Introduction to the Mathematics of Finance

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Introduction to theMathematics of Finance

http://dx.doi.org/10.1090/gsm/072

Introduction to theMathematics of Finance

R. J.Williams

American Mathematical SocietyProvidence, Rhode Island

Graduate Studies in Mathematics

Volume 72

Editorial Board

Walter CraigNikolai Ivanov

Steven G. KrantzDavid Saltman (Chair)

2000 Mathematics Subject Classification. Primary 60G42, 60G44, 60J65, 62P05, 91B28;Secondary 58J65, 60J60.

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Williams, R. J. (Ruth J.), 1955–.Introduction to the mathematics of finance / R. J. Williams.

p. cm. — (Graduate studies in mathematics ; v. 72)Includes bibliographical references (p. ) and index.ISBN 0-8218-3903-9 (hard : alk. paper)1. Finance—Mathematical models—Textbooks. I. Title. II. Series.

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Contents

Preface vii

Chapter 1. Financial Markets and Derivatives 1

1.1. Financial Markets 1

1.2. Derivatives 2

1.3. Exercise 5

Chapter 2. Binomial Model 7

2.1. Binomial or CRR Model 7

2.2. Pricing a European Contingent Claim 10

2.3. Pricing an American Contingent Claim 19

2.4. Exercises 28

Chapter 3. Finite Market Model 31

3.1. Definition of the Finite Market Model 32

3.2. First Fundamental Theorem of Asset Pricing 34

3.3. Second Fundamental Theorem of Asset Pricing 39

3.4. Pricing European Contingent Claims 44

3.5. Incomplete Markets 47

3.6. Separating Hyperplane Theorem 51

3.7. Exercises 52

Chapter 4. Black-Scholes Model 55

4.1. Preliminaries 56

4.2. Black-Scholes Model 57

v

vi Contents

4.3. Equivalent Martingale Measure 614.4. European Contingent Claims 634.5. Pricing European Contingent Claims 654.6. European Call Option — Black-Scholes Formula 694.7. American Contingent Claims 744.8. American Call Option 804.9. American Put Option 834.10. Exercises 86

Chapter 5. Multi-dimensional Black-Scholes Model 895.1. Preliminaries 915.2. Multi-dimensional Black-Scholes Model 925.3. First Fundamental Theorem of Asset Pricing 995.4. Form of Equivalent Local Martingale Measures 1015.5. Second Fundamental Theorem of Asset Pricing 1105.6. Pricing European Contingent Claims 1165.7. Incomplete Markets 1205.8. Exercises 121

Appendix A. Conditional Expectation and Lp-Spaces 123

Appendix B. Discrete Time Stochastic Processes 127

Appendix C. Continuous Time Stochastic Processes 131

Appendix D. Brownian Motion and Stochastic Integration 135D.1. Brownian Motion 135D.2. Stochastic Integrals (with respect to Brownian motion) 136D.3. Ito Process 139D.4. Ito Formula 141D.5. Girsanov Transformation 142D.6. Martingale Representation Theorem 143

Bibliography 145

Index 149

Preface

This monograph is intended as an introduction to some elements of mathe-matical finance. It begins with the development of the basic ideas of hedg-ing and pricing of European and American derivatives in the discrete (i.e.,discrete time and discrete state) setting of binomial tree models. Then ageneral discrete finite market model is defined and the fundamental theo-rems of asset pricing are proved in this setting. Tools from probability suchas conditional expectation, filtration, (super)martingale, equivalent martin-gale measure, and martingale representation are all used first in this simplediscrete framework. This is intended to provide a bridge to the continuous(time and state) setting which requires the additional concepts of Brownianmotion and stochastic calculus. The simplest model in the continuous set-ting is the Black-Scholes model. For this, pricing and hedging of Europeanand American derivatives are developed. The book concludes with a descrip-tion of the fundamental theorems of asset pricing for a continuous marketmodel that generalizes the simple Black-Scholes model in several directions.

The modern subject of mathematical finance has undergone considerabledevelopment, both in theory and practice, since the seminal work of Blackand Scholes appeared a third of a century ago. The material presented hereis intended to provide students and researchers with an introduction thatwill enable them to go on to read more advanced texts and research pa-pers. Examples of topics for such further study include incomplete markets,interest rate models and credit derivatives.

