# Derivative definition – What are derivatives

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## What are derivatives?

Derivatives are financial products that derive their value from the price of an underlying asset. Derivatives are often used by traders as a device to speculate on the future price movements of an asset, whether that be up or down, without having to buy the asset itself.

As no physical assets are being traded when derivative positions are opened, they normally exist as a contract between two parties, which can be traded over-the-counter or on a stock exchange. A large range of underlying assets can be traded using derivatives, including forex, shares, indices, bonds and commodities.

## Discover the differences between spread bets and CFDs

Find out the key differences between these two derivative products.

## Examples of derivatives

There is a wide range of derivative products that you can choose from. Two of the most popular derivative products are contracts for difference (CFDs) and spread bets.

When you trade CFDs, you are entering into a contract to exchange the difference in the price of an asset from the time your position is opened to when it is closed. Being a derivative, you never take ownership of any assets when trading CFDs, instead you are speculating on the underlying price.

Spread betting is similar to trading CFDs in that you don’t own the assets themselves. When you spread bet, you are placing a bet on the direction in which an underlying asset’s price will move.

Other examples of derivatives include options, forward contracts and futures contract.

## Pros and cons of derivatives

### Pros of derivatives

Trading derivatives can be used to hedge: a method of minimising losses to other positions. This is because derivative products offer a larger amount of flexibility when compared to trading the underlying asset directly.

With traditional investing, you open a long position – buying an asset in the hope that it rises in value. But with derivatives, you can also speculate on markets that are falling in price – this is done by opening a short position.

Some derivative products are traded on margin, which means that you only need to put down a fraction of the value of a position to receive full market exposure. Any profit to the position is calculated using the full value of the trade, which can mean that returns on successful trades are magnified. However, it can also amplify your losses. This makes it important to consider your trade in terms of its full value and downside potential.

### Cons of derivatives

Derivatives are sometimes criticised for adding to market volatility. In the past, speculators have been accused of greed during times of increasing fuel and food prices, and for causing drastic swings in the markets. Price movements fuelled by speculation can lead to speculative bubbles, which push the intrinsic value of an asset above its normal market price.

When speculative bubbles burst, the effects are often devastating on the markets and even on the economies of countries around the world. This happened in 2008 when the American housing bubble burst.

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## What are Derivatives in Finance?

Derivatives in Finance are financial instruments that derive their value from the value of the underlying asset. The underlying asset can be bonds, stocks, currency, commodities, etc.

### Top 4 Types of Derivatives in Finance

The following are the top 4 types of derivatives in finance.

#### # 1- Future

A futures derivative contract in Finance is an agreement between two parties to buy/sell the commodity or financial instrument at a predetermined price on a specified date.

#### #2 – Forward

A forward contract works in the same way as the futures, the only difference being, it is traded over the counter. So there is a benefit of customization.

#### #3 – Option

Options in Finance also work on the same principle, the however biggest advantage of options are that they give the buyer a right and not an obligation to buy or sell an asset, unlike other agreements where exchanging is an obligation.

#### #4 – Swap

A swap is a derivative contract in Finance where the buyer and seller settle the cash flows on predetermined dates.

There are investors/investment managers in the market who are called the market makers, they maintain the bid and offer prices in a given security and stand ready to buy or sell lots of those securities at quoted prices.

### Examples of Use of Derivatives in Finance

The following are examples of use if derivatives on finance.

#### #1 – Forward Derivative Contract Example

Suppose a company from the United States is going to receive payment of €15M in 3 months. The company is worried that the euro will depreciate and is thinking of using a forward contract to hedge the risk. This effectively means they fear they will receive less $when they go out to exchange their € in the market. Therefore by using a forward contract the company can sell the euro right now at a predetermined rate and avoid the risk of receiving less$.

#### #2 – Future Derivatives Contract Example

To keep it simple and clear the same example as above can be taken to explain the futures contract. However, the futures contract has some major Differences as compared to forwards. Futures are Exchange-traded, therefore they are governed and regulated by the exchange. Unlike forwards which can be customized and structured as per the parties’ needs. Which is why there is much less credit, counterparty risk in forwards as they are designed according to the parties’ needs.

#### #3 – Options Contracts Example

An investor has $10,000 to invest, he believes that the price of stock X will increase in a month’s time. The current price is$30, in order to speculate, the investor can buy a 1-month call option with a strike price of let’s say \$35. He could simply pay the premium and go long call on this particular stock instead of buying the shares. The mechanism of our option is exactly the opposite of a call.

