Amir Sadr: Mathematical Techniques in Finance

1 Cover

2 Title Page Mathematical Techniques in Finance An Introduction AMIR SADR

3 Copyright

4 Dedication

5 Preface BACKGROUND BOOK STRUCTURE

6 Acknowledgments

7 About the Author

8 Acronyms

9 CHAPTER 1: Finance 1.1 FOLLOW THE MONEY 1.2 FINANCIAL MARKETS AND PARTICIPANTS 1.3 QUANTITATIVE FINANCE

10 CHAPTER 2: Rates, Yields, Bond Math 2.1 INTEREST RATES 2.2 ARBITRAGE, LAW OF ONE PRICE 2.3 PRICE‐YIELD FORMULA 2.4 SOLVING FOR YIELD: ROOT SEARCH 2.5 PRICE RISK 2.6 LEVEL PAY LOAN 2.7 YIELD CURVE EXERCISES PYTHON PROJECTS

11 CHAPTER 3: Investment Theory 3.1 UTILITY THEORY 3.2 PORTFOLIO SELECTION 3.3 CAPITAL ASSET PRICING MODEL 3.4 FACTORS 3.5 MEAN‐VARIANCE EFFICIENCY AND UTILITY 3.6 INVESTMENTS IN PRACTICE REFERENCES EXERCISES PYTHON PROJECTS

12 CHAPTER 4: Forwards and Futures 4.1 FORWARDS 4.2 FUTURES CONTRACTS 4.3 STOCK DIVIDENDS 4.4 FORWARD FOREIGN CURRENCY EXCHANGE RATE 4.5 FORWARD INTEREST RATES REFERENCES EXERCISES

13 CHAPTER 5: Risk‐Neutral Valuation 5.1 CONTINGENT CLAIMS 5.2 BINOMIAL MODEL 5.3 FROM ONE TIME‐STEP TO TWO 5.4 RELATIVE PRICES REFERENCES EXERCISES

14 CHAPTER 6: Option Pricing 6.1 RANDOM WALK AND BROWNIAN MOTION 6.2 BLACK‐SCHOLES‐MERTON CALL FORMULA 6.3 IMPLIED VOLATILITY 6.4 GREEKS 6.5 DIFFUSIONS, ITO 6.6 CRR BINOMIAL MODEL 6.7 AMERICAN‐STYLE OPTIONS 6.8 PATH‐DEPENDENT OPTIONS 6.9 EUROPEAN OPTIONS IN PRACTICE REFERENCES EXERCISES PYTHON PROJECTS

15 CHAPTER 7: Interest Rate Derivatives 7.1 TERM STRUCTURE OF INTEREST RATES 7.2 INTEREST RATE SWAPS 7.3 INTEREST RATE DERIVATIVES 7.4 INTEREST RATE MODELS 7.5 BERMUDAN SWAPTIONS 7.6 TERM STRUCTURE MODELS 7.7 INTEREST RATE DERIVATIVES IN PRACTICE REFERENCES EXERCISES

16 APPENDIX A: Math and Probability ReviewA.1 CALCULUS AND DIFFERENTIATION RULES A.2 PROBABILITY REVIEW A.3 LINEAR REGRESSION ANALYSIS

17 APPENDIX B: Useful Excel Functions

18 About the Companion Website

19 Index

20 End User License Agreement

1 Chapter 1 TABLE 1.1 Balance sheet of the United States at 2020 year end. TABLE 1.2 Households sector balance sheet as of 2020 year end. TABLE 1.3 Market size ($Trillions). TABLE 1.4 U.S. bond market ($Trillions). TABLE 1.5 Global derivatives market size ($Trillions). TABLE 1.6 Market participants and financial products.

2 Chapter 2 TABLE 2.1 Future Value of $100,000 for a 2 year ( ) loan with per annum. TABLE 2.2 Interest for principal of $1,000,000 and interest rate per year ...TABLE 2.3 Sensitivity measures for three different bonds.TABLE 2.4 Discount factor curve construction via bootstrap method.TABLE 2.5 Hint for bootstrap problem.

3 Chapter 3TABLE 3.1 St. Petersburg game.TABLE 3.2 MVP portfolio for two risky assets.TABLE 3.3 Regression statistics.

4 Chapter 4TABLE 4.1 3‐month evolution of an asset and its forward price.

5 Chapter 6TABLE 6.1 Properties of normal and lognormal random variables.TABLE 6.2 Six‐month call option prices with different strikes.TABLE 6.3 Six‐month put option prices with different strikes.TABLE 6.4 BSM formulas and Greeks.TABLE 6.5 10 x 1000‐path simulation runs, random.seed(2021).

