# Mathematical Tools in Finance – Terms, Definition & Topical Outline

Here we look at some terms and definitions that form a broad, and reasonably complete, topical outline of mathematical tools in finance.

## Key Takeaways – Mathematical Tools in Finance

• This overview highlights key mathematical tools in finance.
• Also included are optimization techniques, the Monte Carlo Method for numerical solutions, Real Analysis, and PDEs relevant in finance.
• Probability concepts, various distributions, and Stochastic Calculus with processes like Brownian Motion are discussed.
• Nearly 50 concepts covered.

## Mathematical Tools in Finance

### Asymptotic Analysis

The process of determining the behavior of a function as its arguments approach a limit, often infinity.

### Backward Stochastic Differential Equation (BSDE)

A backward stochastic differential equation might be used in finance to determine the price of a financial derivative, where the final condition (payoff) is known.

We seek the initial price, working backward from the known outcome.

### Calculus

A branch of mathematics dealing with derivatives and integrals, used to analyze the rates of change and accumulation.

### Copulas, including Gaussian

Copulas in finance are used to model and analyze the joint distribution of different asset returns, capturing the way they move together and allowing for more accurate assessment of portfolio risk.

Gaussian copulas specifically involving multivariate normal distributions.

### Differential Equations

Differential equations in finance are used for modeling the continuous change in financial markets, such as the evolution of stock prices over time or interest rate movements.

### Expected Value

The weighted average of all possible values of a random variable, with weights being their respective probabilities.

Related: Expected Value

### Ergodic Theory

A branch of mathematics that studies statistical properties of deterministic dynamical systems through time averages.

We also look at how ergodic economics can challenge traditional expected value models when such a game and its consequences are considered through time in this article.

### Feynman–Kac Formula

A representation of the solution to certain PDEs using the expectation of stochastic processes, particularly Brownian motion.

### Quantitative Finance

The field that applies mathematical and statistical methods to financial markets and securities’ valuation.

### Fourier Transform

A mathematical transform that decomposes functions into their constituent frequencies.

The Fourier Transform in finance is used to evaluate option pricing models more efficiently by transforming differential equations into algebraic equations that are easier to solve.

For example, it allows for fast computation of the characteristic functions of option prices, which can be inverted to obtain the actual option prices.

### Girsanov Theorem

A result in stochastic calculus that describes how to change probability measures to remove the drift from a stochastic process.

It simplifies certain types of probabilistic analysis in finance.

An example of using Girsanov’s Theorem is in the pricing of financial derivatives, where it can be employed to convert the actual probability of stock price movements to a risk-neutral probability.

This assumes no arbitrage opportunities and makes the expected return of the stock the risk-free rate (thus simplifying the pricing model).

### Itô’s Lemma

A key result in stochastic calculus, providing the differential of a function of a stochastic process.

Itô’s Lemma is important because it provides a way to determine the stochastic behavior of a function of a stochastic process.

This is important in financial mathematics for modeling the random movement of options prices over time.

### Martingale Representation Theorem

A theorem stating that every martingale can be represented as an integral against some predictable process with respect to Brownian motion.

### Mathematical Models

Quantitative constructs representing systems via mathematical concepts and language.

### Mathematical Optimization

The selection of the best element, with regard to some criteria, from some set of available alternatives.

For example, in finance, you might want to maximize return per each unit of risk as a type of optimization.

However, there are other forms of optimization as well.

### Linear Programming

An optimization technique for a linear objective function, subject to linear equality and inequality constraints.

In finance, linear programming could be used for investment portfolio optimization where a trader wants to maximize returns from a selection of stocks while adhering to a fixed budget.

The relationship between investment amounts and returns is often assumed linear for simplification in such models.

### Nonlinear Programming

The process of solving optimization problems where the objective function or the constraints are nonlinear.

A finance-related example of nonlinear programming could involve a trader who wants to choose a combination of stocks and bonds to maximize their portfolio’s expected return for a given level of risk.

The risk-return tradeoff is not linear but follows a more complex, nonlinear relationship.

An optimization method to solve problems with a quadratic objective function and linear constraints.

For quadratic programming in finance, consider a situation where a trader aims to minimize the risk of their investment portfolio, where risk is expressed as the variance of portfolio returns – a quadratic function.

The trader will have to balance the portfolio across various assets while maintaining a desired level of expected return

### Monte Carlo Method

A computational algorithm that relies on repeated random sampling to obtain numerical results.

### Numerical Analysis

The study of algorithms that use numerical approximation for problems of mathematical analysis.

In finance, numerical analysis might be used to approximate the fair value of complex financial derivatives when an exact analytical solution is not possible.

A numerical integration method using the Gaussian distribution to choose the sample points and weights.

Gaussian Quadrature could be employed in finance to more efficiently calculate the expected value of a financial derivative’s payoff, which requires integration over certain probability distributions.

