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 |