EC2A 4BQ
UK
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Risk Analysis
in Finance and
Insurance
Second Edition
Alexander Melnikov
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CRC Press
Taylor & Francis Group
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Preface to 2nd edition
This book deals with the notion of “risk” and is devoted to analysis of risks
in finance and insurance. We will study risks associated with financial and
insurance contracts, by which we understand risks to be uncertainties that
may result in financial loss and affect the ability to make payments associ-
atedwiththe correspondingcontracts.Our approachtothis analysisis based
on the development of a methodology for estimating the present value of the
futurepaymentsgivencurrentfinancial,insurance,andotherinformation.Us-
ingthis approach,onecanappropriatelydefine notionsofprice forafinancial
contract, of premium for insurance policy, and of reserve for an insurance
company.Historically,financialrisksweresubjectto elementarymathematics
of finance and were treated separately from insurance risks, which were an-
alyzed in actuarial science. The development of quantitative methods based
on stochastic analysis is a key achievement of modern financial mathematics.
Thesemethodscanbenaturallyextendedandappliedinthe areaofactuarial
science, thus leading to unified methods of risk analysis and management.
Theaimofthisbookistogiveanaccessiblecomprehensiveintroductionto
themainideas,methods,andtechniquesthattransformriskmanagementinto
a quantitative science. Because of the interdisciplinary nature of this book,
many important notions and facts from mathematics, finance, and actuarial
science are discussed in an appropriately simplified manner. Our goal is to
presentinterconnections among these disciplines and to encourageour reader
towardfurtherstudy ofthe subject.We indicate someinitialdirectionsinthe
Bibliographic Remarks.
This edition is reorganized in a way that allows a natural flow of topics
covered in the first edition to be combined together with new additions such
as: financial markets with stochastic volatility, risk measures, risk-adjusted
performance measures,equity-linked insurance, and so forth. The substantial
extensionofthesectionregardingthefoundationsofProbabilityandStochas-
ticAnalysismakesthisbookself-contained.Furthermore,anincreasednumber
ofworkedexamplesandacollectionofsome140problems,whichisaccompa-
nied by the Instructor’s Solutions Manual, make this edition more attractive
bothfromaresearchanda pedagogicalperspective.This bookcanbe readily
used as a textbook for a Mathematical Finance course, both at introductory
undergraduate and advanced graduate levels. It has been used for teaching
Mathematical Finance at both levels at the University of Alberta, and many
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vi Preface
student comments and recommendations are taken into account in this edi-
tion.
The author thanks his graduate students Anna Evstafyeva, Hao Li, and
Henry Heung for their help in introducing some new worked examples and
problemsinthisedition.TheauthorisalsoverygratefultoDr.AlexeiFilinkov
for translating, editing, and preparing the manuscript.
Alexander Melnikov
University of Alberta, Edmonton, Canada
October 2010
Introduction
Financial and insurance markets always operate under various types of un-
certaintiesthatcanaffectthe financialpositionofcompaniesandindividuals.
In financial and insurance theories,these uncertainties are usually referredto
as risks. Given certain states of the market, and the economy in general, one
can talk about risk exposure. It is expected that individuals, companies, and
public establishments that aim to accumulate wealth should examine their
riskexposure.Theprocessofriskmanagementconsistsofasequenceofcorre-
sponding actions overa period of time that are designed to mitigate the level
of risk exposure. Some of the main principles and ingredients of risk man-
agement are qualitative identification of risk, estimation of possible losses,
choosingtheappropriatestrategiesforavoidinglossesandforshiftingtherisk
tootherpartsofthefinancialsystem,including analysisofthe involvedcosts,
and using feedback for developing adequate controls.
The first six chapters of this book are devoted to the financial market
risks. We aim to give an elementary and yet comprehensive introduction to
themainideas,methods,andstochasticmodelsoffinancialmathematics.The
probabilistic approach appears to be one of the most efficient ways of mod-
eling uncertainties in financial markets. Risks (or uncertainties of financial
marketoperations)aredescribedintermsofstatisticallystablestochasticex-
periments, and therefore estimation of risks is reduced to the construction
of financial forecasts adapted to these experiments. Using conditional expec-
tations, one can quantitatively describe these forecasts given the observable
marketpricesandevents.Thus,itcanbepossibletoconstructdynamichedg-
ing strategies and those for optimal investment. The foundations and key
conceptsofthemodernmethodologyofquantitativefinancialanalysisarethe
main focus of Chapters 1–6.
Insurance against possible financial losses is one of the key ingredients of
risk management. However, the insurance business is an integral part of the
financial system. Chapters 7–8 focus on the problems of managing insurance
risks.Multipleintrinsicconnectionsbetweeninsurancerisksandfinancialrisks
are also considered.
Our treatment of insurance risk management demonstrates that methods
of risk evaluation and management in insurance and finance are interrelated
and can be treated using a single integrated approach. Estimations of future
payments and their corresponding risks are key operational tasks of financial
and insurance companies. Management of these risks requires an accurate
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evaluationofthepresentvaluesoffuturepayments,andthereforetheadequate
modelingof(financialandinsurance)riskprocesses.Stochasticanalysisisone
of the most powerful tools for such modeling, and it is the fundamental basis
of our presentation.
Finally, we note that probabilistic methods were used in finance and in-
surance since the early 1950s. They were developed extensively over the past
decades, especially after the seminal papers by F. Black and M. Scholes and
R. C. Merton, published in 1973.
Contents
1 Introductory concepts of Financial Risk Management and
Related Mathematical Tools 1
1.1 Introductory concepts of the securities market . . . . . . . . 1
1.2 Probabilistic foundations of financial modeling and pricing of
contingent claims . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Elements of probability theory and stochastic analysis . . . . 12
2 Financial Risk Management in the Binomial Model 33
2.1 Thebinomialmodelofafinancialmarket.Absenceofarbitrage,
uniqueness of a risk-neutral probability measure, martingale
representation. . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Hedging contingent claims in the binomial market model. The
Cox-Ross-Rubinstein formula . . . . . . . . . . . . . . . . . . 39
2.3 Pricing and hedging American options . . . . . . . . . . . . . 48
2.4 Utility functionsandSt.Petersburg’sparadox.Theproblemof
optimal investment. . . . . . . . . . . . . . . . . . . . . . . . 52
2.5 Thetermstructureofprices,hedging,andinvestmentstrategies
in the Ho-Lee model . . . . . . . . . . . . . . . . . . . . . . . 57
2.6 Thetransitionfromthebinomialmodelofafinancialmarketto
a continuous model. The Black-Scholes formula and equation. 63
3 AdvancedAnalysisofFinancialRisks:DiscreteTimeModels 69
3.1 Fundamental theorems on arbitrage and completeness. Pric-
ing and hedging contingent claims in complete and incomplete
markets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 The structure of options prices in incomplete markets and in
markets with constraints. . . . . . . . . . . . . . . . . . . . . 77
3.3 Hedging contingent claims in mean square . . . . . . . . . . 85
3.4 Gaussianmodelofafinancialmarketindiscretetime.Insurance
appreciation and discrete version of the Black-Scholes formula. 91
4 Analysis of Risks: Continuous Time Models 101
4.1 The Black-Scholesmodel. “Greek” parametersin risk manage-
ment, hedging, and optimal investment. . . . . . . . . . . . . 101
4.2 Beyond of the Black-Scholes model . . . . . . . . . . . . . . 112
4.3 Imperfect hedging and risk measures . . . . . . . . . . . . . . 126
ix