Derivat

a

We will take \(G(P(t),t)\) to figure the price of an option, with \(t\) being time, strike price \(X\) (not introduced yet), and expiration date \(T > t\) on a stock with price \(P(t)\) at time \(t\).

This representation does something really important: it expresses \(G\) as a function of only the current stock price \(P(t)\).

A derived variable is a mapping between states to a set, usually the natural numbers. Remember, if we can, given a state and match it to a number and show a relation which would iterate the state and decrease the states’ number. We can show that the algorithm terminates.

For a matrix, for instance, like:

\begin{equation} \begin{bmatrix} a & b \\ c & d \end{bmatrix} \end{equation}

We wish to find the matrix’s determinant; we write it down as:

\begin{equation} \begin{vmatrix} a & b \\ c & d \end{vmatrix} \end{equation}

geometric interpretation of determinants

Geometrically, determinants are how matrices send a unit object after its mapping; i.e. how does it transform the area of a unit square.

determinants can be computed along any axes

You can pick any row or column as the “axes”, and expand the matrix along any direction