# The Mathematics of Finance: Modeling and Hedging

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*Victor Goodman; Joseph Stampfli*

This book is ideally suited for an introductory undergraduate course on
financial engineering. It explains the basic concepts of financial
derivatives, including put and call options, as well as more complex
derivatives such as barrier options and options on futures contracts. Both
discrete and continuous models of market behavior are developed in this
book. In particular, the analysis of option prices developed by Black and
Scholes is explained in a self-contained way, using both the probabilistic
Brownian Motion method and the analytical differential equations method.

The book begins with binomial stock price models, moves on to
multistage models, then to the Cox–Ross–Rubinstein option
pricing process, and then to the Black–Scholes formula. Other
topics presented include Zero Coupon Bonds, forward rates, the yield
curve, and several bond price models. The book continues with foreign
exchange models and the Keynes Interest Rate Parity Formula, and
concludes with the study of country risk, a topic not inappropriate
for the times.

In addition to theoretical results, numerical models are presented in much
detail. Each of the eleven chapters includes a variety of exercises.

An instructor's manual for this title is available electronically. Please
send email to textbooks@ams.org for more
information.

#### Readership

Undergraduate students interested in financial engineering.

#### Reviews & Endorsements

[T]he book is a worthwhile contribution to the literature...Its main strength is that it provides an introduction to mathematical finance at a level that is not too technical. Indeed, it is very successful in achieving this outcome...[P]rospective undergraduate students of financial mathematics will find life much easier by reading [this] book.

-- Kam Fong Chan, Pacific Accounting Review

#### Table of Contents

# Table of Contents

## The Mathematics of Finance: Modeling and Hedging

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents ix10 free
- 1 Financial Markets 118 free
- 2 Binomial Trees, Replicating Portfolios, and Arbitrage 2542
- 3 Tree Models for Stocks and Options 4461
- 3.1 A Stock Model 4461
- 3.2 Pricing a Call Option with the Tree Model 4966
- 3.3 Pricing an American Option 5269
- 3.4 Pricing an Exotic Option—Knockout Options 5572
- 3.5 Pricing an Exotic Option—Lookback Options 5976
- 3.6 Adjusting the Binomial Tree Model to Real-World Data 6178
- 3.7 Hedging and Pricing the N-Period Binomial Model 6683

- 4 Using Spreadsheets to Compute Stock and Option Trees 7188
- 5 Continuous Models and the Black-Scholes Formula 8198
- 5.1 A Continuous-Time Stock Model 8198
- 5.2 The Discrete Model 8299
- 5.3 An Analysis of the Continuous Model 87104
- 5.4 The Black-Scholes Formula 90107
- 5.5 Derivation of the Black-Scholes Formula 92109
- 5.6 Put-Call Parity 97114
- 5.7 Trees and Continuous Models 98115
- 5.8 The GBM Stock Price Model—A Cautionary Tale 103120
- 5.9 Appendix: Construction of a Brownian Path 106123

- 6 The Analytic Approach to Black-Scholes 109126
- 6.1 Strategy for Obtaining the Differential Equation 110127
- 6.2 Expanding V(S, t) 110127
- 6.3 Expanding and Simplifying V(S[sub(t)],t) 111128
- 6.4 Finding a Portfolio 112129
- 6.5 Solving the Black-Scholes Differential Equation 114131
- 6.6 Options on Futures 116133
- 6.7 Appendix: Portfolio Differentials 120137

- 7 Hedging 122139
- 8 Bond Models and Interest Rate Options 137154
- 8.1 Interest Rates and Forward Rates 137154
- 8.2 Zero-Coupon Bonds 140157
- 8.3 Swaps 144161
- 8.4 Pricing and Hedging a Swap 152169
- 8.5 Interest Rate Models 157174
- 8.5.1 Discrete Interest Rate Models 158175
- 8.5.2 Pricing ZCBs from the Interest Rate Model 162179
- 8.5.3 The Bond Price Paradox 165182
- 8.5.4 Can the Expected Value Pricing Method Be Arbitraged? 166183
- 8.5.5 Continuous Models 171188
- 8.5.6 A Bond Price Model 171188
- 8.5.7 A Simple Example 174191
- 8.5.8 The Vasicek Model 178195

- 8.6 Bond Price Dynamics 180197
- 8.7 A Bond Price Formula 181198
- 8.8 Bond Prices, Spot Rates, and HJM 183200
- 8.9 The Derivative Approach to HJM: The HJM Miracle 186203
- 8.10 Appendix: Forward Rate Drift 188205

- 9 Computational Methods for Bonds 190207
- 10 Currency Markets and Foreign Exchange Risks 207224
- 11 International Political Risk Analysis 221238
- Answers to Selected Exercises 241258
- Index 247264
- Back Cover Back Cover1268