The Modern Gentleman

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. Cover 1 Title Page 5 Copyright 6 Contents 9 Preface 15 Acknowledgments 21 About the Author 23 Acronyms 25 CHAPTER 1 Finance 31 1.1 Follow the Money 31 1.2 Financial Markets and Participants 33 1.3 Quantitative Finance 35 CHAPTER 2 Rates, Yields, Bond Math 37 2.1 Interest Rates 37 2.1.1 Fractional Periods 38 2.1.2 Continuous Compounding 39 2.1.3 Discount Factor, PV, FV 39 2.1.4 Yield, Internal Rate of Return 40 2.2 Arbitrage, Law of One Price 41 2.3 Price‐Yield Formula 42 2.3.1 Clean Price 45 2.3.2 Zero‐Coupon Bond 47 2.3.3 Annuity 47 2.3.4 Fractional Years, Day Counts 49 2.3.5 U.S. Treasury Securities 50 2.4 Solving for Yield: Root Search 51 2.4.1 Newton‐Raphson Method 51 2.4.2 Bisection Method 51 2.5 Price Risk 52 2.5.1 PV01, PVBP 53 2.5.2 Convexity 54 2.5.3 Taylor Series Expansion 55 2.5.4 Expansion Around C 57 2.5.5 Numerical Derivatives 57 2.6 Level Pay Loan 58 2.6.1 Interest and Principal Payments 60 2.6.2 Average Life 61 2.6.3 Pool of Loans 62 2.6.4 Prepayments 62 2.6.5 Negative Convexity 64 2.7 Yield Curve 66 2.7.1 Bootstrap Method 67 2.7.2 Interpolation Method 67 2.7.3 Rich/Cheap Analysis 69 2.7.4 Yield Curve Trades 70 Exercises 71 Python Projects 78 CHAPTER 3 Investment Theory 85 3.1 Utility Theory 86 3.1.1 Risk Appetite 87 3.1.2 Risk versus Uncertainty, Ranking 88 3.1.3 Utility Theory Axioms 89 3.1.4 Certainty‐Equivalent 91 3.1.5 X‐ARRA 93 3.2 Portfolio Selection 94 3.2.1 Asset Allocation 95 3.2.2 Markowitz Mean‐Variance Portfolio Theory 95 3.2.3 Risky Assets 96 3.2.4 Portfolio Risk 96 3.2.5 Minimum Variance Portfolio 98 3.2.6 Leverage, Short Sales 101 3.2.7 Multiple Risky Assets 101 3.2.8 Efficient Frontier 105 3.2.9 Minimum Variance Frontier 106 3.2.10 Separation: Two‐Fund Theorem 108 3.2.11 Risk‐Free Asset 109 3.2.12 Capital Market Line 109 3.2.13 Market Portfolio 110 3.3 Capital Asset Pricing Model 111 3.3.1 CAPM Pricing 114 3.3.2 Systematic and Diversifiable Risk 114 3.4 Factors 115 3.4.1 Arbitrage Pricing Theory 115 3.4.2 Fama‐French Factors 117 3.4.3 Factor Investing 118 3.4.4 PCA 118 3.5 Mean‐Variance Efficiency and Utility 120 3.5.1 Parabolic Utility 121 3.5.2 Jointly Normal Returns 122 3.6 Investments in Practice 123 3.6.1 Rebalancing 123 3.6.2 Performance Measures 124 3.6.3 Z‐Scores, Mean‐Reversion, Rich‐Cheap 125 3.6.4 Pairs Trading 125 3.6.5 Risk Management 127 3.6.5.1 Gambler's Ruin 127 3.6.5.2 Kelly's Ratio 128 References 129 Exercises 130 Python Projects 136 CHAPTER 4 Forwards and Futures 139 4.1 Forwards 139 4.1.1 Forward Price 140 4.1.2 Cash and Carry 141 4.1.3 Interim Cash Flows 141 4.1.4 Valuation of Forwards 141 4.1.5 Forward Curve 142 4.2 Futures Contracts 144 4.2.1 Futures versus Forwards 145 4.2.2 Zero‐Cost, Leverage 146 4.2.3 Mark‐to‐Market Loss 146 4.3 Stock Dividends 147 4.4 Forward Foreign Currency Exchange Rate 147 4.5 Forward Interest Rates 149 References 150 Exercises 150 CHAPTER 5 Risk‐Neutral Valuation 155 5.1 Contingent Claims 155 5.2 Binomial Model 157 5.2.1 Probability‐Free Pricing 160 5.2.2 No Arbitrage 160 5.2.3 Risk‐Neutrality 161 5.3 From One Time‐Step to Two 162 5.3.1 Self‐Financing, Dynamic Hedging 164 5.3.2 Iterated Expectation 165 5.4 Relative Prices 167 5.4.1 Risk‐Neutral Valuation 168 5.4.2 Fundamental