#### #4 – Swaps Example

Let’s say a company wants to borrow € 1,000,000 at a fixed rate in the market but ends up buying at the floating rate due to some research-based factors and comparative advantage. Another company in the market wants to buy € 1,000,000 at the floating rate but ends up buying at a fixed rate due to some internal constraints or simply because of low ratings. This is where the market for swap is created, both the companies can enter into a swap agreement promising to pay each other their agreed obligation.

### Calculation Mechanism of Derivatives Instruments in Finance

• The payoff for a forward derivative contract in finance is calculated as the difference between the spot price and the delivery price, St-K. Where St is the price at the time contract was initiated and k is the price the parties have agreed to expire the contract at.
• The payoff for a futures contract is calculated as the difference between the closing price of yesterday and the closing price of today. Based on the difference it is determined who has gained, the buyer or the seller. If the prices have decreased the seller gains, whereas if the prices increased the buyer gains. This is known as the mark to market payment model where the gains and losses are calculated on a daily basis and the parties notified of their obligation accordingly.
• The payoff schedule for options is a little more complicated.
• Call Options: Gives the buyer a right but not an obligation to buy the underlying asset as per the agreement in exchange of a premium, it is calculated as- max (0, St – X). Where St is the stock price at maturity and X is the strike price agreed between by the parties and the 0 whichever is greater. To calculate the profit from this position the buyer will have to remove the premium from the payoff.
• Put Options: Gives the buyer a right but not an obligation to sell the underlying asset as per the agreement in exchange for a premium. The calculation schedule for these options is exactly the reverse of calls, i.e. strike minus the spot
• The payoff for swap contracts is calculated by netting the cash flow for both the counterparties. An example of a simple vanilla swap will help solidify the concept.

Some of the advantages of derivatives are as follows:

• It allows the parties to take ownership of the underlying asset through minimum investment.
• It allows to play around in the market and transfer the risk to other parties.
• It allows for speculating in the market, as such anyone having an opinion or intuition with some amount to invest, can take positions in the market with a possibility of reaping high rewards.
• In case of options, one can buy OTC (over the counter) customized option that suits their need and make an investment as per their intuition. The same applies to forward contracts.
• Similarly, in the case of futures contracts counterparty trades with the exchange, so it’s highly regulated and organized.

Some of the disadvantages of derivatives are as follows:

• The underlying assets in the contracts are exposed to high risk due to various factors like volatility in the market, economic instability, political inefficiency, etc. Therefore as much as they provide ownership, they are severely exposed to risk.
• Dealing in derivatives contracts in Finance requires a high level of expertise because of the complex nature of the instruments. Therefore a layman is better off investing in easier avenues like mutual funds/ stocks or fixed income.
• Famous Investor and philanthropist, Warren Buffet once called derivatives ‘weapons of mass destruction’ because of its inextricable link to other assets/product classes.

### Conclusion

The bottom line is although it gives exposure to high-value investment, in real sense it is very risky and requires a great level of expertise and juggling techniques to avoid and shift the risk. The number of risks it exposes you to is multiple. Therefore unless one can measure and sustain the risk involved, investing in big position is not advisable. Conversely, a well-calibrated approach with calculated risk structure can take an investor a long way in the world of financial derivatives.

This has been a guide to what are Derivatives in Finance & its definition. Here we discuss the top 4 types of derivatives in Finance along with examples, advantages, and disadvantages. You can learn more about accounting from following articles –

## Derivative definition – What are derivatives?

In the first section of the Limits chapter we saw that the computation of the slope of a tangent line, the instantaneous rate of change of a function, and the instantaneous velocity of an object at $$x = a$$ all required us to compute the following limit.

We also saw that with a small change of notation this limit could also be written as,

This is such an important limit and it arises in so many places that we give it a name. We call it a derivative. Here is the official definition of the derivative.

#### Defintion of the Derivative

Note that we replaced all the a’s in $$\eqref$$ with x’s to acknowledge the fact that the derivative is really a function as well. We often “read” $$f’\left( x \right)$$ as “f prime of x”.

Let’s compute a couple of derivatives using the definition.

So, all we really need to do is to plug this function into the definition of the derivative, $$\eqref$$, and do some algebra. While, admittedly, the algebra will get somewhat unpleasant at times, but it’s just algebra so don’t get excited about the fact that we’re now computing derivatives.