For reading this book, a basic knowledge of probability theory at thelevel of the book by Chung [10] or D. Williams [38], plus for the chapterson continuous models, an acquaintance with stochastic calculus at the levelof the book by Chung and Williams [11] or Karatzas and Shreve [27], is

vii

viii Preface

desirable. To assist the reader in reviewing this material, a summary ofsome of the key concepts and results relating to conditional expectation,martingales, discrete and continuous time stochastic processes, Brownianmotion and stochastic calculus is provided in the appendices. In particular,the basic theory of continuous time martingales and stochastic calculus forBrownian motion should be briefly reviewed before commencing Chapter 4.Appendices C and D may be used for this purpose.

Most of the results in the main body of the book are proved in detail.Notable exceptions are results from linear programming used in Section 3.5,results used for pricing American contingent claims based on continuousmodels in Sections 4.7 through 4.9, and several results related to the fun-damental theorems of asset pricing for the multi-dimensional Black-Scholesmodel treated in Chapter 5.

I benefited from reading treatments of various topics in other bookson mathematical finance, including those by Pliska [33], Lamberton andLapeyre [30], Elliott and Kopp [15], Musiela and Rutkowski [32], Bing-ham and Kiesel [4], and Karatzas and Shreve [28], although the treatmentpresented here does not parallel any one of them.

This monograph is based on lectures I gave in a graduate course at theUniversity of California, San Diego. The material in Chapters 1–3 was alsoused in adapted form for part of a junior/senior-level undergraduate courseon discrete models in mathematical finance at UCSD. The students in bothcourses came principally from mathematics and economics. I would like tothank the students in the graduate course for taking notes which formedthe starting point for this monograph. Special thanks go to Nick Loehr andAmber Puha for assistance in preparing and reading the notes, and to JudyGregg and Zelinda Collins for technical typing of some parts of the manu-script. Thanks also go to Steven Bell, Sumit Bhardwaj, Jonathan Goodmanand Raphael de Santiago, for providing helpful comments on versions of thenotes. Finally, I thank Bill Helton for his continuous encouragement andgood humor.

R. J. WilliamsLa Jolla, CA

Bibliography

[1] Artzner, P., and Heath, D. (1995). Approximate completeness with multiple martin-gale measures, Mathematical Finance, 5, 1–11.

[2] Bank for International Settlements (2005). OTC derivatives market activity in thesecond half of 2004, report of the Monetary and Economic Department, Basel,Switzerland.

[3] Bertsimas, D., and Tsitsiklis, J. N. (1997). Introduction to Linear Optimization,Athena Scientific, Belmont, MA.

[4] Bingham, N. H., and Kiesel, R. (1998). Risk-Neutral Valuation, Springer-Verlag, NewYork.

[5] Black, F., and Scholes, M. (1973). The pricing of options and corporate liabilities, J.Political Economy, 81, 637–659.

[6] Cameron, R. H., and Martin, W. T. (1944). Transformations of Wiener integralsunder translations, Annals of Mathematics, 45, 386–396.

[7] Cameron, R. H., and Martin, W. T. (1945). Transformations of Wiener integrals undera general class of linear transformations, Trans. Amer. Math. Soc., 58, 184–219.

[8] Cameron, R. H., and Martin, W. T. (1949). The transformation of Wiener integralsby nonlinear transformations, Trans. Amer. Math. Soc., 66, 253–283.

[9] Chung, K. L. (1982). Lectures from Markov Processes to Brownian Motion, Springer-Verlag, New York.

[10] Chung, K. L. (2000). A Course in Probability Theory Revised, Academic Press, SanDiego.

[11] Chung, K. L., and Williams, R. J. (1990). Introduction to Stochastic Integration,Second Edition, Birkhauser, Boston.

[12] Cox, J. C., Ross, S. A., and Rubinstein, M. (1979). Option pricing: a simplifiedapproach, J. Finan. Econom., 7, 229–263.

