Here are 100 chapter titles for a book on Financial Mathematics, progressing from beginner to advanced:
I. Foundations (1-20)
- Introduction to Financial Mathematics
- Time Value of Money: Present and Future Value
- Simple Interest: Calculations and Applications
- Compound Interest: The Power of Growth
- Effective Interest Rates: Comparing Investments
- Discounting: Bringing Future Value to Present
- Annuities: Regular Payments
- Ordinary Annuities: Payments at the End of the Period
- Annuities Due: Payments at the Beginning of the Period
- Perpetuities: Payments that Last Forever
- Growing Annuities and Perpetuities
- Loan Amortization: Paying Back a Debt
- Mortgage Calculations: A Practical Application
- Bond Valuation: Pricing Fixed Income Securities
- Yield to Maturity: Measuring Bond Returns
- Introduction to Financial Markets
- Stocks and Bonds: Basic Concepts
- Risk and Return: A Fundamental Trade-off
- Portfolio Diversification: Reducing Risk
- Review and Preview: Looking Ahead
II. Intermediate Techniques (21-40)
- The Term Structure of Interest Rates
- Spot Rates and Forward Rates
- Yield Curves: Interpreting Market Expectations
- Duration: Measuring Interest Rate Risk
- Convexity: Refining Duration Estimates
- Immunization: Protecting Portfolios from Interest Rate Changes
- Introduction to Derivatives: Options and Futures
- Options: Call and Put Options
- Option Pricing: Basic Principles
- Futures Contracts: Hedging and Speculation
- Forward Contracts: Locking in Prices
- Swaps: Exchanging Cash Flows
- Interest Rate Swaps: Managing Interest Rate Risk
- Currency Swaps: Managing Exchange Rate Risk
- Pricing Derivatives: An Introduction
- The Binomial Option Pricing Model: A Simple Approach
- The Black-Scholes Model: A Cornerstone of Option Pricing
- Volatility: A Key Input to Option Pricing
- Greeks: Measuring Option Sensitivities
- Review and Practice: Intermediate Techniques
III. Advanced Topics (41-60)
- Stochastic Calculus: The Mathematics of Random Processes
- Brownian Motion: A Foundation for Financial Models
- Ito's Lemma: A Powerful Tool for Derivative Pricing
- The Black-Scholes Model: Derivation and Extensions
- Exotic Options: Beyond Vanilla Calls and Puts
- Asian Options: Averaging Prices
- Barrier Options: Path-Dependent Payoffs
- Lookback Options: Finding the Best Price
- Monte Carlo Simulation: Pricing Complex Derivatives
- Numerical Methods: Approximating Solutions
- Finite Difference Methods: Solving Partial Differential Equations
- Trees and Lattices: Discrete-Time Models
- Credit Risk: Measuring the Risk of Default
- Credit Derivatives: Managing Credit Risk
- Credit Default Swaps: Transferring Credit Risk
- Collateralized Debt Obligations (CDOs): Structured Finance Products
- Securitization: Packaging and Selling Assets
- Mortgage-Backed Securities (MBS): Investing in Mortgages
- Asset-Backed Securities (ABS): Diversifying Investments
- Review and Practice: Advanced Topics
IV. Special Topics and Applications (61-80)
- Financial Time Series Analysis
- Statistical Properties of Financial Data
- Volatility Modeling: ARCH and GARCH Models
- Risk Management: Measuring and Managing Risk
- Value at Risk (VaR): Quantifying Potential Losses
- Expected Shortfall (ES): A More Robust Risk Measure
- Capital Budgeting: Evaluating Investment Projects
- Net Present Value (NPV) and Internal Rate of Return (IRR)
- Real Options: Valuing Flexibility
- Mergers and Acquisitions: Financial Considerations
- Corporate Finance: Raising Capital and Investing
- Portfolio Management: Constructing and Managing Portfolios
- Asset Allocation: Diversifying Across Asset Classes
- Performance Measurement: Evaluating Investment Returns
- Behavioral Finance: The Psychology of Investing
- Market Efficiency: The EMH Hypothesis
- Algorithmic Trading: Automated Trading Strategies
- High-Frequency Trading: Ultra-Fast Trading
- Financial Regulation: Oversight of Financial Markets
- Advanced Applications: A Survey
V. Deeper Dive and Extensions (81-100)
- Stochastic Differential Equations: Advanced Modeling
- Martingales: A Key Concept in Financial Modeling
- Filtering Theory: Estimating Hidden Variables
- Kalman Filters: Applications in Finance
- Jump Processes: Modeling Sudden Changes in Prices
- Lévy Processes: Generalizing Brownian Motion
- Copulas: Modeling Dependence Between Variables
- Credit Risk Modeling: Structural and Reduced-Form Models
- Model Risk: The Risk of Using Incorrect Models
- Numerical Methods: Advanced Techniques
- Finite Element Methods: Solving Complex Problems
- Optimization Techniques: Portfolio Optimization
- Dynamic Programming: Solving Sequential Decision Problems
- Game Theory: Applications in Finance
- Information Asymmetry: The Role of Information in Markets
- Behavioral Finance: Advanced Topics
- Market Microstructure: The Mechanics of Trading
- Financial Engineering: Designing New Financial Products
- History of Financial Mathematics: A Detailed Account
- Open Problems and Future Directions in Financial Mathematics