First plug the function into the definition of the derivative.

$\beginf’\left( x \right) & = \mathop <\lim >\limits_ \frac <\right) – f\left( x \right)>>\\ & = \mathop <\lim >\limits_ \frac <<2<<\left( \right)>^2> – 16\left( \right) + 35 – \left( <2– 16x + 35> \right)>>\end$

Be careful and make sure that you properly deal with parenthesis when doing the subtracting.

Now, we know from the previous chapter that we can’t just plug in $$h = 0$$ since this will give us a division by zero error. So, we are going to have to do some work. In this case that means multiplying everything out and distributing the minus sign through on the second term. Doing this gives,

$\beginf’\left( x \right) & = \mathop <\lim >\limits_ \frac <<2+ 4xh + 2 – 16x – 16h + 35 – 2 + 16x – 35>>\\ & = \mathop <\lim >\limits_ \frac <<4xh + 2– 16h>>\end$

Notice that every term in the numerator that didn’t have an h in it canceled out and we can now factor an h out of the numerator which will cancel against the h in the denominator. After that we can compute the limit.

$\beginf’\left( x \right) & = \mathop <\lim >\limits_ \frac <\right)>>\\ & = \mathop <\lim >\limits_ 4x + 2h – 16\\ & = 4x – 16\end$

So, the derivative is,

$f’\left( x \right) = 4x – 16$

This one is going to be a little messier as far as the algebra goes. However, outside of that it will work in exactly the same manner as the previous examples. First, we plug the function into the definition of the derivative,

Note that we changed all the letters in the definition to match up with the given function. Also note that we wrote the fraction a much more compact manner to help us with the work.

As with the first problem we can’t just plug in $$h = 0$$. So, we will need to simplify things a little. In this case we will need to combine the two terms in the numerator into a single rational expression as follows.

Before finishing this let’s note a couple of things. First, we didn’t multiply out the denominator. Multiplying out the denominator will just overly complicate things so let’s keep it simple. Next, as with the first example, after the simplification we only have terms with h’s in them left in the numerator and so we can now cancel an h out.

So, upon canceling the h we can evaluate the limit and get the derivative.

The derivative is then,

First plug into the definition of the derivative as we’ve done with the previous two examples.

In this problem we’re going to have to rationalize the numerator. You do remember rationalization from an Algebra class right? In an Algebra class you probably only rationalized the denominator, but you can also rationalize numerators. Remember that in rationalizing the numerator (in this case) we multiply both the numerator and denominator by the numerator except we change the sign between the two terms. Here’s the rationalizing work for this problem,

Again, after the simplification we have only h’s left in the numerator. So, cancel the h and evaluate the limit.

And so we get a derivative of,

Let’s work one more example. This one will be a little different, but it’s got a point that needs to be made.

Since this problem is asking for the derivative at a specific point we’ll go ahead and use that in our work. It will make our life easier and that’s always a good thing.

So, plug into the definition and simplify.

We saw a situation like this back when we were looking at limits at infinity. As in that section we can’t just cancel the h’s. We will have to look at the two one sided limits and recall that

The two one-sided limits are different and so

doesn’t exist. However, this is the limit that gives us the derivative that we’re after.

If the limit doesn’t exist then the derivative doesn’t exist either.

In this example we have finally seen a function for which the derivative doesn’t exist at a point. This is a fact of life that we’ve got to be aware of. Derivatives will not always exist. Note as well that this doesn’t say anything about whether or not the derivative exists anywhere else. In fact, the derivative of the absolute value function exists at every point except the one we just looked at, $$x = 0$$.

The preceding discussion leads to the following definition.

#### Definition

A function $$f\left( x \right)$$ is called differentiable at $$x = a$$ if $$f’\left( a \right)$$ exists and $$f\left( x \right)$$ is called differentiable on an interval if the derivative exists for each point in that interval.

The next theorem shows us a very nice relationship between functions that are continuous and those that are differentiable.

#### Theorem

If $$f\left( x \right)$$ is differentiable at $$x = a$$ then $$f\left( x \right)$$ is continuous at $$x = a$$.

Note that this theorem does not work in reverse. Consider $$f\left( x \right) = \left| x \right|$$ and take a look at,

$\mathop <\lim >\limits_ f\left( x \right) = \mathop <\lim >\limits_ \left| x \right| = 0 = f\left( 0 \right)$

So, $$f\left( x \right) = \left| x \right|$$ is continuous at $$x = 0$$ but we’ve just shown above in Example 4 that $$f\left( x \right) = \left| x \right|$$ is not differentiable at $$x = 0$$.

