designed To Form The Basis Of An Undergraduate Course In Mathematical Finance, This Book Builds On Mathematical Models Of Bond And Stock Prices And Covers Three Major Areas Of Mathematical Finance That All Have An Enormous Impact On The Way Modern Financial Markets Operate, Namely: Black-scholes’ Arbitrage Pricing Of Options And Other Derivative Securities; Markowitz Portfolio Optimization Theory And The Capital Asset Pricing Model; And Interest Rates And Their Term Structure. Assuming Only A Basic Knowledge Of Probability And Calculus, It Covers The Material In A Mathematically Rigorous And Complete Way At A Level Accessible To Second Or Third Year Undergraduate Students. The Text Is Interspersed With A Multitude Of Worked Examples And Exercises, So It Is Ideal For Self-study And Suitable Not Only For Students Of Mathematics, But Also Students Of Business Management, Finance And Economics, And Anyone With An Interest In Finance Who Needs To Understand The Underlying Theory.

lgli/Mathematics for Finance an introduction to Financial Engineering.pdf

lgrsnf/Mathematics for Finance an introduction to Financial Engineering.pdf

zlib/Business & Economics/Others/Marek Capinski Tomasz Zastawniak/Mathematics for Finance an introduction to Financial Engineering_3616108.pdf

Capinski, Marek, Zastawniak, Tomasz

Marek Capiski; Tomasz J. Zastawniak

Marek Capiński; Tomasz Zastawniak

Springer undergraduate mathematics series, London Berlin Heidelberg New York Hong Kong Milan Paris Tokyo, 2003

Springer Nature (Textbooks & Major Reference Works), London, 2003

Springer undergraduate mathematics series, 3. print, London, 2005

1st ed. 2003. Corr. 4th printing edition, July 6, 2003

United Kingdom and Ireland, United Kingdom

0

lg2279818

{"edition":"1951","isbns":["1852333308","9781852333300"],"last_page":321,"publisher":"springer"}

1. Introduction: A Simple Market Model . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Basic Notions and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 No-Arbitrage Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 One-Step Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Risk and Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Forward Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Call and Put Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Managing Risk with Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2. Risk-Free Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Time Value of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.2 Periodic Compounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.3 Streams of Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.4 Continuous Compounding . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.5 How to Compare Compounding Methods . . . . . . . . . . . . . . 35
2.2 Money Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.1 Zero-Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.2 Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.3 Money Market Account . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3. Risky Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1 Dynamics of Stock Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.1 Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.2 Expected Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Binomial Tree Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
vii
viii Contents
3.2.1 Risk-Neutral Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.2 Martingale Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.1 Trinomial Tree Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.2 Continuous-Time Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4. Discrete Time Market Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1 Stock and Money Market Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.1 Investment Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.2 The Principle of No Arbitrage . . . . . . . . . . . . . . . . . . . . . . . 79
4.1.3 Application to the Binomial Tree Model . . . . . . . . . . . . . . . 81
4.1.4 Fundamental Theorem of Asset Pricing . . . . . . . . . . . . . . . 83
4.2 Extended Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5. Portfolio Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1 Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Two Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.1 Risk and Expected Return on a Portfolio . . . . . . . . . . . . . . 97
5.3 Several Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.1 Risk and Expected Return on a Portfolio . . . . . . . . . . . . . . 107
5.3.2 Efficient Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4 Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4.1 Capital Market Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4.2 Beta Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.3 Security Market Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6. Forward and Futures Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.1 Forward Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.1.1 Forward Price. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.1.2 Value of a Forward Contract . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.2 Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2.1 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.2.2 Hedging with Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7. Options: General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.2 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.3 Bounds on Option Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.3.1 European Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.3.2 European and American Calls on Non-Dividend Paying
Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.3.3 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Contents ix
7.4 Variables Determining Option Prices . . . . . . . . . . . . . . . . . . . . . . . . 159
7.4.1 European Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.4.2 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.5 Time Value of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8. Option Pricing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.1 European Options in the Binomial Tree Model . . . . . . . . . . . . . . . 174
8.1.1 One Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.1.2 Two Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.1.3 General N-Step Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.1.4 Cox–Ross–Rubinstein Formula . . . . . . . . . . . . . . . . . . . . . . . 180
8.2 American Options in the Binomial Tree Model . . . . . . . . . . . . . . . 181
8.3 Black–Scholes Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9. Financial Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
9.1 Hedging Option Positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
9.1.1 Delta Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
9.1.2 Greek Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.2 Hedging Business Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.2.1 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9.2.2 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.3 Speculating with Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
9.3.1 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
9.3.2 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
10. Variable Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
10.1 Maturity-Independent Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
10.1.1 Investment in Single Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . 217
10.1.2 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
10.1.3 Portfolios of Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
10.1.4 Dynamic Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
10.2 General Term Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
10.2.1 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
10.2.2 Money Market Account . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
11. Stochastic Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
11.1 Binomial Tree Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
11.2 Arbitrage Pricing of Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
11.2.1 Risk-Neutral Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
11.3 Interest Rate Derivative Securities . . . . . . . . . . . . . . . . . . . . . . . . . . 253
11.3.1 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
x Contents
11.3.2 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
11.3.3 Caps and Floors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
11.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Glossary of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

"Assuming only a basic knowledge of probability and calculus the book combines financial motivation with mathematical style. It covers the material in a mathematically rigorous and complete way at a level accessible to second or third year undergraduate students." "The text is interspersed with a multitude of worked examples and exercises, which provides ample material for tutorials, and makes the book ideal for self-study. It is suitable not only for students of mathematics, but also students of business management, finance and economics, and anyone with an interest in finance who needs to understand the underlying theory."--Jacket

This textbook contains the fundamentals for an undergraduate course in mathematical finance aimed primarily at students of mathematics. Assuming only a basic knowledge of probability and calculus, the material is presented in a mathematically rigorous and complete way. The book covers the time value of money, including the time structure of interest rates, bonds and stock valuation; derivative securities (futures, options), modelling in discrete time, pricing and hedging, and many other core topics. With numerous examples, problems and exercises, this book is ideally suited for independent study.

Contains the fundamentals for an undergraduate course in mathematical finance aimed primarily at students of mathematics. This book covers the time value of money, including the time structure of interest rates, bonds and stock valuation; derivative securities (futures, options), and modelling in discrete time, pricing and hedging.

2018-11-02