Mathematics of Finance: Modeling and Hedging

by Victor Goodman, Joseph G. Stampfli

Overview

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.

Product Details

ISBN-13:	9780821847930
Publisher:	American Mathematical Society
Publication date:	03/10/2009
Series:	Pure and Applied Undergraduate Texts Series , #7
Edition description:	New Edition
Pages:	250
Product dimensions:	7.00(w) x 10.20(h) x 0.80(d)

Table of Contents

1	Financial Markets	1
1.1	Markets and Math	1
1.2	Stocks and Their Derivatives	2
1.2.1	Forward Stock Contracts	3
1.2.2	Call Options	7
1.2.3	Put Options	9
1.2.4	Short Selling	11
1.3	Pricing Futures Contracts	12
1.4	Bond Markets	15
1.4.1	Rates of Return	16
1.4.2	The U.S. Bond Market	17
1.4.3	Interest Rates and Forward Interest Rates	18
1.4.4	Yield Curves	19
1.5	Interest Rate Futures	20
1.5.1	Determining the Futures Price	20
1.5.2	Treasury Bill Futures	21
1.6	Foreign Exchange	22
1.6.1	Currency Hedging	22
1.6.2	Computing Currency Futures	23
2	Binomial Trees, Replicating Portfolios, and Arbitrage	25
2.1	Three Ways to Price a Derivative	25
2.2	The Game Theory Method	26
2.2.1	Eliminating Uncertainty	27
2.2.2	Valuing the Option	27
2.2.3	Arbitrage	27
2.2.4	The Game Theory Method--A General Formula	28
2.3	Replicating Portfolios	29
2.3.1	The Context	30
2.3.2	A Portfolio Match	30
2.3.3	Expected Value Pricing Approach	31
2.3.4	How to Remember the Pricing Probability	32
2.4	The Probabilistic Approach	34
2.5	Risk	36
2.6	Repeated Binomial Trees and Arbitrage	39
2.7	Appendix: Limits of the Arbitrage Method	41
3	Tree Models for Stocks and Options	44
3.1	A Stock Model	44
3.1.1	Recombining Trees	46
3.1.2	Chaining and Expected Values	46
3.2	Pricing a Call Option with the Tree Model	49
3.3	Pricing an American Option	52
3.4	Pricing an Exotic Option--Knockout Options	55
3.5	Pricing an Exotic Option--Lookback Options	59
3.6	Adjusting the Binomial Tree Model to Real-World Data	61
3.7	Hedging and Pricing the N-Period Binomial Model	66
4	Using Spreadsheets to Compute Stock and Option Trees	71
4.1	Some Spreadsheet Basics	71
4.2	Computing European Option Trees	74
4.3	Computing American Option Trees	77
4.4	Computing a Barrier Option Tree	79
4.5	Computing N-Step Trees	80
5	Continuous Models and the Black-Scholes Formula	81
5.1	A Continuous-Time Stock Model	81
5.2	The Discrete Model	82
5.3	An Analysis of the Continuous Model	87
5.4	The Black-Scholes Formula	90
5.5	Derivation of the Black-Scholes Formula	92
5.5.1	The Related Model	92
5.5.2	The Expected Value	94
5.5.3	Two Integrals	94
5.5.4	Putting the Pieces Together	96
5.6	Put-Call Parity	97
5.7	Trees and Continuous Models	98
5.7.1	Binomial Probabilities	98
5.7.2	Approximation with Large Trees	100
5.7.3	Scaling a Tree to Match a GBM Model	102
5.8	The GBM Stock Price Model--A Cautionary Tale	103
5.9	Appendix: Construction of a Brownian Path	106
6	The Analytic Approach to Black-Scholes	109
6.1	Strategy for Obtaining the Differential Equation	110
6.2	Expanding V(S, t)	110
6.3	Expanding and Simplifying V(S[subscript t], t)	111
6.4	Finding a Portfolio	112
6.5	Solving the Black-Scholes Differential Equation	114
6.5.1	Cash or Nothing Option	114
6.5.2	Stock-or-Nothing Option	115
6.5.3	European Call	116
6.6	Options on Futures	116
6.6.1	Call on a Futures Contract	117
6.6.2	A PDE for Options on Futures	118
6.7	Appendix: Portfolio Differentials	120
7	Hedging	122
7.1	Delta Hedging	122
7.1.1	Hedging, Dynamic Programming, and a Proof that Black-Scholes Really Works in an Idealized World	123
7.1.2	Why the Foregoing Argument Does Not Hold in the Real World	124
7.1.3	Earlier [Delta] Hedges	125
7.2	Methods for Hedging a Stock or Portfolio	126
7.2.1	Hedging with Puts	126
7.2.2	Hedging with Collars	127
7.2.3	Hedging with Paired Trades	127
7.2.4	Correlation-Based Hedges	127
7.2.5	Hedging in the Real World	128
