Derivative

In mathematics, the derivative is a mapping associated to another mapping, which measures the instantaneous rate of change at a point. Calculus is mostly the study of the derivative and its implications. When calculus is all dressed up in its fancy clothes, it is called analysis and constitutes a very large branch of mathematics.

See noaseboar's writeup below for the definition. I might add that the nth derivative of a mapping f: X → Y can be regarded as living in the space of symmetric n-multilinear maps X → Y, which is naturally isomorphic to Hom(Symⁿ X, Y) where Symⁿ X is the nth symmetric power of X. See Geometric measure theory by H. Federer for a careful treatment of the multilinear algebra of geometric calculus, which can get tricky.

Derivative
In Financial Markets a derivative is an instrument that derives it's value from another, underlying instrument.

Derivatives have no value on their own, unlike, for example, a share in a company (more commonly called stock) or a bond.

Derivatives are interesting since they have no existince in a tangible, or physical form; they are virtual instruments.

Derivatives are used for what is known as Risk Management . They accomplish this because of their intrinsic property of leverage .

This leverage accounts for their bad reputation as well as the rather spectacular failure of some financial institutions trading them.

The derivative of a function is the graph of its slopes at various points. An antiderivative or integral is the opposite of a derivative, the derivative of the antiderivative is the original function. The derivative of position is velocity, the derivative of velocity is acceleration, the derivative of acceleration is jerk.

The derivative of a function is denoted by f^I(x) (Which is read f prime of x). And can be solved in many ways. One form of the derivative of a function is:

f^I(x) = lim_x->a(f(x) - f(a))/(x - a)

There are also many formulas for finding derivatives of common functions without having to go through that whole limit nonsense.

When f(x) = xⁿ
f^I(x) = nx^n-1

When f(x) = c (Where c is a constant)
f^I(x) = 0

When f(x) = e^x
f^I(x) = e^x

When f(x) = a^x
f^I(x) = ln(a) * a^x

When f(x) = ln(x)
f^I(x) = 1/x

When f(x) = sin(x)
f^I(x) = cos(x)

When f(x) = cos(x)
f^I(x) = -sin(x)

When f(x) = tan(x)
f^I(x) = sec²(x)

When f(x) = tan^-1(x)
f^I(x) = 1/(1 + x²)

Also you can use the rules on functions like: f(x) = xⁿ + x or n^x/xⁿ thinking of it as several functions combined using the following rules:

(f(x) + g(x))^I = f^I(x) + g^I(x)

(f(x) - g(x))^I = f^I(x) - g^I(x)

(f(x) * g(x))^I = f(x) * g^I(x) + f^I(x) * g(x)

(f(x) / g(x))^I = ((g(x) * f^I(x)) - (f(x) * g^I(x)))/g(x)²

Some examples:
f(x) = sin(x) + x⁵
f^I(x) = cos(x) + 5x⁴

f(x) = x⁴2x⁵
f^I(x) = (x⁴10x⁴) + (4x³2x⁵)

Let X,Y be normed spaces with norms || ||_X and || ||_Y.
A function f: X -> Y is called differentiable in the point x of X, iff there is a continuous linear map A and the limit

            f(x+h) - A(h) - f(x)
lim         ---------------------------
||h|| -> 0            ||h||

exists and equals 0 (||h||=||h||_X, h of X).
A is called the derivative of f in x and is usually written as Df(x). (This is a linear map, to get any values you would have to write Df(x)(z).)

Let L(X,Y) the normed space of continuous linear functions from X to Y.
If the derivative of f exists in an open neighborhood of x, then the derivative of the map Df: X -> L(X,Y), x |-> Df(x) might exist. It's called the second derivative of f in x, written as D²f(x).
If the second derivative exists in an open neighborhood of x, then the derivative of D²f : X -> L(X,L(X,Y)) might exist and is called the third derivative.
In fact the n-th derivative is the derivative of the function D^n-1f : X -> L(X,...L(X,Y))...)), x-> D^n-1f(x).
However these "stacked" spaces of linear functions L(X,...L(X,Y))..)) are difficult to use. Therefore one uses the fact that L(X,..L(X,Y)...) with n L's stacked is isometric to B(X,Y,n) is the space of n-linear continuous functions (Note: B(X,Y,n) is not canonical for this space, I just made it up)
The isomorphism is defined per: h of L(X,...L(X,Y)..) goes to g of B(X,Y,n) via g(x₁,...,x_n):= h(x₁)...(x_n).
So one takes as the n-th derivative the function Dⁿf(x₁)...(x_n) instead of Dⁿf(x₁) of L(X,...L(X,Y)..) (n-1 L's stacked)

The function f is called n times continuous differentiable in x iff the map Dⁿf: X -> B(X,Y,n), x |-> Dⁿf(x) is continuous (Note: this is not a linear map !)

Now comes the question: "What has this to do with the usual derivative of R¹ -> R¹ ?"
The derivative of R¹ is scalar. Multiplication with a scalar is the form of linear maps from R¹ to R¹. Set Df(x)(y) := f'(x)· y and you get the above form.