[13] Davis, M. H. A. (2004). Valuation, hedging and investment in incomplete financialmarkets, in Applied Mathematics Entering the 21st Century, eds. J. M. Hill and R.Moore, Society for Industrial and Applied Mathematics, 49–70.

[14] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamentaltheorem of asset pricing, Math. Annalen, 300, 463–520.

145

146 Bibliography

[15] Elliott, R. J., and Kopp, P. E. (2004). Mathematics of Financial Markets, SecondEdition, Springer-Verlag, New York.

[16] Follmer, H., and Schied, A. (2002). Stochastic Finance – an Introduction in DiscreteTime, de Gruyter, Berlin.

[17] Fouque, J.-P., Papanicolaou, G., and Sircar, K. R. (2000). Derivatives in FinancialMarkets with Stochastic Volatility, Cambridge University Press, Cambridge, U.K.

[18] Girsanov, I. V. (1960). On transforming a certain class of stochastic processes byabsolutely continuous substitutions of measures, Theory of Probability and Its Appli-cations, 5, 285–301.

[19] Harris, L. (2003). Trading and Exchanges, Oxford University Press, Oxford, U.K.

[20] Harrison, J. M., and Kreps, D. M. (1979). Martingales and arbitrage in multiperiodsecurities markets, J. Economic Theory, 20, 381–408.

[21] Harrison, J. M., and Pliska, S. R. (1981). Martingales and stochastic integrals inthe theory of continuous trading, Stochastic Processes and their Applications, 11,215–260.

[22] Harrison, J. M., and Pliska, S. R. (1983). A stochastic calculus model of continuoustrading: complete markets, Stochastic Processes and their Applications, 15, 313–316.

[23] Hull, J. C. (2005). Options, Futures and other Derivative Securities, Prentice Hall,Sixth Edition, Upper Saddle River, New Jersey.

[24] Jacka, S. D. (1991). Optimal stopping and the American put, Mathematical Finance,1, 1–14.

[25] Jacod, J., and Yor, M. (1977). Etude des solutions extremales et representationintegrales des solutions pour certains problemes de martingales, Zeitschrift furWahrscheinlichkeitstheorie und verwandte Gebeite, 38, 83–125.

[26] Jarrow, R. A., Jin, X., and Madan, D. B. (1999). The second fundamental theoremof asset pricing, Mathematical Finance, 9, 255–273.

[27] Karatzas, I., and Shreve, S. E. (1997). Brownian Motion and Stochastic Calculus,Second Edition, Springer-Verlag, New York.

[28] Karatzas, I., and Shreve, S. E. (1998). Methods of Mathematical Finance, Springer-Verlag, New York.

[29] Kreps, D. M. (1981). Arbitrage and equilibrium in economies with infinitely manycommodities, J. of Mathematical Economics, 8, 15–35.

[30] Lamberton, D., and Lapeyre, B. (1996). Introduction to Stochastic Calculus Appliedto Finance, Chapman and Hall, London.

[31] Merton, R. C. (1973). Theory of rational option pricing, Bell J. Econ. Mgt. Science,4, 141–183.

[32] Musiela, M., and Rutkowski, M. (1998). Martingale Methods in Financial Modelling,Springer, New York.

[33] Pliska, S. (1997). Introduction to Mathematical Finance: Discrete Time Models,Blackwell, Oxford, UK.

[34] Protter, P. (2001). A partial introduction to financial asset pricing theory, StochasticProcesses and their Applications, 91, 169–203.

[35] Revuz, D., and Yor, M. (1999). Continuous Martingales and Brownian Motion, ThirdEdition, Springer-Verlag, New York.

[36] Rogers, L. C. G., and Williams, D. (1987). Diffusions, Markov Processes, and Mar-tingales, Volume 2, John Wiley and Sons, Chichester, U.K.

Bibliography 147

[37] Rogers, L. C. G., and Williams, D. (1994). Diffusions, Markov Processes, and Mar-tingales, Volume 1, Second Edition, John Wiley and Sons, Chichester, U.K.

[38] Williams, D. (1991). Probability with Martingales, Cambridge University Press, Cam-bridge, U.K.