Theorems of Asset Pricing 170 References 170 Exercises 171 CHAPTER 6 Option Pricing 173 6.1 Random Walk and Brownian Motion 173 6.1.1 Random Walk 173 6.1.2 Brownian Motion 174 6.1.3 Lognormal Distribution, Geometric Brownian Motion 175 6.2 Black‐Scholes‐Merton Call Formula 175 6.2.1 Put‐Call Parity 181 6.2.2 Black's Formula: Options on Forwards 182 6.2.3 Call Is All You Need 183 6.3 Implied Volatility 184 6.3.1 Skews, Smiles 185 6.4 Greeks 186 6.4.1 Greeks Formulas 187 6.4.2 Gamma versus Theta 187 6.4.3 Delta, Gamma versus Time 190 6.5 Diffusions, Ito 191 6.5.1 Black‐Scholes‐Merton PDE 192 6.5.2 Call Formula and Heat Equation 193 6.6 CRR Binomial Model 195 6.6.1 CRR Greeks 197 6.7 American‐Style Options 197 6.7.1 American Call Options 198 6.7.2 Backward Induction 199 6.8 Path‐Dependent Options 200 6.9 European Options in Practice 203 References 203 Exercises 204 Python Projects 209 CHAPTER 7 Interest Rate Derivatives 217 7.1 Term Structure of Interest Rates 217 7.1.1 Zero Curve 217 7.1.2 Forward Rate Curve 218 7.2 Interest Rate Swaps 219 7.2.1 Swap Valuation 220 7.2.2 Swap = Bone − 100% 223 7.2.3 Discounting the Forwards 223 7.2.4 Swap Rate as Average Forward Rate 223 7.3 Interest Rate Derivatives 224 7.3.1 Black's Normal Model 224 7.3.2 Caps and Floors 226 7.3.3 European Swaptions 227 7.3.4 Constant Maturity Swaps 230 7.4 Interest Rate Models 230 7.4.1 Money Market Account, Short Rate 231 7.4.2 Short Rate Models 232 7.4.3 Mean Reversion, Vasicek and Hull‐White Models 232 7.4.4 Short Rate Lattice Model 234 7.4.5 Pure Securities 236 7.5 Bermudan Swaptions 238 7.6 Term Structure Models 241 7.7 Interest Rate Derivatives in Practice 242 7.7.1 Interest Rate Risk 242 7.7.2 Value at Risk (VaR) 243 References 243 Exercises 244 APPENDIX A Math and Probability Review 247 A.1 Calculus and Differentiation Rules 247 A.1.1 Taylor Series 248 A.2 Probability Review 248 A.2.1 Density and Distribution Functions 249 A.2.2 Expected Values, Moments 250 A.2.3 Conditional Probability and Expectation 252 A.2.4 Jensen's Inequality 253 A.2.5 Normal Distribution 253 A.2.6 Central Limit Theorem 254 A.3 Linear Regression Analysis 255 A.3.1 Regression Distributions 256 APPENDIX B Useful Excel Functions 259 About the Companion Website 261 Index 263 EULA 267 "Finance as a distinct field from economics is generally defined as the science or study of the management of funds. The creation of credit, savings, investments, banking institutions, financial markets and products, risk management all fall under the purview of finance. The unifying themes in finance are time, risk and money. Mathematical or quantitative finance is the application of mathematics to these core areas. While simple arithmetic was enough for accounting and keeping ledgers and double-entry bookkeeping, Louis Bachelier's doctoral thesis, Theorie de la speculation published in 1900 used Brownian motion to study stock prices, and is widely recognized as the beginning of quantitative finance. Since then, the use of increasingly sophisticated and specialized mathematics has created the modern field of quantitative finance encompassing investment theory, asset pricing, derivatives, financial data science and the emerging area of crypto assets and Decentralized Finance (DeFi)"-- Provided by publisher