Mathematical Finance

Mathematical finance is a field of applied mathematics concerned with the modeling of financial markets. It employs sophisticated tools from probability theory, stochastic calculus, partial differential equations, and statistics to address complex problems in pricing, hedging, and risk management. Core applications include the valuation of derivative securities such as options and futures, the construction of optimal investment portfolios, and the quantification of market risk to inform financial decision-making.

1. Foundations of Mathematical Finance
   1. Basic Financial Instruments and Markets
      1. Asset Classes
         1. Equities
            1. Common Stocks
            2. Preferred Stocks
            3. Stock Indices
            4. Equity Market Structure
            5. Trading Mechanisms
         2. Fixed Income Securities
            1. Government Bonds
            2. Corporate Bonds
            3. Municipal Bonds
            4. Treasury Bills
            5. Commercial Paper
            6. Certificates of Deposit
            7. Mortgage-Backed Securities
            8. Asset-Backed Securities
            9. Bond Pricing Fundamentals
            10. Credit Ratings
            11. Duration and Convexity
         3. Currencies
            1. Foreign Exchange Markets
            2. Currency Pairs and Quotes
            3. Exchange Rate Mechanisms
            4. Currency Risk
         4. Commodities
            1. Physical Commodities
            2. Commodity Futures
            3. Commodity Indices
            4. Storage Costs and Convenience Yield
      2. Financial Derivatives
         1. Forward Contracts
            1. Definition and Payoff Structure
            2. Settlement and Delivery
            3. Forward Price Determination
            4. Uses and Risks
            5. Marking to Market
         2. Futures Contracts
            1. Exchange-Traded vs OTC
            2. Margin Requirements
            3. Initial Margin
            4. Maintenance Margin
            5. Mark-to-Market Process
            6. Payoff Diagrams
            7. Basis and Basis Risk
         3. Options
            1. Call Options
               1. Payoff Structure
               2. Profit Diagrams
               3. Moneyness Concepts
            2. Put Options
               1. Payoff Structure
               2. Profit Diagrams
               3. Moneyness Concepts
            3. Option Styles
               1. European Options
               2. American Options
               3. Bermudan Options
               4. Asian Options
            4. Option Strategies
               1. Protective Put
               2. Covered Call
               3. Straddles
               4. Strangles
               5. Spreads
         4. Swaps
            1. Interest Rate Swaps
            2. Currency Swaps
            3. Equity Swaps
            4. Credit Default Swaps
            5. Swap Valuation Principles
      3. Market Participants and Structure
         1. Individual Investors
         2. Institutional Investors
            1. Pension Funds
            2. Insurance Companies
            3. Mutual Funds
            4. Hedge Funds
         3. Market Makers
         4. Brokers and Dealers
         5. Investment Banks
         6. Commercial Banks
         7. Exchanges and Trading Venues
         8. Over-the-Counter Markets
         9. Clearinghouses and Settlement Systems
         10. Regulatory Framework
   2. Time Value of Money
      1. Interest Rates and Compounding
         1. Simple Interest
            1. Formula and Calculation
         2. Compound Interest
            1. Annual Compounding
            2. Semi-Annual Compounding
            3. Quarterly Compounding
            4. Monthly Compounding
            5. Daily Compounding
         3. Effective Annual Rate
            1. Calculation
            2. Comparison Across Frequencies
         4. Continuous Compounding
            1. Mathematical Foundation
            2. Exponential Growth Formula
            3. Applications in Finance
      2. Present and Future Value
         1. Present Value Calculations
            1. Single Cash Flow
            2. Multiple Cash Flows
         2. Future Value Calculations
            1. Single Cash Flow
            2. Multiple Cash Flows
         3. Discount Factors
            1. Definition and Calculation
         4. Net Present Value
            1. Calculation
            2. Decision Rules
      3. Annuities and Perpetuities
         1. Ordinary Annuities
            1. Present Value Formula
            2. Future Value Formula
         2. Annuities Due
            1. Present Value Formula
            2. Future Value Formula
         3. Growing Annuities
            1. Constant Growth Rate
            2. Variable Growth Rate
         4. Perpetuities
            1. Present Value Formula
            2. Growing Perpetuities
      4. Yield Curves and Term Structure
         1. Spot Rates
            1. Calculation from Bond Prices
         2. Forward Rates
            1. Calculation from Spot Rates
            2. Forward Rate Agreements
         3. Yield to Maturity
            1. Calculation Methods
         4. Term Structure Theories
            1. Pure Expectations Theory
            2. Liquidity Preference Theory
            3. Market Segmentation Theory
   3. Arbitrage and Market Efficiency
      1. Law of One Price
         1. Theoretical Foundation
         2. Practical Implications
      2. Arbitrage Opportunities
         1. Pure Arbitrage
         2. Statistical Arbitrage
         3. Risk Arbitrage
         4. Detection Methods
         5. Exploitation Strategies
      3. No-Arbitrage Principle
         1. Theoretical Foundation
         2. Pricing Implications
         3. Replication Strategies
         4. Hedging Applications
      4. Market Completeness
         1. Complete Markets
         2. Incomplete Markets
         3. Spanning and Linear Independence
         4. Implications for Pricing
         5. Implications for Hedging