Mathematics of Finance

• Mathematics of finance, also known as quantitative finance, applies mathematical methods to financial problems.

Core Concepts

• Time value of money is a foundational concept, recognizing that money available today is worth more than the same amount in the future because of its potential earning capacity.
• Interest rates are a key factor in finance, representing the cost of borrowing money or the return on an investment.
• Simple interest is calculated only on the principal amount.
• Compound interest is calculated on the principal amount and the accumulated interest from previous periods.
• Present value is the current worth of a future sum of money or stream of cash flows, given a specified rate of return.
• Future value is the value of an asset or investment at a specified date in the future, based on an assumed rate of growth.
• Annuities are a series of equal payments made at regular intervals.
• Ordinary annuity payments are made at the end of each period.
• Annuity due payments are made at the beginning of each period.
• Perpetuities are annuities that continue indefinitely.
• Amortization is the process of paying off a debt over time by making regular payments.

Financial Instruments

• Stocks (equities) represent ownership in a corporation.
• Bonds (fixed income) are debt instruments issued by corporations or governments.
• Derivatives are contracts whose value is derived from the performance of an underlying asset, index, or interest rate.
• Options give the buyer the right, but not the obligation, to buy (call option) or sell (put option) an asset at a specified price on or before a specified date.
• Futures are contracts obligating the buyer to purchase an asset, or the seller to sell an asset at a predetermined future date and price.
• Swaps are contracts in which two parties exchange cash flows or liabilities from two different financial instruments.
• Mortgages are loans secured by real property.
• Mutual funds are investment vehicles that pool money from multiple investors to purchase a diversified portfolio of assets.
• Exchange-Traded Funds (ETFs) are similar to mutual funds but are traded on stock exchanges like individual stocks.

Portfolio Theory

• Portfolio diversification allocates investments among various financial instruments to reduce risk.
• Modern Portfolio Theory (MPT) is a framework for constructing portfolios that maximize expected return for a given level of risk.
• Efficient frontier represents the set of portfolios that offer the highest expected return for each level of risk.
• Capital Asset Pricing Model (CAPM) describes the relationship between systematic risk and expected return for assets, particularly stocks.
• Beta measures an asset's volatility relative to the overall market.
• Sharpe ratio measures risk-adjusted return, calculated as the excess return per unit of total risk.

Risk Management

• Risk is the uncertainty associated with future outcomes.
• Market risk is the risk of losses due to factors that affect the overall performance of the financial markets.
• Credit risk is the risk that a borrower will default on their debt obligations.
• Liquidity risk is the risk that an asset cannot be bought or sold quickly enough to prevent or minimize a loss.
• Operational risk is the risk of losses resulting from inadequate or failed internal processes, people, and systems, or from external events.
• Value at Risk (VaR) is a statistical measure of the potential loss in value of an asset, or portfolio, over a defined period for a given confidence interval.
• Stress testing involves simulating extreme market conditions to assess the potential impact on a portfolio or financial institution.

Option Pricing Theory

• Black-Scholes Model is a mathematical model used to calculate the theoretical price of European-style options.
• The Black-Scholes model relies on factors such as the current stock price, strike price, time to expiration, risk-free interest rate, and volatility.
• Volatility is a measure of the degree of variation of a trading price series over time, usually measured by standard deviation.
• Implied volatility is the volatility implied by the market price of an option, reflecting the market's expectation of future price fluctuations.
• Option Greeks are measures of the sensitivity of an option's price to changes in underlying parameters.
• Delta measures the change in an option's price for a one-unit change in the underlying asset's price.
• Gamma measures the rate of change of delta with respect to changes in the underlying asset's price.
• Vega measures the change in an option's price for a one-unit change in volatility.
• Theta measures the change in an option's price with respect to time.
• Rho measures the change in an option's price with respect to changes in the risk-free interest rate.

Interest Rate Models

• Term structure of interest rates describes the relationship between interest rates and maturities of debt securities.
• Yield curve is a graphical representation of the term structure of interest rates.
• Spot rate is the yield on a zero-coupon bond.
• Forward rate is the interest rate agreed upon today for a loan to be made in the future.
• Vasicek Model is a mathematical model used to describe the evolution of interest rates.
• Cox-Ingersoll-Ross (CIR) Model is another model for describing the evolution of interest rates, which ensures that interest rates remain positive.
• Heath-Jarrow-Morton (HJM) Model is a framework for modeling the entire yield curve.

Stochastic Calculus

• Stochastic processes are mathematical models that describe the evolution of random variables over time.
• Brownian motion is a continuous-time stochastic process used to model random phenomena, such as stock prices.
• Ito's Lemma is a fundamental result in stochastic calculus that allows one to calculate the differential of a function of a stochastic process.
• Stochastic differential equations (SDEs) are differential equations in which one or more of the terms is a stochastic process, used to model various financial phenomena.
• Monte Carlo methods are computational algorithms that rely on repeated random sampling to obtain numerical results, often used to simulate complex financial models.

Credit Risk Models

• Structural models of credit risk relate the probability of default to the asset value of a company.
• Merton Model is a structural model where default occurs when a firm's asset value falls below its debt obligations.
• Reduced-form models of credit risk model the probability of default directly, without explicitly modeling the firm's assets.
• Credit Default Swaps (CDS) are contracts that provide insurance against the risk of default by a particular company or sovereign entity.

Time Series Analysis

• Time series analysis involves analyzing data points indexed in time order.
• Autocorrelation measures the correlation between a time series and its lagged values.
• Moving averages are used to smooth out short-term fluctuations in a time series.
• ARIMA models (Autoregressive Integrated Moving Average) are a class of statistical models for analyzing and forecasting time series data.
• GARCH models (Generalized Autoregressive Conditional Heteroskedasticity) are used to model volatility clustering in financial time series.

Numerical Methods

• Numerical methods are used to solve mathematical problems that cannot be solved analytically.
• Root-finding algorithms, such as the Newton-Raphson method, are used to find the roots of equations.
• Numerical integration techniques, such as the trapezoidal rule and Simpson's rule, are used to approximate definite integrals.
• Finite difference methods are used to approximate the solutions of differential equations.
• Optimization algorithms, such as gradient descent and dynamic programming, are used to find the optimal solutions to financial problems.