This definition allows you to differentiate in really sick spaces like function space, spaces of matrices etc.
The derivatives are quite difficult to determine there but some simple laws still hold:

• The derivative of a continuous linear map is the map itself at any point of X.
• The chain rule always holds: D(f(g))(x)(y) = (Df)(g(x))(Dg(x)(y)) where (Df)(g(x)) is the derivative of f at point g(x), the formula means: to get the image of y under D(f(g)) first apply Dg(x) on y and then apply (Df)(g(x)) ( f(g) is the function you get when you apply f to the images of g)

Definition

The derivative of a function of one variable, f(x), is another function f'(x). Geometrically, the derivative represents the slope of a line tangent to the graph of f at x.

The derivative is one of the fundamental concepts of the calculus, developed around the same time by both Sir Isaac Newton and Gottfried Wilhelm von Leibniz. Leibniz used a different notation to represent the derivative: "df/dx" which is read as "the derivative of f with respect to x."

The derivative of the derivative of a function is referred to as the second derivative (and this can go on to third, fourth, fifth, etc.).

Functions of more than one variable do not have a derivative, but have partial derivatives with respect to each independent variable. A partial derivative is obtained by treating all other independent variables as constants and then performing normal differentiation.

Derivation of the formula for the derivative of f(x)
(This is not a rigorous proof, but is typical of what you'd see in a Freshman level Calculus class)

The slope of a secent line through two points, (x1,y1) and (x2,y2), on a the graph of f(x) is given by:

f(x2)-f(x1)
-----------
x2-x1

The difference in x between the two points is change in x or more commonly: deltaX. Thus, this formula can be rewritten as:

f(x1+deltaX)-f(x1)
------------------
deltaX


The tangent line is the same as the secent line, except there is no change in x. So if we take the limit of this expression as deltaX approaches 0, then we get a new function of x which represents the slope of a line tangent to f at x. This is the definition of the derivative.

In the world of finance, derivatives are instruments where the performance is based on the behaviour of the price of an underlying asset.

A derivative is a contract between a pair of counterparties, due to be executed on some future date, apart from options, where the execution (exercise) is optional.

The Underlying

The underlying can be one of the following physical assets:

• equities. These are stocks and shares in individual companies
• bonds
• commodities, including gold, petroleum, sugar, coffee, bananas etc.

The underlying can also be something non-physical, which will always be cash settled:

• exchange rate, such as USD -> EUR
• index, such as the Dow Jones industrial average, FTSE 100
• interest rate

The underlying can sometimes itself be a derivative. You can have options on futures

Types of derivative

• futures

  Futures are an agreement to trade an underlying asset at a future date, based on today's price. Futures are exchange traded derivatives, and require payments of margin. Expiry of futures contracts happens on a 3 monthly cycle, usually March, June, September and December.

• Forwards

  Forwards are an OTC equivalent of futures. As such, the issuing bank can be flexible, making a tailor made contract. However, there is the need to provide collateral, and/or credit references. One popular type of forward is the forward rate agreement, or FRA, offering forwards on interest rates.

• Options

  Options are concerned with buying and selling the right to buy or sell an underlying. The right to buy is called a call option, and the right to sell is called a put option. Each option has two counterparties, the buyer of the option, the long party, and the seller (writer), the short party. There are four combinations:

  • buying the right to buy. Long on call.
  • selling the right to buy. Short on call.
  • buying the right to sell. Long on put.
  • selling the right to sell. Short on put.

  In exchange for the options contract, the long party pays a premium to the short party, much in the manner of insurance. The short party is the one which is exposed to the risk, as Nick Leeson demonstrated spectacularly.

  Options are traded both OTC and on stock exchanges.

• Swaps

  Similar to a playground scenario, a swap involves the exchange between counterparties of one underlying for another underlying. An everyday example of a swap transaction is exchanging a variable rate property mortgage for a fixed rate mortgage.

  Swaps are usually OTC derivatives. Sometimes they involve currencies, for instance USD may have a different exposure to risks than EUR. Also popular are interest rate swaps.

What derivatives are used for

Contrary to popular belief, derivatives are most commonly used in risk management, hedging, i.e. reducing the exposure to risk. We have much to thank the press for in giving a bad name to derivatives trading and markets. It is comparable to the equally fallacious argument "The insurance company never pays up". If this were the case, the insurance company would not be in business.

However, derivatives can also be used for speculation, as Enron have demonstrated. Many countries require financial insitutions taking risks to account for their transactions on a daily basis, and to put hedging in place for risky transactions - as a legal requirement.

Source: Mastering Derivatives Markets - Francesca Taylor. Prentice Hall

The point of a derivative, in elementary calculus at least, is to find the gradient of a function at any particular point P. Before we learn about calculus, the problem is usually solved by drawing an approximation to the tangent at the point P, forming a right-angled triangle and dividing the length of the vertical side by the length of the horizontal side. This gives the gradient of the hypotenuse of the triangle and hence an approximation to the gradient of the function at the point P.