6 Chapter 7TABLE 7.1 Fixed leg's cash flows of a $100M 1‐year 4% fixed versus floating ...TABLE 7.2 Floating leg's cash flows of a $100M 1‐year 4% fixed versus floati...TABLE 7.3 Discount factor curve, forward 6‐month rate curve, and swap rates ...TABLE 7.4 1‐year forward start 1‐year quarterly cap with strike .TABLE 7.5 Discount factor curve at each node.TABLE 7.6 Value of $1M 6 month into a 1.5‐year Bermudan 4% p.a. semiannual s...

1 Chapter 2 FIGURE 2.1 Bond price versus yield with coupon rate = 2% p.a. FIGURE 2.2 Price of bond versus remaining years to maturity. FIGURE 2.3 Pull to par effect for a 2‐year, 4% semiannual coupon bond. FIGURE 2.4 Cash flows of a 2‐year 4% semiannual coupon bond versus a 2‐year ...FIGURE 2.5 Newton‐Raphson method.FIGURE 2.6 PV01, PVBP, and modified duration of a coupon bond.FIGURE 2.7 Interest and principal payments of a level pay loan.FIGURE 2.8 A pool of loans with low prepayment speed.FIGURE 2.9 A pool of loans with high prepayment speed.FIGURE 2.10 Negative convexity due to increased prepayments when rates are l...FIGURE 2.11 U.S. Treasury yield curve.

2 Chapter 3FIGURE 3.1 Compound lottery.FIGURE 3.2 Risk attitude and utility function.FIGURE 3.3 Risk premium for a risk‐averse investor.FIGURE 3.4 Conic sections.FIGURE 3.5 Feasible region for two risky assets.FIGURE 3.6 Feasible regions for different correlations.FIGURE 3.7 Feasible region for three or more risky assets.FIGURE 3.8 Method of Lagrange multipliers.FIGURE 3.9 Relationship between CML and the feasible region of risky assets....FIGURE 3.10 Proof of the CAPM formula.FIGURE 3.11 PCA identification of eigenvectors.FIGURE 3.12 Utility indifference curves.FIGURE 3.13 5‐year monthly price history of Amazon (AMZN), Walmart (WMT).FIGURE 3.14 Walmart (WMT) versus Amazon (AMZN).FIGURE 3.15 Z‐Score for Amazon‐Walmart pair trade.FIGURE 3.16 10,000 risky portfolios.

3 Chapter 4FIGURE 4.1 Contango versus backwardation.FIGURE 4.2 3‐month evolution of an asset's spot and forward prices.

4 Chapter 5FIGURE 5.1 Economic value of European‐style call and put options at expirati...FIGURE 5.2 One‐step binomial model.FIGURE 5.3 Lack of arbitrage.FIGURE 5.4 Two‐step binomial model.FIGURE 5.5 Two‐period evolution of the replicating portfolio for a call opti...FIGURE 5.6 A symmetric random walk is a martingale.

5 Chapter 6FIGURE 6.1 Standard Brownian motion.FIGURE 6.2 Geometric Brownian motion.FIGURE 6.3 Random walk with drift (top); exponentiated random walk with drif...FIGURE 6.4 Normal and lognormal random variables with same mean and variance...FIGURE 6.5 European‐style option payoffs.FIGURE 6.6 Skew and smile effect for out‐of‐the‐money options.FIGURE 6.7 Call option value and its delta.FIGURE 6.8 Put option value and its delta.FIGURE 6.9 Convexity PnL versus time decay for a delta‐hedged call option.FIGURE 6.10 Delta as a function of time to expiration, .FIGURE 6.11 CRR binomial model.FIGURE 6.12 Backward induction algorithm.FIGURE 6.13 Two antithetic sample paths in a random walk.FIGURE 6.14 Convergence of CRR model to BSM Formula.FIGURE 6.15 Convergence of backward induction model to the American option....

6 Chapter 7FIGURE 7.1 Forward‐rate, zero‐coupon, and discount‐factor curves.FIGURE 7.2 Cash flows of a 1‐year USD fixed versus floating interest rate sw...FIGURE 7.3 Payoff of a into ‐year payer swaption.FIGURE 7.4 Typical implementation of the Hull‐White model.FIGURE 7.5 Navigating the sublattice originating from each node to extract t...FIGURE 7.6 Forward induction and pure security prices.FIGURE 7.7 Two‐year evolution of the 6‐month rate.FIGURE 7.8 Full term structure model.