7.3	Implied Volatility	128
7.3.1	Computing [sigma subscript 1] with Maple	128
7.3.2	The Volatility Smile	129
7.4	The Parameters [Delta], [Gamma], and [Theta]	130
7.4.1	The Role of [Gamma]	131
7.4.2	A Further Role for [Delta], [Gamma], [Theta]	133
7.5	Derivation of the Delta Hedging Rule	134
7.6	Delta Hedging a Stock Purchase	135
8	Bond Models and Interest Rate Options	137
8.1	Interest Rates and Forward Rates	137
8.1.1	Size	138
8.1.2	The Yield Curve	138
8.1.3	How Is the Yield Curve Determined?	139
8.1.4	Forward Rates	139
8.2	Zero-Coupon Bonds	140
8.2.1	Forward Rates and ZCBs	140
8.2.2	Computations Based on Y(t) or P(t)	142
8.3	Swaps	144
8.3.1	Another Variation on Payments	147
8.3.2	A More Realistic Scenario	148
8.3.3	Models for Bond Prices	149
8.3.4	Arbitrage	150
8.4	Pricing and Hedging a Swap	152
8.4.1	Arithmetic Interest Rates	153
8.4.2	Geometric Interest Rates	155
8.5	Interest Rate Models	157
8.5.1	Discrete Interest Rate Models	158
8.5.2	Pricing ZCBs from the Interest Rate Model	162
8.5.3	The Bond Price Paradox	165
8.5.4	Can the Expected Value Pricing Method Be Arbitraged?	166
8.5.5	Continuous Models	171
8.5.6	A Bond Price Model	171
8.5.7	A Simple Example	174
8.5.8	The Vasicek Model	178
8.6	Bond Price Dynamics	180
8.7	A Bond Price Formula	181
8.8	Bond Prices, Spot Rates, and HJM	183
8.8.1	Example: The Hall-White Model	184
8.9	The Derivative Approach to HJM: The HJM Miracle	186
8.10	Appendix: Forward Rate Drift	188
9	Computational Methods for Bonds	190
9.1	Tree Models for Bond Prices	190
9.1.1	Fair and Unfair Games	190
9.1.2	The Ho-Lee Model	192
9.2	A Binomial Vasicek Model: A Mean Reversion Model	200
9.2.1	The Base Case	201
9.2.2	The General Induction Step	202
10	Currency Markets and Foreign Exchange Risks	207
10.1	The Mechanics of Trading	207
10.2	Currency Forwards: Interest Rate Parity	209
10.3	Foreign Currency Options	211
10.3.1	The Garman-Kohlhagen Formula	211
10.3.2	Put-Call Parity for Currency Options	213
10.4	Guaranteed Exchange Rates and Quantos	214
10.4.1	The Bond Hedge	215
10.4.2	Pricing the GER Forward on a Stock	216
10.4.3	Pricing the GER Put or Call Option	219
10.5	To Hedge or Not to Hedge--and How Much	220
11	International Political Risk Analysis	221
11.1	Introduction	221
11.2	Types of International Risks	222
11.2.1	Political Risk	222
11.2.2	Managing International Risk	223
11.2.3	Diversification	223
11.2.4	Political Risk and Export Credit Insurance	224
11.3	Credit Derivatives and the Management of Political Risk	225
11.3.1	Foreign Currency and Derivatives	225
11.3.2	Credit Default Risk and Derivatives	226
11.4	Pricing International Political Risk	228
11.4.1	The Credit Spread or Risk Premium on Bonds	229
11.5	Two Models for Determining the Risk Premium	230
11.5.1	The Black-Scholes Approach to Pricing Risky Debt	230
11.5.2	An Alternative Approach to Pricing Risky Debt	234
11.6	A Hypothetical Example of the JLT Model	238
	Answers to Selected Exercises	241
	Index	247

Editorial Reviews

Explains basic financial and mathematical concepts used in modeling and hedging. Each topic is introduced with the assumption that the reader has little to no previous exposure to financial matters or to the activities that are common to major equity markets. Contains chapters on financial markets, binomial trees, tree models for stocks and options, using spreadsheets to compute stock and option trees, and continuous models and the Black-Scholes formula. Other chapter topics are hedging, bond models and interest rate options, computational methods for bonds, currency markets and foreign exchange risks, and international political risk analysis. Includes exercises and selected answers. The authors are affiliated with Indiana University. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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