Mathematical Techniques in Finance: An Introduction

Cover
Title Page
Copyright
Contents
Preface
Acknowledgments
About the Author
Acronyms
CHAPTER 1 Finance
	1.1 Follow the Money
	1.2 Financial Markets and Participants
	1.3 Quantitative Finance
CHAPTER 2 Rates, Yields, Bond Math
	2.1 Interest Rates
		2.1.1 Fractional Periods
		2.1.2 Continuous Compounding
		2.1.3 Discount Factor, PV, FV
		2.1.4 Yield, Internal Rate of Return
	2.2 Arbitrage, Law of One Price
	2.3 Price‐Yield Formula
		2.3.1 Clean Price
		2.3.2 Zero‐Coupon Bond
		2.3.3 Annuity
		2.3.4 Fractional Years, Day Counts
		2.3.5 U.S. Treasury Securities
	2.4 Solving for Yield: Root Search
		2.4.1 Newton‐Raphson Method
		2.4.2 Bisection Method
	2.5 Price Risk
		2.5.1 PV01, PVBP
		2.5.2 Convexity
		2.5.3 Taylor Series Expansion
		2.5.4 Expansion Around C
		2.5.5 Numerical Derivatives
	2.6 Level Pay Loan
		2.6.1 Interest and Principal Payments
		2.6.2 Average Life
		2.6.3 Pool of Loans
		2.6.4 Prepayments
		2.6.5 Negative Convexity
	2.7 Yield Curve
		2.7.1 Bootstrap Method
		2.7.2 Interpolation Method
		2.7.3 Rich/Cheap Analysis
		2.7.4 Yield Curve Trades
	Exercises
	Python Projects
CHAPTER 3 Investment Theory
	3.1 Utility Theory
		3.1.1 Risk Appetite
		3.1.2 Risk versus Uncertainty, Ranking
		3.1.3 Utility Theory Axioms
		3.1.4 Certainty‐Equivalent
		3.1.5 X‐ARRA
	3.2 Portfolio Selection
		3.2.1 Asset Allocation
		3.2.2 Markowitz Mean‐Variance Portfolio Theory
		3.2.3 Risky Assets
		3.2.4 Portfolio Risk
		3.2.5 Minimum Variance Portfolio
		3.2.6 Leverage, Short Sales
		3.2.7 Multiple Risky Assets
		3.2.8 Efficient Frontier
		3.2.9 Minimum Variance Frontier
		3.2.10 Separation: Two‐Fund Theorem
		3.2.11 Risk‐Free Asset
		3.2.12 Capital Market Line
		3.2.13 Market Portfolio
	3.3 Capital Asset Pricing Model
		3.3.1 CAPM Pricing
		3.3.2 Systematic and Diversifiable Risk
	3.4 Factors
		3.4.1 Arbitrage Pricing Theory
		3.4.2 Fama‐French Factors
		3.4.3 Factor Investing
		3.4.4 PCA
	3.5 Mean‐Variance Efficiency and Utility
		3.5.1 Parabolic Utility
		3.5.2 Jointly Normal Returns
	3.6 Investments in Practice
		3.6.1 Rebalancing
		3.6.2 Performance Measures
		3.6.3 Z‐Scores, Mean‐Reversion, Rich‐Cheap
		3.6.4 Pairs Trading
		3.6.5 Risk Management
			3.6.5.1 Gambler's Ruin
			3.6.5.2 Kelly's Ratio
	References
	Exercises
	Python Projects
CHAPTER 4 Forwards and Futures
	4.1 Forwards
		4.1.1 Forward Price
		4.1.2 Cash and Carry
		4.1.3 Interim Cash Flows
		4.1.4 Valuation of Forwards
		4.1.5 Forward Curve
	4.2 Futures Contracts
		4.2.1 Futures versus Forwards
		4.2.2 Zero‐Cost, Leverage
		4.2.3 Mark‐to‐Market Loss
	4.3 Stock Dividends
	4.4 Forward Foreign Currency Exchange Rate
	4.5 Forward Interest Rates
	References
	Exercises
CHAPTER 5 Risk‐Neutral Valuation
	5.1 Contingent Claims
	5.2 Binomial Model
		5.2.1 Probability‐Free Pricing
		5.2.2 No Arbitrage
		5.2.3 Risk‐Neutrality
	5.3 From One Time‐Step to Two
		5.3.1 Self‐Financing, Dynamic Hedging
		5.3.2 Iterated Expectation
	5.4 Relative Prices
		5.4.1 Risk‐Neutral Valuation
		5.4.2 Fundamental Theorems of Asset Pricing
	References
	Exercises
CHAPTER 6 Option Pricing
	6.1 Random Walk and Brownian Motion
		6.1.1 Random Walk
		6.1.2 Brownian Motion
		6.1.3 Lognormal Distribution, Geometric Brownian Motion
	6.2 Black‐Scholes‐Merton Call Formula
		6.2.1 Put‐Call Parity
		6.2.2 Black's Formula: Options on Forwards
		6.2.3 Call Is All You Need
	6.3 Implied Volatility
		6.3.1 Skews, Smiles
	6.4 Greeks
		6.4.1 Greeks Formulas
		6.4.2 Gamma versus Theta
		6.4.3 Delta, Gamma versus Time
	6.5 Diffusions, Ito
		6.5.1 Black‐Scholes‐Merton PDE
		6.5.2 Call Formula and Heat Equation
	6.6 CRR Binomial Model
		6.6.1 CRR Greeks
	6.7 American‐Style Options
		6.7.1 American Call Options
		6.7.2 Backward Induction
	6.8 Path‐Dependent Options
	6.9 European Options in Practice
	References
	Exercises
	Python Projects
CHAPTER 7 Interest Rate Derivatives
	7.1 Term Structure of Interest Rates
		7.1.1 Zero Curve
		7.1.2 Forward Rate Curve
	7.2 Interest Rate Swaps
		7.2.1 Swap Valuation
		7.2.2 Swap = Bone − 100%
		7.2.3 Discounting the Forwards
		7.2.4 Swap Rate as Average Forward Rate
	7.3 Interest Rate Derivatives
		7.3.1 Black's Normal Model
		7.3.2 Caps and Floors
		7.3.3 European Swaptions
		7.3.4 Constant Maturity Swaps
	7.4 Interest Rate Models
		7.4.1 Money Market Account, Short Rate
		7.4.2 Short Rate Models
		7.4.3 Mean Reversion, Vasicek and Hull‐White Models
		7.4.4 Short Rate Lattice Model
		7.4.5 Pure Securities
	7.5 Bermudan Swaptions
	7.6 Term Structure Models
	7.7 Interest Rate Derivatives in Practice
		7.7.1 Interest Rate Risk
		7.7.2 Value at Risk (VaR)
	References
	Exercises
APPENDIX A Math and Probability Review
	A.1 Calculus and Differentiation Rules
		A.1.1 Taylor Series
	A.2 Probability Review
		A.2.1 Density and Distribution Functions
		A.2.2 Expected Values, Moments
		A.2.3 Conditional Probability and Expectation
		A.2.4 Jensen's Inequality
		A.2.5 Normal Distribution
		A.2.6 Central Limit Theorem
	A.3 Linear Regression Analysis
		A.3.1 Regression Distributions
APPENDIX B Useful Excel Functions
About the Companion Website
Index
EULA