Mathematical Techniques in Finance An Introduction

Explore the foundations of modern finance with this intuitive mathematical guide

In Mathematical Techniques in Finance: An Introduction, distinguished finance professional Amir Sadr delivers an essential and practical guide to the mathematical foundations of various areas of finance, including corporate finance, investments, risk management, and more.

Readers will discover a wealth of accessible information that reveals the underpinnings of business and finance. You’ll learn about:

• Investment theory, including utility theory, mean-variance theory and asset allocation, and the Capital Asset Pricing Model
• Derivatives, including forwards, options, the random walk, and Brownian Motion
• Interest rate curves, including yield curves, interest rate swap curves, and interest rate derivatives

Complete with math reviews, useful Excel functions, and a glossary of financial terms, Mathematical Techniques in Finance: An Introduction is required reading for students and professionals in finance.

Stephen McKeever is an Emeritus Regents Professor in the Department of Physics at Oklahoma State University in the United States. He has published over 200 peer-reviewed papers in the field of luminescence measurements for radiation dosimetry.

Preface xiii

Background xiii

Book Structure xiv

Bonds xiv

Stocks, Investments xv

Forwards, Futures xv

Risk-Neutral Option Pricing xv

Interest Rate Derivatives xvi

Problems and Python Projects xvi

Acknowledgments xix

Acronyms xxiii

1 Finance 1

1.1 Follow the Money 1

1.2 Financial Markets and Participants 2

1.3 Quantitative Finance 4

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 16

2.3.3 Annuity 17

2.3.4 Fractional Years, Day Counts 17

2.3.5 US Treasury Securities 19

2.4 Solving for Yield: Root Search 20

2.4.1 Newton-Raphson Method 21

2.4.2 Bisection Method 22

2.5 Price Risk 22

2.5.1 PV01, PVBP 22

2.5.2 Convexity 23

2.5.3 Taylor Series Expansion 24

2.5.4 Expansion Around C 26

2.5.5 Numerical Derivatives 27

2.6 Level Pay Loan 27

2.6.1 Interest and Principal Payments 29

2.6.2 Average Life 30

2.6.3 Pool of Loans 30

2.6.4 Prepayments 31

2.6.5 Negative Convexity 33

2.7 Yield Curve 35

2.7.1 Bootstrap Method 36

2.7.2 Interpolation Method 36

2.7.3 Rich/Cheap Analysis 38

2.7.4 Yield Curve Trades 38

Problems 39

Python Projects 46

3 Investment Theory 53

3.1 Utility Theory 54

3.1.1 Risk Appetite 54

3.1.2 Risk versus Uncertainty, Ranking 56

3.1.3 Utility Theory Axioms 58

3.1.4 Certainty-Equivalent 58

3.1.5 X-ARRA 60

3.2 Portfolio Selection 62

3.2.1 Asset Allocation 62

3.2.2 Markowitz Mean-Variance Theory 63

3.2.3 Risky Assets 64

3.2.4 Portfolio Risk 64

3.2.5 Minimum Variance Portfolio 65

3.2.6 Leverage, Short Sales 67

3.2.7 Multiple Risky Assets 69

3.2.8 Efficient Frontier 73

3.2.9 Minimum Variance Frontier 73

3.2.10 Separation: Two Fund Theorem 75

3.2.11 Risk-Free Asset 76

3.2.12 Capital Market Line 76

3.2.13 Market Portfolio 77

3.3 Capital Asset Pricing Model 78

3.3.1 CAPM Pricing 81

3.3.2 Systematic and Diversifiable Risk 81

3.4 Factors 82

3.4.1 Arbitrage Pricing Theory 82

3.4.2 Fama-French Factors 84

3.4.3 Factor Investing 85