If we think about the triangle drawing idea above and bring it to a limit, so that we're drawing an arbitrarily small triangle, then we approach the true gradient at the point. This lets us derive the formula for a derivative from first principles.

Say we have the function f and we want to find the gradient at the point x₁. Then we 'draw' a horizontal line from the point (x₁, f(x₁)) to the point (x₁ + δx, f(x₁)) and a vertical line from the point (x₁ + δx, f(x₁)) to the point (x₁ + δx, f(x₁ + δx)), where δx is a small change in the x-coordinate.

Now, dividing the length of the vertical line by the length of the horizontal line, and cancelling, gives us the approximation to the gradient at the point x₁:

                          f(x1 + δx) - f(x1)
                        ---------------------
                                  δx

Now, this should give us the exact gradient if we let δx go to zero, so the formula we eventually obtain for the gradient of a function f at a point x₁ is:

                          f(x1 + δx) - f(x1)
       f|(x1) = limδx->0 ---------------------
                                  δx

---

Now for a quick example. Anyone who's taken even the most elementary of elementary calculus courses knows that the derivative of xⁿ is nx^n-1. Pumping f(x) = xⁿ into the formula derived above gives:

                          (x1 + δx)n - x1n
       f|(x1) = limδx->0 ---------------------
                                  δx

Now, by the binomial theorem: (x₁ + δx)ⁿ = x₁ⁿ + nx₁^n-1δx + (n(n-1)/2)x₁^n-2δx² + o(δx³)

where o(δx³) means terms where δx is present in at least the third power.

So:

                          x1n + nx1n-1δx + (n(n-1)/2)x1n-2δx2 + o(δx3) - x1n
       f|(x1) = limδx->0 ----------------------------------------------------
                                                  δx
                         nx1n-1δx + (n(n-1)/2)x1n-2δx2 + o(δx3)
              = limδx->0 --------------------------------------
                                        δx
                        nx1n-1δx       (n(n-1)/2)x1n-2δx2        o(δx3)
              = limδx->0 -------   +   -----------------   +   -------
                           δx                 δx                 δx

lim_δx->0 (δx²/δx) and lim_δx->0 (o(δx³)/δx) are both zero, so, after cancelling δx's, we are left with:

f^|(x₁) = nx₁^n-1

which is exactly the result we were looking for. Similar arguments (using expansions and cancelling) can be used to derive the derivatives of lots of functions including the trigonometric functions.

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I've used the <pre> tag lots and lots and it looks fine on my system. If it looks all funny, please /msg me.

Derivatives can be understood in another way by using infinitesimal calculus, which relies on algebra and infinitesimals, rather than limits, to define calculus. It's not as easy to extend as limit calculus, but it's more intuitive, and the results are the same.

Suppose you have an equation, such as y = x². If you increase x by a minute amount, what is the increase on y? To determine that increase, you use calculus.

Let's call the minute increase on y dy, and the minute increase on x dx. So:

y + dy = (x + dx)²

Multiply out (x + dx)² to get:

y + dy = x² + 2x*dx + (dx)²

Now, we defined dx as being minutely small. Much like how a second is minute compared to a minute, it is even more minute compared to an hour. So (dx)² is so small, we can disregard it.

y + dy = x² + 2x*dx

I know, that sounds like real bullshit, but bear with me.

The original equation was y = x². Subtract that out.

dy = 2x*dx

Now, we divide by dx. (Again, sounds like bullshit, but...)

dy/dx = 2x

And there we go. The derivative of x² is 2x. When you increase x² by a certain amount, it is increased by 2x. Similarly with decreases.

A derivative is the value by which change is determined.

De*riv"a*tive (?), a. [L. derivativus: cf. F. d'erivatif.]

Obtained by derivation; derived; not radical, original, or fundamental; originating, deduced, or formed from something else; secondary; as, a derivative conveyance; a derivative word.

Derivative circulation, a modification of the circulation found in some parts of the body, in which the arteries empty directly into the veins without the interposition of capillaries.

Flint.

-- De*riv"a*tive*ly, adv. -- De*riv"a*tive*ness, n.

 

© Webster 1913.

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De*riv"a*tive, n.

1.

That which is derived; anything obtained or deduced from another.

2. Gram.

A word formed from another word, by a prefix or suffix, an internal modification, or some other change; a word which takes its origin from a root.

3. Mus.

A chord, not fundamental, but obtained from another by inversion; or, vice versa, a ground tone or root implied in its harmonics in an actual chord.

4. Med.

An agent which is adapted to produce a derivation (in the medical sense).

5. Math.

A derived function; a function obtained from a given function by a certain algebraic process.

⇒ Except in the mode of derivation the derivative is the same as the differential coefficient. See Differential coefficient, under Differential.

6. Chem.

A substance so related to another substance by modification or partial substitution as to be regarded as derived from it; thus, the amido compounds are derivatives of ammonia, and the hydrocarbons are derivatives of methane, benzene, etc.

 

© Webster 1913.