#### Alternate Notation

Next, we need to discuss some alternate notation for the derivative. The typical derivative notation is the “prime” notation. However, there is another notation that is used on occasion so let’s cover that.

Given a function $$y = f\left( x \right)$$ all of the following are equivalent and represent the derivative of $$f\left( x \right)$$ with respect to x.

Because we also need to evaluate derivatives on occasion we also need a notation for evaluating derivatives when using the fractional notation. So, if we want to evaluate the derivative at $$x = a$$ all of the following are equivalent.

Note as well that on occasion we will drop the $$\left( x \right)$$ part on the function to simplify the notation somewhat. In these cases the following are equivalent.

$f’\left( x \right) = f’$

As a final note in this section we’ll acknowledge that computing most derivatives directly from the definition is a fairly complex (and sometimes painful) process filled with opportunities to make mistakes. In a couple of sections we’ll start developing formulas and/or properties that will help us to take the derivative of many of the common functions so we won’t need to resort to the definition of the derivative too often.

This does not mean however that it isn’t important to know the definition of the derivative! It is an important definition that we should always know and keep in the back of our minds. It is just something that we’re not going to be working with all that much.

## Introduction to Derivatives

Slope = Change in YChange in X

We can find an average slope between two points.

But how do we find the slope at a point?

There is nothing to measure!

But with derivatives we use a small difference .

. then have it shrink towards zero.

## Let us Find a Derivative!

To find the derivative of a function y = f(x) we use the slope formula:

Slope = Change in Y Change in X = ΔyΔx

And (from the diagram) we see that:

 x changes from x to x+Δx y changes from f(x) to f(x+Δx)

• Fill in this slope formula: ΔyΔx = f(x+Δx) − f(x)Δx
• Simplify it as best we can
• Then make Δx shrink towards zero.

### Example: the function f(x) = x 2

We know f(x) = x 2 , and we can calculate f(x +Δx ) :

 Start with: f(x +Δx ) = (x +Δx ) 2 Expand (x + Δx) 2 : f(x +Δx ) = x 2 + 2x Δx + (Δx) 2

Result: the derivative of x 2 is 2x

In other words, the slope at x is 2x

And “the derivative of” is commonly written :

x 2 = 2x
“The derivative of x 2 equals 2x
or simply “d dx of x 2 equals 2x

### What does x 2 = 2x mean?

It means that, for the function x 2 , the slope or “rate of change” at any point is 2x.

So when x=2 the slope is 2x = 4, as shown here:

Or when x=5 the slope is 2x = 10, and so on.

Note: sometimes f’(x) is also used for “the derivative of”:

f’(x) = 2x
“The derivative of f(x) equals 2x”
or simply “f-dash of x equals 2x”

Let’s try another example.

### Example: What is x 3 ?

We know f(x) = x 3 , and can calculate f(x +Δx ) :

 Start with: f(x +Δx ) = (x +Δx ) 3 Expand (x + Δx) 3 : f(x +Δx ) = x 3 + 3x 2 Δx + 3x (Δx) 2 + (Δx) 3

Result: the derivative of x 3 is 3x 2

Have a play with it using the Derivative Plotter.

## Derivatives of Other Functions

We can use the same method to work out derivatives of other functions (like sine, cosine, logarithms, etc).

But in practice the usual way to find derivatives is to use:

### Example: what is the derivative of sin(x) ?

On Derivative Rules it is listed as being cos(x)

Using the rules can be tricky!

### Example: what is the derivative of cos(x)sin(x) ?

You can’t just find the derivative of cos(x) and multiply it by the derivative of sin(x) . you must use the “Product Rule” as explained on the Derivative Rules page.

It actually works out to be cos 2 (x) − sin 2 (x)

So that is your next step: learn how to use the rules.

## Notation

“Shrink towards zero” is actually written as a limit like this:

“The derivative of f equals the limit as Δ x goes to zero of f(x+Δx) – f(x) over Δx ”

Or sometimes the derivative is written like this (explained on Derivatives as dy/dx ):

The process of finding a derivative is called “differentiation”.

You do differentiation . to get a derivative.

## Where to Next?

Go and learn how to find derivatives using Derivative Rules, and get plenty of practice:

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