Method for doing a task, often in the form of a programming code such as C++ or VBA. In finance, it is commonly used to refer to programming code we opposed to manually solving various formulae.

One of the more popular probability distributions, the Binomial model shows the number of occurences of an event within a number of observations. The binomial concept can be illustrated via the use of a coin, in that the model represents the number of times that the outcome of flipping a coin is either a head, or a tail.

Main characteristics of the distribution / model:

1. Underlying variable is discrete (in that it is not continuous)
2. There can only be 2 outcomes
3. The event must be mutually exclusive
4. The variable is randomly selected

> It can also be applied within option pricing via the aptly named Binomial Pricing Model which gives the price of an asset (an option for example) given a rise or fall in the underlying asset.

One of the important applications of physics within finance is the concept of Brownian Motion. It is a stochastic process which has stationary independent increments which follows a Normal distribution and also has continuous sample paths. In physics, Brownian motion has been used to show the movement of atomic particles which are subject to many "shocks".

It is a Markovian process, and is also known as a Wiener process.

> It is often applied to stock returns as many conclude that log of the returns follow a normal distribution. (see Geometric Brownian Motion

A variable z follows this process if

1. The change in z (dz) is = .

Where is a random pick from a normal distribution and dt is a small change in time.

2. The values of dz for two separate time intervals of dt are independent.

In other words, the change in the variable z is not dependent on any 2 individual intervals of time (dt).

A joint distribution function on two or more variables where the marginals are unit uniform random variables. Copula functions contain the entire information about the variables' dependency structure.

Copulas were first introduced in 1959 in surveys on random metric spaces, but they were not applied to finance until 1999.

Copulas offer a more 'dynamic' representation of correlation between two or more assets, and although often more difficult to implement than other traditional distributions, when put in place, provides a more complete picture of events.

Correlation measures the strength of a relationship between 2 variables. It varies from 0 (random relationship) to 1 (perfect linear relationship) or -1 (perfect negative linear relationship). A high correlation (near 1) reflects the fact that the 2 variables have a strong relationship in movement together, whereas a low correlation denotes a weak relationship between 2 variables.

For instance, if r^2 is 0.25, then the independent variable is said to explain 25% of the variance in the dependent variable.

A greek letter which, in derivatives is commonly used as a measure of the sensitivity of an option's price to its underlying price.

Delta for a call is defined as being (delta_call):

Where dc is change in call price, and dS is the change in underlying stock price. The equation also equals the standard normal distribution of d1, where d1 is part of the Black-Scholes formula for pricing options.

For a put, the delta is the (delta_call - 1).

We can see how the delta of a call changes with time to maturity:

You might have come across this in your high-school / college mathematics class, and did some simple differentiation. In calculus, a differential equation shows how a change in one or more variables are represented by a change in another. The equation relates one or more unknown functions and their derivatives. For example:

A standard differential equation comes in the form of dy/dx.

To consider this in a more broad sense, imagine that dy was the change in the volume of water in a bathtub which was having water poured into it. As the tap continues to run the volume of the bathtub continues to rise the change in the volume is determined by the time the tap is left running; which is denoted by dx.

Differential equations can be generalised to being either ordinary or not ordinary. An ordinary differential equation only involves a single unknown function and its derivative, whereas non-ordinary differential equations, such as partial differential equations have an unknown function dependent on more than variable.

Most of the differential equations found in financial engineering are of the partial differential form and can come in non-linear types which often offer no definitive solution.

By now, you might have linked the relationship of derivatives in general and differential equations. The term derivative is directly associated with differential equations in the sense that one function, for example the delta of a call option is determined by the change in the call price and the change in the underlying asset price. The full proof of delta will be here shortly.

Find an overview of partial differential equations here.

Drift is a statistical term refering to the bias which exists within pricing of various derivative products. The drift variable is strongly time-dependent and is often used to adjust derivative prices to try and reflect random behaviour. A drift rate of 0 implies that the expected value of a variable at any time in the future is equal to its current value.

The drift term is found in many parts of finance, particularly at a foundation level when explaining geometric brownian motion or a wiener process. Common drift comes in the form of risk free rate minus the dividend yield (r-D) in the Black-Scholes European option pricing formula.

Continuing on from differential equations, if we consider K as a linear operator, by which we mean that F transforms a function into another function by means of partial derivatives or multiplication by other functions, y is a real number within a linear space , f is a function whereby Kf = yf, then f is considered to be an eigenfunction with an eigenvalue y.

Eigenvalue

Within linear algebra, eigenvalues are a special set of scalars associated with a linear system of equations (matrix equation) that are sometimes also known as characteristic roots, proper values, or latent roots. In other words, given a square matrix M, a column vector (i.e. a vector which is represented in a vertical form) X, a scalar vector Y and MX = YX, then we can say that Y is an eigenvalue of matrix M.

Eigenvectors are a set of vectors associated with a matrix equation within linear. Also known as characteristic vectors, proper vectors, or latent vectors. In other words, given a square matrix M, a column vector (i.e. a vector which is represented in a vertical form) X, a scalar vector Y and MX = YX, then we can say that X is an eigenvector of matrix M.

A Euclidean space is less exotic than it probably sounds. It is the common n-dimensional space where containing the standard mathematical topology. It is generally denoted as , where n is the number of dimensions and R are a set of real numbers. For example, a Euclidean space of is a 2-dimensional space containing 2 elements within, in the form of:

Where u is a vector, and a1 and a2 are the 2 elements of R which describe the size and direction of the vector.

An exponential (function) refers to a function in which Euler's number e is raised to the power x. For example, a function is said to be exponential if f(x) = exp^x.