The Mathematics of Finance: Modeling and Hedging
Victor Goodman, Joseph Stampfli
180 x 240 mm
Year of Publishing
Territorial Rights
American Mathematical Society

Out Of Stock


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 for more information.

Victor Goodman and Joseph Stampfli, Indiana University, Bloomington, IN

1 Financial Markets
1.1 Markets and Math
1.2 Stocks and Their Derivatives
    1.2.1 Forward Stock Contracts
    1.2.2 Call Options
    1.2.3 Put Options
    1.2.4 Short Selling
1.3 Pricing Futures Contracts
1.4 Bond Markets
    1.4.1 Rates of Return
    1.4.2 The U.S. Bond Market
    1.4.3 Interest Rates and Forward Interest Rates
    1.4.4 Yield Curves
1.5 Interest Rate Futures
    1.5.1 Determining the Futures Price
    1.5.2 Treasury Bill Futures
1.6 Foreign Exchange
    1.6.1 Currency Hedging
    1.6.2 Computing Currency Futures

2 Binomial Trees, Replicating Portfolios, and Arbitrage
2.1 Three Ways to Price a Derivative
2.2 The Game Theory Method
    2.2.1 Eliminating Uncertainty
    2.2.2 Valuing the Option
    2.2.3 Arbitrage
    2.2.4 The Game Theory Method—A General Formula
2.3 Replicating Portfolios
    2.3.1 The Context
    2.3.2 A Portfolio Match
    2.3.3 Expected Value Pricing Approach
    2.3.4 How to Remember the Pricing Probability
2.4 The Probabilistic Approach
2.5 Risk
2.6 Repeated Binomial Trees and Arbitrage
2.7 Appendix: Limits of the Arbitrage Method

3 Tree Models for Stocks and Options
3.1 A Stock Model
    3.1.1 Recombining Trees
    3.1.2 Chaining and Expected Values
3.2 Pricing a Call Option with the Tree Model
3.3 Pricing an American Option
3.4 Pricing an Exotic Option—Knockout Options
3.5 Pricing an Exotic Option—Lookback Options
3.6 Adjusting the Binomial Tree Model to Real-World Data
3.7 Hedging and Pricing the Af-Period Binomial Model

4 Using Spreadsheets to Compute Stock and Option Trees
4.1 Some Spreadsheet Basics
4.2 Computing European Option Trees
4.3 Computing American Option Trees
4.4 Computing a Barrier Option Tree
4.5 Computing AT-Step Trees

5 Continuous Models and the Black-Scholes Formula
5.1 A Continuous-Time Stock Model
5.2 The Discrete Model
5.3 An Analysis of the Continuous Model
5.4 The Black-Scholes Formula
5.5 Derivation of the Black-Scholes Formula
    5.5.1 The Related Model
    5.5.2 The Expected Value
    5.5.3 Two Integrals
    5.5.4 Putting the Pieces Together
5.6 Put-Call Parity
5.7 Trees and Continuous Models
    5.7.1 Binomial Probabilities
    5.7.2 Approximation with Large Trees
    5.7.3 Scaling a Tree to Match a GBM Model
5.8 The GBM Stock Price Model—A Cautionary Tale
5.9 Appendix: Construction of a Brownian Path

6 The Analytic Approach to Black-Scholes
6.1 Strategy for Obtaining the Differential Equation
6.2 Expanding V(S, t)
6.3 Expanding and Simplifying V(St, t)
6.4 Finding a Portfolio
6.5 Solving the Black-Scholes Differential Equation
    6.5.1 Cash or Nothing Option
    6.5.2 Stock-or-Nothing Option
    6.5.3 European Call
6.6 Options on Futures
    6.6.1 Call on a Futures Contract
    6.6.2 A PDE for Options on Futures
6.7 Appendix: Portfolio Differentials

