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.
Author(s): Amir Sadr
Series: Wiley Finance
Edition: 1
Publisher: Wiley
Year: 2022
Language: English
Pages: 272
Tags: Finance; Investments; Risk Management; Investment Theory; Utility Theory; Mean-Variance Theory; Asset Allocation; Capital Asset Pricing Model; Derivatives; Random Walk; Brownian Motion; Interest Rate Curves
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
