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Contents 1

Introduction 1

1 Preliminary Mathematics 5

1.1 Random Walk 5

1.2 Another Take on Volatility and Time 8

1.3 A First Glance at It？'s Lemma 9

1.4 Continuous Time:Brownian Motion;More on It？'s Lemma 11

1.5 Two-Dimensional Brownian Motion 14

1.6 Bivariate It？'s Lemma 15

1.7 Three Paradoxes of Finance 16

1.7.1 Paradox 1:Siegel's Paradox 16

1.7.2 Paradox 2:The Stock,Free-Lunch Paradox 18

1.7.3 Paradox 3:The Skill Versus Luck Paradox 19

2 Principles of Financial Valuation 22

2.1 Uncertainty,Utility Theory,and Risk 22

2.2 Risk and the Equilibrium Pricing of Securities 28

2.3 The Binomial Option-Pricing Model 41

2.4 Limiting Option-Pricing Formula 46

2.5 Continuous-Time Models 47

2.5.1 The Black-Scholes/Merton Model-Pricing Kernel Approach 48

2.5.2 The Black-Scholes/Merton Model-Probabilistic Approach 57

2.5.3 The Black-Scholes/Merton Model-Hedging Approach 61

2.6 Exotic Options 63

2.6.1 Digital Options 64

2.6.2 Power Options 65

2.6.3 Asian Options 67

2.6.4 Barrier Options 71

3 Interest Rate Models 78

3.1 Interest Rate Derivatives:Not So Simple 78

3.2 Bonds and Yields 80

3.2.1 Prices and Yields to Maturity 80

3.2.2 Discount Factors,Zero-Coupon Rates, and Coupon Bias 82

3.2.3 Forward Rates 85

3.3 Naive Models of Interest Rate Risk 88

3.3.1 Duration 88

3.3.2 Convexity 99

3.3.3 The Free Lunch in the Duration Model 104

3.4 An Overview of Interest Rate Derivatives 108

3.4.1 Bonds with Embedded Options 109

3.4.2 Forward Rate Agreements 110

3.4.3 Eurostrip Futures 112

3.4.4 The Convexity Adjustment 113

3.4.5 Swaps 118

3.4.6 Caps and Floors 120

3.4.7 Swaptions 121

3.5 Yield Curve Swaps 122

3.5.1 The CMS Swap 122

3.5.2 The Quanto Swap 127

3.6 Factor Models 131

3.6.1 A General Single-Factor Model 131

3.6.2 The Merton Model 135

3.6.3 The Vasicek Model 139

3.6.4 The Cox-Ingersoll-Ross Model 142

3.6.5 Risk-Neutral Valuation 144

3.7 Term-Structure-Consistent Models 147

3.7.1 "Equilibrium"Versus"Fitting" 147

3.7.2 The Ho-Lee Model 153

3.7.3 The Ho-Lee Model with Time-Varying Volatility 157

3.7.4 The Black-Derman-Toy Model 162

3.8 Risky Bonds and Their Derivatives 166

3.8.1 The Merton Model 167

3.8.2 The Jarrow-Turnbull Model 168

3.9 The Heath,Jarrow,and Morton Approach 172

3.10 Interest Rates as Options 180

4 Mathematics of Asset Pricing 184

4.1 Random Walks 184

4.1.1 Description 184

4.1.2 Gambling Recreations 186

4.2 Arithmetic Brownian Motion 192

4.2.1 Arithmetic Brownian Motion as a Limit of a Simple Random Walk 192

4.2.2 Moments of an Arithmetic Brownian Motion 196

4.2.3 Why Sample Paths Are Not Differentiable 198

4.2.4 Why Sample Paths Are Continuous 198

4.2.5 Extreme Values and Hitting Times 199

4.2.6 The Arcsine Law Revisited 203

4.3 Geometric Brownian Motion 204

4.3.1 Description 204

4.3.2 Moments of a Geometric Brownian Motion 207

4.4 It？ Calculus 209

4.4.1 Riemann-Stieljes,Stratonovitch,and It？ Integrals 209

4.4.2 It？'s Lemma 214

4.4.3 Multidimensional It？'s Lemma 222

4.5 Mean-Reverting Processes 225

4.5.1 Introduction 225

4.5.2 The Ornstein-Uhlenbeck Process 225

4.5.3 Calculations of Moments with the Dynkin Operator 226

4.5.4 The Square-Root Process 228

4.6 Jump Process 229

4.6.1 Pure Jumps 229

4.6.2 Time Between Two Jumps 231

4.6.3 Jump Diffusions 232

4.6.4 It？'s Lemma for Jump Diffusions 233

4.7 Kolmogorov Equations 234

4.7.1 The Kolmogorov Forward Equation 234

4.7.2 The Dirac Delta Function 236

4.7.3 The Kolmogorov Backward Equation 236

4.8 Martingales 239

4.8.1 Definitions and Examples 239

4.8.2 Some Useful Facts About Martingales 241

4.8.3 Martingales and Brownian Motion 242

4.9 Dynamic Programming 245

4.9.1 The Traveling Salesman 245

4.9.2 Optimal Control of It？ Processes:Finite Horizon 247

4.9.3 Optimal Control of It？ Processes:Infinite Horizon 248

4.10 Partial Differential Equations 253

4.10.1 The Kolmogorov Forward Equation Revisited 253

4.10.2 Risk-Neutral Pricing Equation 256

4.10.3 The Laplace Transform 257

4.10.4 Resolution of the Kolmogorov Forward Equation 262

4.10.5 Resolution of the Risk-Neutral Pricing Equation 265

Bibliography 269

Index 327