Mathematics of Finance: Modeling and Hedging

Mathematics of Finance: Modeling and Hedging

by Victor Goodman, Joseph G. Stampfli
     
 

ISBN-10: 0821847937

ISBN-13: 9780821847930

Pub. Date: 03/10/2009

Publisher: American Mathematical Society

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

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

1Financial Markets1
1.1Markets and Math1
1.2Stocks and Their Derivatives2
1.2.1Forward Stock Contracts3
1.2.2Call Options7
1.2.3Put Options9
1.2.4Short Selling11
1.3Pricing Futures Contracts12
1.4Bond Markets15
1.4.1Rates of Return16
1.4.2The U.S. Bond Market17
1.4.3Interest Rates and Forward Interest Rates18
1.4.4Yield Curves19
1.5Interest Rate Futures20
1.5.1Determining the Futures Price20
1.5.2Treasury Bill Futures21
1.6Foreign Exchange22
1.6.1Currency Hedging22
1.6.2Computing Currency Futures23
2Binomial Trees, Replicating Portfolios, and Arbitrage25
2.1Three Ways to Price a Derivative25
2.2The Game Theory Method26
2.2.1Eliminating Uncertainty27
2.2.2Valuing the Option27
2.2.3Arbitrage27
2.2.4The Game Theory Method--A General Formula28
2.3Replicating Portfolios29
2.3.1The Context30
2.3.2A Portfolio Match30
2.3.3Expected Value Pricing Approach31
2.3.4How to Remember the Pricing Probability32
2.4The Probabilistic Approach34
2.5Risk36
2.6Repeated Binomial Trees and Arbitrage39
2.7Appendix: Limits of the Arbitrage Method41
3Tree Models for Stocks and Options44
3.1A Stock Model44
3.1.1Recombining Trees46
3.1.2Chaining and Expected Values46
3.2Pricing a Call Option with the Tree Model49
3.3Pricing an American Option52
3.4Pricing an Exotic Option--Knockout Options55
3.5Pricing an Exotic Option--Lookback Options59
3.6Adjusting the Binomial Tree Model to Real-World Data61
3.7Hedging and Pricing the N-Period Binomial Model66
4Using Spreadsheets to Compute Stock and Option Trees71
4.1Some Spreadsheet Basics71
4.2Computing European Option Trees74
4.3Computing American Option Trees77
4.4Computing a Barrier Option Tree79
4.5Computing N-Step Trees80
5Continuous Models and the Black-Scholes Formula81
5.1A Continuous-Time Stock Model81
5.2The Discrete Model82
5.3An Analysis of the Continuous Model87
5.4The Black-Scholes Formula90
5.5Derivation of the Black-Scholes Formula92
5.5.1The Related Model92
5.5.2The Expected Value94
5.5.3Two Integrals94
5.5.4Putting the Pieces Together96
5.6Put-Call Parity97
5.7Trees and Continuous Models98
5.7.1Binomial Probabilities98
5.7.2Approximation with Large Trees100
5.7.3Scaling a Tree to Match a GBM Model102
5.8The GBM Stock Price Model--A Cautionary Tale103
5.9Appendix: Construction of a Brownian Path106
6The Analytic Approach to Black-Scholes109
6.1Strategy for Obtaining the Differential Equation110
6.2Expanding V(S, t)110
6.3Expanding and Simplifying V(S[subscript t], t)111
6.4Finding a Portfolio112
6.5Solving the Black-Scholes Differential Equation114
6.5.1Cash or Nothing Option114
6.5.2Stock-or-Nothing Option115
6.5.3European Call116
6.6Options on Futures116
6.6.1Call on a Futures Contract117
6.6.2A PDE for Options on Futures118
6.7Appendix: Portfolio Differentials120
7Hedging122
7.1Delta Hedging122
7.1.1Hedging, Dynamic Programming, and a Proof that Black-Scholes Really Works in an Idealized World123
7.1.2Why the Foregoing Argument Does Not Hold in the Real World124
7.1.3Earlier [Delta] Hedges125
7.2Methods for Hedging a Stock or Portfolio126
7.2.1Hedging with Puts126
7.2.2Hedging with Collars127
7.2.3Hedging with Paired Trades127
7.2.4Correlation-Based Hedges127
7.2.5Hedging in the Real World128
7.3Implied Volatility128
7.3.1Computing [sigma subscript 1] with Maple128
7.3.2The Volatility Smile129
7.4The Parameters [Delta], [Gamma], and [Theta]130
7.4.1The Role of [Gamma]131
7.4.2A Further Role for [Delta], [Gamma], [Theta]133
7.5Derivation of the Delta Hedging Rule134
7.6Delta Hedging a Stock Purchase135
8Bond Models and Interest Rate Options137
8.1Interest Rates and Forward Rates137
8.1.1Size138
8.1.2The Yield Curve138
8.1.3How Is the Yield Curve Determined?139
8.1.4Forward Rates139
8.2Zero-Coupon Bonds140
8.2.1Forward Rates and ZCBs140
8.2.2Computations Based on Y(t) or P(t)142
8.3Swaps144
8.3.1Another Variation on Payments147
8.3.2A More Realistic Scenario148
8.3.3Models for Bond Prices149
8.3.4Arbitrage150
8.4Pricing and Hedging a Swap152
8.4.1Arithmetic Interest Rates153
8.4.2Geometric Interest Rates155
8.5Interest Rate Models157
8.5.1Discrete Interest Rate Models158
8.5.2Pricing ZCBs from the Interest Rate Model162
8.5.3The Bond Price Paradox165
8.5.4Can the Expected Value Pricing Method Be Arbitraged?166
8.5.5Continuous Models171
8.5.6A Bond Price Model171
8.5.7A Simple Example174
8.5.8The Vasicek Model178
8.6Bond Price Dynamics180
8.7A Bond Price Formula181
8.8Bond Prices, Spot Rates, and HJM183
8.8.1Example: The Hall-White Model184
8.9The Derivative Approach to HJM: The HJM Miracle186
8.10Appendix: Forward Rate Drift188
9Computational Methods for Bonds190
9.1Tree Models for Bond Prices190
9.1.1Fair and Unfair Games190
9.1.2The Ho-Lee Model192
9.2A Binomial Vasicek Model: A Mean Reversion Model200
9.2.1The Base Case201
9.2.2The General Induction Step202
10Currency Markets and Foreign Exchange Risks207
10.1The Mechanics of Trading207
10.2Currency Forwards: Interest Rate Parity209
10.3Foreign Currency Options211
10.3.1The Garman-Kohlhagen Formula211
10.3.2Put-Call Parity for Currency Options213
10.4Guaranteed Exchange Rates and Quantos214
10.4.1The Bond Hedge215
10.4.2Pricing the GER Forward on a Stock216
10.4.3Pricing the GER Put or Call Option219
10.5To Hedge or Not to Hedge--and How Much220
11International Political Risk Analysis221
11.1Introduction221
11.2Types of International Risks222
11.2.1Political Risk222
11.2.2Managing International Risk223
11.2.3Diversification223
11.2.4Political Risk and Export Credit Insurance224
11.3Credit Derivatives and the Management of Political Risk225
11.3.1Foreign Currency and Derivatives225
11.3.2Credit Default Risk and Derivatives226
11.4Pricing International Political Risk228
11.4.1The Credit Spread or Risk Premium on Bonds229
11.5Two Models for Determining the Risk Premium230
11.5.1The Black-Scholes Approach to Pricing Risky Debt230
11.5.2An Alternative Approach to Pricing Risky Debt234
11.6A Hypothetical Example of the JLT Model238
Answers to Selected Exercises241
Index247

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