3.4.4 PCA 85

3.5 Mean-Variance Efficiency and Utility 87

3.5.1 Parabolic Utility 89

3.5.2 Jointly Normal Returns 89

3.6 Investments in Practice 90

3.6.1 Re-balancing 91

3.6.2 Performance Measures 91

3.6.3 Z-Scores, Mean-Reversion, Rich-Cheap 92

3.6.4 Pairs Trading 92

3.6.5 Risk Management 94

Bibliography 96

Problems 98

Python Projects 103

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 113

4.2.1 Futures versus Forwards 115

4.2.2 Zero-Cost, Leverage 116

4.2.3 Mark-To-Market Loss 117

4.3 Stock Dividends 117

4.4 Forward Foreign Currency Exchange Rate 118

4.5 Forward Interest Rates 119

Bibliography 120

Problems 120

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 130

5.3 From One time-step to Two 132

5.3.1 Self-Financing, Dynamic Hedging 133

5.3.2 Iterated Expectation 136

5.4 Relative Prices 137

5.4.1 Risk-Neutral Valuation 138

5.4.2 Fundamental Theorems of Asset Pricing 140

Bibliography 141

Problems 141

6 Option Pricing 145

6.1 Random Walk and Brownian Motion 145

6.1.1 Random Walk 145

6.1.2 Brownian Motion 146

6.1.3 Log-normal Distribution, Geometric Brownian Motion 147

6.2 Black-Scholes-Merton Call Formula 149

6.2.1 Put-Call Parity 153

6.2.2 Black’s Formula: Options on Forwards 154

6.2.3 Call Is All You Need 154

6.3 Implied Volatility 156

6.3.1 Skews, Smiles 156

6.4 Greeks 157

6.4.1 Greeks Formulae 158

6.4.2 Gamma versus Theta 158

6.4.3 Delta, Gamma versus Time 161

6.5 Diffusions, Ito 162

6.5.1 Black-Scholes-Merton PDE 163

6.5.2 Call Formula and Heat Equation 165

6.6 CRR Binomial Model 166

6.6.1 CRR Greeks 168

6.7 American Style Options 169

6.7.1 American Call Options 169

6.7.2 Backward Induction 170

6.8 Path-Dependent Options 171

6.9 European Options in Practice 174

Bibliography 175

Problems 175

Python Projects 181

7 Interest Rate Derivatives 189

7.1 Term Structure of Interest Rates 189

7.1.1 Zero Curve 189

7.1.2 Forward Rate Curve 190

7.2 Interest Rate Swaps 190

7.2.1 Swap Valuation 193

7.2.2 Swap=Bond-100% 195

7.2.3 Discounting the Forwards 195

7.2.4 Swap Rate as Average Forward Rate 195

7.3 Interest Rate Derivatives 196

7.3.1 Black’s Normal Model 196

7.3.2 Caps and Floors 198

7.3.3 European Swaptions 199

7.3.4 Constant Maturity Swaps 201

7.4 Interest Rate Models 202

7.4.1 Money Market Account, Short Rate 202

7.4.2 Short Rate Models 203

7.4.3 Mean-Reversion, Vasicek and Hull-White Models 204

7.4.4 Short Rate Lattice Model 205

7.4.5 Pure Securities 208

7.5 Bermudan Swaptions 211

7.6 Term Structure Models 212

7.7 Interest Rate Derivatives in Practice 213

7.7.1 Interest Rate Risk 213

7.7.2 Value at Risk (VaR) 214

Bibliography 214

Problems 215

A Math and Probability Review 219

A.1 Calculus and Differentiation Rules 219

A.1.1 Taylor Series 220

A.2 Probability Review 220

A.2.1 Density and Distribution Functions 221

A.2.2 Expected Values, Moments 222

A.2.3 Conditional Probability and Expectation 223

A.2.4 Jensen’s Inequality 225

A.2.5 Normal Distribution 225

A.2.6 Central Limit Theorem 226

A.3 Linear Regression Analysis 226

A.3.1 Regression Distributions 227

B Useful Excel Functions 231

Index 232

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