AMIR SADR, Ph.D. is a highly sought-after expert in fixed income and interest rate derivatives. He is a university lecturer at NYU Courant and a consultant to banks and hedge funds, with a current focus on crypto derivatives. He has held senior management roles in quantitative research and trading at major banks and hedge funds including Morgan Stanley, Greenwich Capital, and Brevan Howard.

Preface xiii

Acknowledgments xix

About the Author xxi

Acronyms xxiii

Chapter 1 Finance 1

1.1 Follow the Money 1

1.2 Financial Markets and Participants 3

1.3 Quantitative Finance 5

Chapter 2 Rates, Yields, Bond Math 7

2.1 Interest Rates 7

2.1.1 Fractional Periods 8

2.1.2 Continuous Compounding 9

2.1.3 Discount Factor, PV, FV 9

2.1.4 Yield, Internal Rate of Return 10

2.2 Arbitrage, Law of One Price 11

2.3 Price-Yield Formula 12

2.3.1 Clean Price 15

2.3.2 Zero-Coupon Bond 17

2.3.3 Annuity 17

2.3.4 Fractional Years, Day Counts 19

2.3.5 U.S. Treasury Securities 20

2.4 Solving for Yield: Root Search 21

2.4.1 Newton-Raphson Method 21

2.4.2 Bisection Method 21

2.5 Price Risk 22

2.5.1 Pv01, Pvbp 23

2.5.2 Convexity 24

2.5.3 Taylor Series Expansion 25

2.5.4 Expansion Around c 27

2.5.5 Numerical Derivatives 27

2.6 Level Pay Loan 28

2.6.1 Interest and Principal Payments 30

2.6.2 Average Life 31

2.6.3 Pool of Loans 32

2.6.4 Prepayments 32

2.6.5 Negative Convexity 34

2.7 Yield Curve 36

2.7.1 Bootstrap Method 37

2.7.2 Interpolation Method 37

2.7.3 Rich/Cheap Analysis 39

2.7.4 Yield Curve Trades 40

Exercises 41

Python Projects 48

Chapter 3 Investment Theory 55

3.1 Utility Theory 56

3.1.1 Risk Appetite 57

3.1.2 Risk versus Uncertainty, Ranking 58

3.1.3 Utility Theory Axioms 59

3.1.4 Certainty-Equivalent 61

3.1.5 X-arra 63

3.2 Portfolio Selection 64

3.2.1 Asset Allocation 65

3.2.2 Markowitz Mean-Variance Portfolio Theory 65

3.2.3 Risky Assets 66

3.2.4 Portfolio Risk 66

3.2.5 Minimum Variance Portfolio 68

3.2.6 Leverage, Short Sales 71

3.2.7 Multiple Risky Assets 71

3.2.8 Efficient Frontier 75

3.2.9 Minimum Variance Frontier 76

3.2.10 Separation: Two-Fund Theorem 78

3.2.11 Risk-Free Asset 79

3.2.12 Capital Market Line 79

3.2.13 Market Portfolio 80

3.3 Capital Asset Pricing Model 81

3.3.1 CAPM Pricing 84

3.3.2 Systematic and Diversifiable Risk 84

3.4 Factors 85

3.4.1 Arbitrage Pricing Theory 85

3.4.2 Fama-French Factors 87

3.4.3 Factor Investing 88

3.4.4 Pca 88

3.5 Mean-Variance Efficiency and Utility 90

3.5.1 Parabolic Utility 91

3.5.2 Jointly Normal Returns 92

3.6 Investments in Practice 93

3.6.1 Rebalancing 93

3.6.2 Performance Measures 94

3.6.3 Z-Scores, Mean-Reversion, Rich-Cheap 95

3.6.4 Pairs Trading 95

3.6.5 Risk Management 97

3.6.5.1 Gambler's Ruin 97

3.6.5.2 Kelly's Ratio 98

References 99

Exercises 100

Python Projects 106

Chapter 4 Forwards and Futures 109

4.1 Forwards 109

4.1.1 Forward Price 110

4.1.2 Cash and Carry 111

4.1.3 Interim Cash Flows 111

4.1.4 Valuation of Forwards 111

4.1.5 Forward Curve 112

4.2 Futures Contracts 114

4.2.1 Futures versus Forwards 115

4.2.2 Zero-Cost, Leverage 116

4.2.3 Mark-to-Market Loss 116

4.3 Stock Dividends 117

4.4 Forward Foreign Currency Exchange Rate 117

4.5 Forward Interest Rates 119

References 120

Exercises 120

Chapter 5 Risk-Neutral Valuation 125

5.1 Contingent Claims 125

5.2 Binomial Model 127

5.2.1 Probability-Free Pricing 130

5.2.2 No Arbitrage 130

5.2.3 Risk-Neutrality 131

5.3 From One Time-Step to Two 132

5.3.1 Self-Financing, Dynamic Hedging 134

5.3.2 Iterated Expectation 135

5.4 Relative Prices 137

5.4.1 Risk-Neutral Valuation 138

5.4.2 Fundamental Theorems of Asset Pricing 140

References 140

Exercises 141

Chapter 6 Option Pricing 143

6.1 Random Walk and Brownian Motion 143

6.1.1 Random Walk 143

6.1.2 Brownian Motion 144

6.1.3 Lognormal Distribution, Geometric Brownian Motion 145

6.2 Black-Scholes-Merton Call Formula 145

6.2.1 Put-Call Parity 151

6.2.2 Black's Formula: Options on Forwards 152

6.2.3 Call Is All You Need 153

6.3 Implied Volatility 154

6.3.1 Skews, Smiles 155

6.4 Greeks 156

6.4.1 Greeks Formulas 157

6.4.2 Gamma versus Theta 157

6.4.3 Delta, Gamma versus Time 160

6.5 Diffusions, Ito 161

6.5.1 Black-Scholes-Merton PDE 162

6.5.2 Call Formula and Heat Equation 163

6.6 CRR Binomial Model 165

6.6.1 CRR Greeks 167

6.7 American-Style Options 167

6.7.1 American Call Options 168

6.7.2 Backward Induction 169

6.8 Path-Dependent Options 170

[...]opean Options in Practice 173

References 173

Exercises 174

Python Projects 179

Chapter 7 Interest Rate Derivatives 187

7.1 Term Structure of Interest Rates 187

7.1.1 Zero Curve 187

7.1.2 Forward Rate Curve 188

7.2 Interest Rate Swaps 189

7.2.1 Swap Valuation 190

7.2.2 Swap = Bone 100% 193

7.2.3 Discounting the Forwards 193

7.2.4 Swap Rate as Average Forward Rate 193

7.3 Interest Rate Derivatives 194

7.3.1 Black's Normal Model 194

7.3.2 Caps and Floors 196

[...]opean Swaptions 197

7.3.4 Constant Maturity Swaps 200

7.4 Interest Rate Models 200

7.4.1 Money Market Account, Short Rate 201

7.4.2 Short Rate Models 202

7.4.3 Mean Reversion, Vasicek and Hull-White Models 202

7.4.4 Short Rate Lattice Model 204

7.4.5 Pure Securities 206

7.5 Bermudan Swaptions 208

7.6 Term Structure Models 211

7.7 Interest Rate Derivatives in Practice 212

7.7.1 Interest Rate Risk 212

7.7.2 Value at Risk (VaR) 213

References 213

Exercises 214

Appendix A Math and Probability Review 217

A. 1 Calculus and Differentiation Rules 217

A.2 Probability Review 218

A.3 Linear Regression Analysis 225

Appendix B Useful Excel Functions 229

About the Companion Website 231

Index 233