7 Hedging
7.1 Delta Hedging
    7.1.1 Hedging, Dynamic Programming, and a Proof that Black-Scholes Really Works in an Idealized World
    7.1.2 Why the Foregoing Argument Does Not Hold in the Real World
    7.1.3 Earlier A Hedges
7.2 Methods for Hedging a Stock or Portfolio
    7.2.1 Hedging with Puts
    7.2.2 Hedging with Collars
    7.2.3 Hedging with Paired Trades
    7.2.4 Correlation-Based Hedges
    7.2.5 Hedging in the Real World
7.3 Implied Volatility
    7.3.1 Computing 07 with Maple
    7.3.2 The Volatility Smile
7.4 The Parameters A, T, and ©
    7.4.1 The Role of T
    7.4.2 A Further Role for A, I\ 0
7.5 Derivation of the Delta Hedging Rule
7.6 Delta Hedging a Stock Purchase

8 Bond Models and Interest Rate Options
8.1 Interest Rates and Forward Rates
    8.1.1 Size
    8.1.2 The Yield Curve
    8.1.3 How Is the Yield Curve Determined?
    8.1.4 Forward Rates
8.2 Zero-Coupon Bonds
    8.2.1 Forward Rates and ZCBs
    8.2.2 Computations Based on Y(t) or P(t)
8.3 Swaps
    8.3.1 Another Variation on Payments
    8.3.2 A More Realistic Scenario
    8.3.3 Models for Bond Prices
    8.3.4 Arbitrage
8.4 Pricing and Hedging a Swap
    8.4.1 Arithmetic Interest Rates
    8.4.2 Geometric Interest Rates
8.5 Interest Rate Models
    8.5.1 Discrete Interest Rate Models
    8.5.2 Pricing ZCBs from the Interest Rate Model
    8.5.3 The Bond Price Paradox
    8.5.4 Can the Expected Value Pricing Method Be Arbitraged?
    8.5.5 Continuous Models
    8.5.6 A Bond Price Model
    8.5.7 A Simple Example
    8.5.8 The Vasicek Model
8.6 Bond Price Dynamics
8.7 A Bond Price Formula
8.8 Bond Prices, Spot Rates, and HJM
    8.8.1 Example: The Hall-White Model
8.9 The Derivative Approach to HJM: The HJM Miracle
8.10 Appendix: Forward Rate Drift

9 Computational Methods for Bonds
9.1 Tree Models for Bond Prices
    9.1.1 Fair and Unfair Games
    9.1.2 The Ho-Lee Model
9.2 A Binomial Vasicek Model: A Mean Reversion Model
    9.2.1 The Base Case 2
    9.2.2 The General Induction Step

10 Currency Markets and Foreign Exchange Risks
10.1 The Mechanics of Trading
10.2 Currency Forwards: Interest Rate Parity
10.3 Foreign Currency Options
    10.3.1 The Garman-Kohlhagen Formula
    10.3.2 Put-Call Parity for Currency Options
10.4 Guaranteed Exchange Rates and Quantos
    10.4.1 The Bond Hedge
    10.4.2 Pricing the GER Forward on a Stock
    10.4.3 Pricing the GER Put or Call Option
10.5 To Hedge or Not to Hedge—and How Much

11 International Political Risk Analysis
11.1 Introduction
11.2 Types of International Risks
    11.2.1 Political Risk
    11.2.2 Managing International Risk
    11.2.3 Diversification
    11.2.4 Political Risk and Export Credit Insurance
11.3 Credit Derivatives and the Management of Political Risk
    11.3.1 Foreign Currency and Derivatives
    11.3.2 Credit Default Risk and Derivatives
11.4 Pricing International Political Risk
    11.4.1 The Credit Spread or Risk Premium on Bonds
11.5 Two Models for Determining the Risk Premium
    11.5.1 The Black-Scholes Approach to Pricing Risky Debt
    11.5.2 An Alternative Approach to Pricing Risky Debt
11.6 A Hypothetical Example of the JLT Model
Answers to Selected Exercises

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