Mathematical Techniques in Finance

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 xiiiAcknowledgments xixAbout the Author xxiAcronyms xxiiiChapter 1 Finance 11.1 Follow the Money 11.2 Financial Markets and Participants 31.3 Quantitative Finance 5Chapter 2 Rates, Yields, Bond Math 72.1 Interest Rates 72.1.1 Fractional Periods 82.1.2 Continuous Compounding 92.1.3 Discount Factor, PV, FV 92.1.4 Yield, Internal Rate of Return 102.2 Arbitrage, Law of One Price 112.3 Price-Yield Formula 122.3.1 Clean Price 152.3.2 Zero-Coupon Bond 172.3.3 Annuity 172.3.4 Fractional Years, Day Counts 192.3.5 U.S. Treasury Securities 202.4 Solving for Yield: Root Search 212.4.1 Newton-Raphson Method 212.4.2 Bisection Method 212.5 Price Risk 222.5.1 Pv01, Pvbp 232.5.2 Convexity 242.5.3 Taylor Series Expansion 252.5.4 Expansion Around c 272.5.5 Numerical Derivatives 272.6 Level Pay Loan 282.6.1 Interest and Principal Payments 302.6.2 Average Life 312.6.3 Pool of Loans 322.6.4 Prepayments 322.6.5 Negative Convexity 342.7 Yield Curve 362.7.1 Bootstrap Method 372.7.2 Interpolation Method 372.7.3 Rich/Cheap Analysis 392.7.4 Yield Curve Trades 40Exercises 41Python Projects 48Chapter 3 Investment Theory 553.1 Utility Theory 563.1.1 Risk Appetite 573.1.2 Risk versus Uncertainty, Ranking 583.1.3 Utility Theory Axioms 593.1.4 Certainty-Equivalent 613.1.5 X-arra 633.2 Portfolio Selection 643.2.1 Asset Allocation 653.2.2 Markowitz Mean-Variance Portfolio Theory 653.2.3 Risky Assets 663.2.4 Portfolio Risk 663.2.5 Minimum Variance Portfolio 683.2.6 Leverage, Short Sales 713.2.7 Multiple Risky Assets 713.2.8 Efficient Frontier 753.2.9 Minimum Variance Frontier 763.2.10 Separation: Two-Fund Theorem 783.2.11 Risk-Free Asset 793.2.12 Capital Market Line 793.2.13 Market Portfolio 803.3 Capital Asset Pricing Model 813.3.1 CAPM Pricing 843.3.2 Systematic and Diversifiable Risk 843.4 Factors 853.4.1 Arbitrage Pricing Theory 853.4.2 Fama-French Factors 873.4.3 Factor Investing 883.4.4 Pca 883.5 Mean-Variance Efficiency and Utility 903.5.1 Parabolic Utility 913.5.2 Jointly Normal Returns 923.6 Investments in Practice 933.6.1 Rebalancing 933.6.2 Performance Measures 943.6.3 Z-Scores, Mean-Reversion, Rich-Cheap 953.6.4 Pairs Trading 953.6.5 Risk Management 973.6.5.1 Gambler’s Ruin 973.6.5.2 Kelly’s Ratio 98References 99Exercises 100Python Projects 106Chapter 4 Forwards and Futures 1094.1 Forwards 1094.1.1 Forward Price 1104.1.2 Cash and Carry 1114.1.3 Interim Cash Flows 1114.1.4 Valuation of Forwards 1114.1.5 Forward Curve 1124.2 Futures Contracts 1144.2.1 Futures versus Forwards 1154.2.2 Zero-Cost, Leverage 1164.2.3 Mark-to-Market Loss 1164.3 Stock Dividends 1174.4 Forward Foreign Currency Exchange Rate 1174.5 Forward Interest Rates 119References 120Exercises 120Chapter 5 Risk-Neutral Valuation 1255.1 Contingent Claims 1255.2 Binomial Model 1275.2.1 Probability-Free Pricing 1305.2.2 No Arbitrage 1305.2.3 Risk-Neutrality 1315.3 From One Time-Step to Two 1325.3.1 Self-Financing, Dynamic Hedging 1345.3.2 Iterated Expectation 1355.4 Relative Prices 1375.4.1 Risk-Neutral Valuation 1385.4.2 Fundamental Theorems of Asset Pricing 140References 140Exercises 141Chapter 6 Option Pricing 1436.1 Random Walk and Brownian Motion 1436.1.1 Random Walk 1436.1.2 Brownian Motion 1446.1.3 Lognormal Distribution, Geometric Brownian Motion 1456.2 Black-Scholes-Merton Call Formula 1456.2.1 Put-Call Parity 1516.2.2 Black’s Formula: Options on Forwards 1526.2.3 Call Is All You Need 1536.3 Implied Volatility 1546.3.1 Skews, Smiles 1556.4 Greeks 1566.4.1 Greeks Formulas 1576.4.2 Gamma versus Theta 1576.4.3 Delta, Gamma versus Time 1606.5 Diffusions, Ito 1616.5.1 Black-Scholes-Merton PDE 1626.5.2 Call Formula and Heat Equation 1636.6 CRR Binomial Model 1656.6.1 CRR Greeks 1676.7 American-Style Options 1676.7.1 American Call Options 1686.7.2 Backward Induction 1696.8 Path-Dependent Options 1706.9 European Options in Practice 173References 173Exercises 174Python Projects 179Chapter 7 Interest Rate Derivatives 1877.1 Term Structure of Interest Rates 1877.1.1 Zero Curve 1877.1.2 Forward Rate Curve 1887.2 Interest Rate Swaps 1897.2.1 Swap Valuation 1907.2.2 Swap = Bone − 100% 1937.2.3 Discounting the Forwards 1937.2.4 Swap Rate as Average Forward Rate 1937.3 Interest Rate Derivatives 1947.3.1 Black’s Normal Model 1947.3.2 Caps and Floors 1967.3.3 European Swaptions 1977.3.4 Constant Maturity Swaps 2007.4 Interest Rate Models 2007.4.1 Money Market Account, Short Rate 2017.4.2 Short Rate Models 2027.4.3 Mean Reversion, Vasicek and Hull-White Models 2027.4.4 Short Rate Lattice Model 2047.4.5 Pure Securities 2067.5 Bermudan Swaptions 2087.6 Term Structure Models 2117.7 Interest Rate Derivatives in Practice 2127.7.1 Interest Rate Risk 2127.7.2 Value at Risk (VaR) 213References 213Exercises 214Appendix A Math and Probability Review 217A. 1 Calculus and Differentiation Rules 217A.2 Probability Review 218A.3 Linear Regression Analysis 225Appendix B Useful Excel Functions 229About the Companion Website 231Index 233