# Derivative definition – What are derivatives

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Contents

## What are Derivatives in Finance?

Derivatives are instruments to manage financial risks. Since risk is an inherent part of any investment, financial markets devised derivatives as their own version of managing financial risk. Derivatives are structured as contracts and derive their returns from other financial instruments.

## Definition of Derivatives

If the market consisted of only simple investments like stocks and bonds, managing risk would be as easy as changing the portfolio allocation among risky stocks and risk-free bonds. However, since that is not the case, risk can be handled in several other ways. Derivatives are one of the ways to insure your investments against market fluctuations. A derivative is defined as a financial instrument designed to earn a market return based on the returns of another underlying asset. It is aptly named after its mechanism; as its payoff is derived from some other financial instrument.

Derivatives are designed as contracts signifying an agreement between two different parties, where both are expected to do something for each other. It could be as simple as one party paying some money to the other and in return, receiving coverage against future financial losses. There also could be a scenario where no money payment is involved up front. In such cases, both the parties agree to do something for each other at a later date. Derivative contracts also have a limited and defined life. Every derivative commences on a certain date and expires on a later date. Generally, the payoff from a certain derivative contract is calculated and/or is made on the termination date, although this can differ in some cases.

As stated in the definition, the performance of a derivative is dependent on the underlying assetвЂ™s performance. Often this underlying asset is simply called as an вЂњunderlyingвЂќ. This asset is traded in a market where both the buyers and the sellers mutually decide its price, and then the seller delivers the underlying to the buyer and is paid in return. Spot or cash price is the price of the underlying if bought immediately.

## Types of Derivatives

Derivative contracts can be differentiated into several types. All the derivative contracts are created and traded in two distinct financial markets, and hence are categorized as following based on the markets:

Exchange-traded contracts trade on a derivatives facility that is organized and referred to as an exchange. These contracts have standard features and terms, with no customization allowed and are backed by a clearinghouse.

## Over The Counter Contract

Over the counter (OTC) contracts are those transactions that are created by both buyers and sellers anywhere else. Such contracts are unregulated and may carry the default risk for the contract owner.

## Derivative Categories

Generally, the derivatives are classified into two broad categories:

• Forward Commitments
• Contingent Claims

## Forward Commitments

Forward commitments are contracts in which the parties promise to execute the transaction at a specific later date at a price agreed upon in the beginning. These contracts are further classified as follows:

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### Over the Counter Contracts

Over the counter contracts are of two types:

### Forward

In this type of contract, one party commits to buy and the other commits to sell an underlying asset at a certain price on a certain future date. The underlying can either be a physical asset or a stock. The loss or gain of a particular party is determined by the price movement of the asset. If the price increases, the buyer incurs a gain as he still gets to buy the asset at the older and lower price. On the other hand, the seller incurs a loss in the same scenario.

For a detailed understanding, you can read our exclusive post on Forward Contract

Swap can be defined as a series of forward derivatives. It is essentially a contract between two parties where they exchange a series of cash flows in the future. One party will consent to pay the floating interest rate on a principal amount while the other party will pay a fixed interest rate on the same amount in return. Currency and equity returns swaps are the most commonly used swaps in the markets.

Exchange traded forward commitments are called futures. A future contract is another version of a forward contract, which is exchange-traded and standardized. Unlike forward contracts, future contracts are actively traded in the secondary market, have the backing of the clearinghouse, follow regulations and involve a daily settlement cycle of gains and losses.

## Contingent Claims

Contingent claims are contracts in which the payoff depends on the occurrence of a certain event. Unlike forward commitments where the contract is bound to be settled on or before the termination date, contingent claims are legally obliged to settle the contract only when a specific event occurs. Contingent claims are also categorized into OTC and exchange-traded contracts, depending on the type of contract. The contingent claims are further sub-divided into the following types of derivatives:

### Options

Options are the type of contingent claims that are dependent on the price of the underlying at a future date. Unlike the forward commitments derivatives where payoffs are calculated keeping the movement of the price in mind, the options have payoffs only if the price of the underlying crosses a certain threshold. Options are of two types: Call and Put. A call option gives the option holder right to buy the underlying asset at exercise or strike price. A put option gives the option holder right to sell the underlying asset at exercise or strike price.

### Interest Rate Options

Options where the underlying is not a physical asset or a stock, but the interest rates. It includes Interest Rate Cap, floor and collar agreement. Further forward rate agreement can also be entered upon.

### Warrants

Warrants are the options which have a maturity period of more than one year and hence, are called long-dated options. These are mostly OTC derivatives.

### Convertible Bonds

Convertible bonds are the type of contingent claims that gives the bondholder an option to participate in the capital gains caused by the upward movement in the stock price of the company, without any obligation to share the losses.

### Callable Bonds

Callable bonds provide an option to the issuer to completely pay off the bonds before their maturity.

### Asset-Backed Securities

Asset-backed securities are also a type of contingent claim as they contain an optional feature, which is the prepayment option available to the asset owners.

### Options on Futures

A type of options that are based on the futures contracts.

### Exotic Options

These are the advanced versions of the standard options, having more complex features.

In addition to the categorization of derivatives on the basis of payoffs, they are also sub-divided on the basis of their underlying asset. Since a derivative will always have an underlying asset, it is common to categorize derivatives on the basis of the asset. Equity derivatives, weather derivatives, interest rate derivatives, commodity derivatives, exchange derivatives, etc. are the most popular ones that derive their name from the asset they are based on. There are also credit derivatives where the underlying is the credit risk of the investor or the government.

Derivatives take their inspiration from the history of mankind. Agreements and contracts have been used for ages to execute commercial transactions and so is the case with derivatives. Likewise, financial derivatives have also become more important and complex to execute smooth financial transactions. This makes it important to understand the basic characteristics and the type of derivatives available to the players in the financial market.

References

• Study Session 17, CFA Level 1 Volume 6 Derivatives and Alternative Investments, 7th Edition

## 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.

## 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 –

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