### Real Analysis

A branch of mathematics dealing with the set of real numbers and functions of real variables.

Real Analysis can be applied in finance to understand the behavior of stock price movements or to examine the convergence of a financial algorithm that predicts market trends.

### Partial Differential Equations

Equations involving unknown multivariable functions and their partial derivatives.

### Heat Equation

A PDE describing the distribution of heat over time in a given space.

The heat equation is analogous to the Black-Scholes equation used to price options, describing how the value of an option diffuses over time as the underlying asset’s price fluctuates.

### Numerical Partial Differential Equations

The study of numerical methods for the approximation of PDEs.

### Crank–Nicolson Method

A numerical method used to solve heat and diffusion equations with second-order accuracy in both time and space.

In finance, the Crank-Nicolson method is used to create stable, accurate pricing models for options and derivatives by simulating changes over time with a focus on thermal or diffusion equations (traditionally the wheelhouse of physics).

### Finite Difference Method

A numerical technique for solving differential equations by approximating derivatives with finite differences.

The Finite Difference Method is applied in finance to solve partial differential equations such as the Black-Scholes equation – i.e., used to price options by approximating the changes in the option’s value over time and underlying asset price movements.

### Probability

The measure quantifying the likelihood that events will occur.

### Probability Distributions

Mathematical functions that give the probabilities of occurrence of different possible outcomes for an experiment.

### Binomial Distribution

A probability distribution that summarizes the likelihood of a value taking one of two independent states across a fixed number of iterations.

The Binomial Distribution can be used to model the price movements of an asset, typically assuming an “up” or “down” movement with fixed probabilities.

Examples include the pricing of binary options or in the binomial tree model for American options.

### Johnson’s SU-distribution

A versatile four-parameter family of probability distributions that can take on the characteristics of most common distributions.

Johnson’s SU-distribution is used in finance to model asset returns with skewness and kurtosis, allowing for a better fit to the actual return distributions of securities than normal distribution, especially for risk management and option pricing.

### Log-normal Distribution

A probability distribution of a random variable whose logarithm is normally distributed.

### Student’s t-distribution

A probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.

### Quantile Functions

The inverse functions to cumulative distribution functions, determining the value below which a given percentage of observations in a group of observations fall.

A derivative of one measure with respect to another measure, used particularly in the context of absolutely continuous measures.

This allows for the pricing of derivatives under a new measure where the discounted asset prices are martingales.

### Risk-Neutral Measure

A probability measure where the discounted present value of a payoff under this measure is equal to its current market price.

### Scenario Optimization

An optimization framework for problems under uncertainty, where the goal is to find solutions that perform well across a range of scenarios.

### Stochastic Calculus

A branch of mathematics that operates on stochastic processes and provides tools for modeling random systems.

### Brownian Motion

A stochastic process that describes the random continuous movement of particles suspended in a fluid.

In finance, Brownian Motion is used to model the random behavior of asset prices over time in the Black-Scholes and other financial models.

### Lévy Process

A stochastic process with stationary, independent increments, generalizing Brownian motion and Poisson processes.

A Lévy Process in finance can model more complex random movements in asset prices, allowing for jumps unlike the continuous Brownian Motion.

Related: Semimartingales in Finance

### Stochastic Differential Equation

An equation used to model systems that evolve over time with a deterministic trend and random fluctuations.

Stochastic differential equations are used to model the price dynamics of financial derivatives and assets.

They help account for the “random” nature of markets.

### Stochastic Optimization

Optimization methods that take into account the inherent uncertainty in the parameters and the environment.

Stochastic optimization in finance involves finding optimal solutions when dealing with unpredictable market conditions – e.g., as in portfolio allocation and derivative pricing.

### Stochastic Volatility

A property of financial markets where the volatility of asset prices is itself a random process.

Stochastic volatility models in finance describe how the unpredictability of an asset’s price movement changes over time – important for pricing options and risk management.

Related

### Survival Analysis

A branch of statistics dealing with death in biological organisms and failure in mechanical systems.

In finance, survival analysis may be used to estimate the “lifespan” of investments or to predict the time until an event such as default or reaching a certain price threshold.

### Value at Risk (VaR)

A statistical technique used to measure and quantify the level of financial risk within a firm or investment portfolio over a specific time frame.

### Volatility

A statistical measure of the dispersion of returns for a given security or market index, often quantified as the standard deviation or variance of returns.

### ARCH Model

An econometric model that provides a way to model changing variance over time as a function of past variances, designed for time series data.

The ARCH model is used in finance to measure and predict the time-varying volatility of financial returns – e.g., for risk management and derivative pricing.

### GARCH Model

A generalized version of the ARCH model that includes lagged forecast variances in addition to lagged actual variances for predicting future volatility.

The GARCH Model is employed in finance to forecast future volatility of financial securities based on both their own past variance and past forecast variances.

Can improve risk and pricing models for financial instruments.