Index

admissible strategy, 63, 66, 110, 118

American contingent claim (ACC), 4, 19, 74

American Stock and Options Exchange (AMEX),2

arbitrage, 2

continuous model, 60, 67, 77, 99, 119

discrete model, 10, 12, 25, 34

Asian option, 87

attainable, 40

augmentation, 136

binomial model, 7

Black-Scholes formula, 70

Black-Scholes model, 57

multi-dimensional, 92

Black-Scholes partial differential equation,73

bond, 8, 32, 57

Brownian filtration, 136

Brownian motion, 135

call-put parity, 29

Chicago Board of Options Exchange (CBOE),

2

Chicago Board of Trade (CBOT), 2

Chicago Mercantile Exchange (CME), 2

complete market model, 40, 114

conditional expectation, 123

contingent claim (see also American contin-gent claim and European contingent claim),

2

covariation process, 105

Cox-Ross-Rubinstein model (CRR), 7

CRR, 7

derivative, 2

discounted asset price, 34, 58, 94

discounted stock price, 16

discounted value, 16, 34, 58, 67, 95, 118

Doob’s inequality, 129, 134

Doob’s stopping theorem, 129, 134

equivalent local martingale measure (ELMM),99, 106

equivalent martingale measure (EMM), 35,

61

equivalent probability measure, 35, 61, 99,

101

European contingent claim (ECC), 4, 10, 39,63, 110

exercise price, 3

filtered probability space, 128, 131

filtration, 9, 127, 131

financial market, 1

first fundamental theorem, 35, 101

forward contract, 2

futures contract, 2

gains process, 33

Girsanov transformation, 62, 108, 142

hedging, 4, 110

hedging strategy, 11, 40, 63

implied volatility, 71

incomplete, 47, 71, 120

Ito formula, 141

Ito process, 93, 139

local martingale, 99, 133

localizing sequence, 134

149

150 Index

look-back option, 52

market price of risk, 106

marking to market, 2

martingale, 128, 133

martingale representation theorem, 43, 63,76, 104, 143

money market asset, 89

mutual variation process, 140

No Free Lunch with Vanishing Risk (NFLVR),

100

Novikov’s criterion, 102, 143

null set, 131

numeraire, 92

option, 3

American option, 3, 19, 75, 80, 83

call option, 3, 10, 19, 69, 80

European option, 3, 10, 40, 69

put option, 3, 11, 83

optional time, 128, 133

OTC, 2

over-the-counter (OTC), 2

predictable

σ-algebra, 137

process, 140

simple, 137

pricing

American contingent claims, 19, 74

European contingent claims, 10, 44, 65,116

quadratic variation process, 141

rate of return, 92

replicating strategy, 11, 40, 63, 110

risk neutral probability, 12, 35, 61, 99

second fundamental theorem, 41, 110

self-financing, 10, 33, 58

separating hyperplane theorem, 51

Snell envelope, 23, 75

spot price, 3

standard filtration, 136

stochastic integral, 136

stochastic process

continuous time, 131

discrete time, 127

stochastic volatility, 71, 121

stock, 8, 32, 57, 89

stopping time, 128, 133

strike price, 3

submartingale, 129, 133

superhedging, 19, 47, 75

supermartingale, 23, 129, 133

tame strategy, 100trading strategy, 9, 12, 17, 32, 57, 94

usual conditions, 132

value, 10, 33, 58, 67, 95, 118viable, 2, 34, 101, 110

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The book begins with the development of the basic ideas of hedging and pricing of European and American derivatives in the discrete (i.e., discrete time and discrete state) setting of binomial tree models. Then a general ������������������������������������������������������������������������������������������������������������������ � ������������������� ������ ������������ ��������� ����� ���������� �!�������� ��������� �������� ��� ��������� ����������� ��� ���� ����� ���� �� ����� �������discrete framework. This provides a bridge to the continuous (time and state) �������� "����� �!����� ���� ���������� �������� ��� #�"��� ������ ���stochastic calculus. The simplest model in the continuous setting is the famous #�����$�����������������"������������������������%���������&������derivatives are developed. The book concludes with a description of the fundamental theorems for a continuous market model that generalizes the �������#�����$�������������������������������

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