Calculus

Question 1

What is a Function?

Answer 1

A Function is a rule that assigns to each number in a set a unique second number. A function is also called a mapping, or transformation; both words
invoke the action of associated one thing with another.
The statement y = f (x) is an example of a generic function, which reads “y is a function of x”.
In the function y = f (x), x is known as the argument of the function or independent variable, while y is the value of the function or dependent variable.

Question 2

Quadratic function

Answer 2

The generic formula for a quadratic function is
f (x) = ax2 + bx + c, where a, b and c are real numbers.
If the coeffcient a > 0 then the quadratic function is U shaped.

If a < 0 then it is n shaped.
The magnitude of a determines the steepness of the curve.
The coecient b determines the horizontal location of the curve.
The coecient c determines the vertical location.

Question 3

Exponential Function

Answer 3

The exponential number is of great importance in nance and is found by computing the limit
e = lim (1+ 1/n)^n =
n->unlimited

= 2,7182818
While the exponential function is e^n

Question 4

Bonds expressed in terms calculus function

Answer 4

A bond’s yield to maturity depends upon the maturity of the bond and is normally a monotonic increasing function, YTM = f (T).
The price of a bond in turn depends upon the yield to maturity of the bond, with a lower price for a higher yield reflecting the risk associated,
P = f (YTM) .

Question 5

Option expressed in terms of calculus

Answer 5

The value of an option at a given
point in time is a function of a
number of factors including:
 1. The price of the underlying
 2. The strike price
 3. The time to maturity
 4. Interest rates
 5. Volatility of the underlying

Question 6

What is Calculus?

Answer 6

Calculus is the branch of Mathematics that studies change and can be divided into two main areas: dierential calculus and integral calculus.

Dierential calculus calculates the rate at which a function changes as a response to a change in one or more other variables.

Integral calculus is used to calculate the areas and volumes of
bounded shapes.

Question 7

calculus Maxima and Minima used in finance

Answer 7

There are many situations in which we need to nd the minimum or maximum of a function. For example, if we need to find the combination of assets to minimize portfolio risk or maximize the
probability of true parameters in a model.

The combination of the first and second derivative can be used to determine if a particular point on a curve is at a peak (maxima) or a trough (minima). Moreover, we can use calculus to determine if
a function changes from convex to concave and vice versa.

Given a function f (x), when f’ (x) = 0 then x is called a stationary point of f . This means the tangent to the function is horizontal at a stationary point, but can be a minima, a maxima or
a point of in inflection.

If the second derivative f” (x) < 0 then the function is said to have a local maxima at point x.

If the second derivative f” (x) > 0 then the function is said to have a local minima at point x.

If the second derivative f” (x) = 0 then the function is said to have a stationary point of in
ection at point x (saddle point).

Question 8

Function of two variables,

partial derivative

Answer 8

Consider a function z = f (x, y), where x and y are independent.
Such a function can be dierentiated with respect to one independent variable, the others assumed to be fixed. This is known as partial dierentiation.
The partial derivative with respect to x is found by holding y constant and is denoted by fx or @z
@x .
Similarly, the partial derivative with respect to y is found by holding y constant and is denoted by fy or @z
@y .

Question 9

Total Derivatives

Answer 9

The partial derivative tells us about what happen to a function, when only one variable changes and the others are kept static.
The total derivative examines what happens to a function, when all the variables change at once and is expressed in terms of the differential operator.

For f (x, y), the total derivative df (x, y) gives the incremental change in the function for a small change in both x and y and is given by
df (x, y) = f (x + dx, y + dy) - f (x, y) ;
where dx and dy are increments in x and y.