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(St,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 △ 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 ？ with Maple 128
7.3.2 The Volatility Smile 129
7."4 The Parameters△,Г,and Θ 130
7.4.1 The Role of Г 131
7."4.2 A Further Role for △,Г,Θ 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