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THE BASICS OF FINANCIAL DERIVATIVES
This note describes the basic elements and pricing of financial derivatives. Financial
derivatives are contracts whose value is derived from the value of some other underlying asset,
such as a share of common stock, a commodity (like coffee, oil, or wheat), or a bond. Each
derivative has its own special features and provisions, and each is used for a special financial
purpose. Derivatives are often used to hedge or provide risk reduction to investments, to lessen
risk exposure to foreign exchange rates, to protect against movement in interest rates and, for
risk-seekers, to speculate in stocks, bonds, and commodities. Derivatives are sometimes traded
with the underlying asset they are based on, but they also trade as separate instruments on their
own. They may trade on the large exchanges, such as the New York Stock Exchange (NYSE), or
on specialized exchanges, like the Chicago Board Options Exchange (CBOE).
As you work through this note, you will learn more about the fundamental features of
today’s most commonly used derivatives: call and put options, warrants, swaps, convertible
securities, and forward and futures contracts.
Options
One common type of derivative is a call option. A call option is a contract conveying the
right to buy a specific asset (the underlying asset) at a specified price (the exercise price or strike
price) over a specified period of time. Such a contract is an agreement between two parties, the
buyer and the seller (sometimes called the writer) of the call option. For example, an investor
holding a call option on IBM stock with a strike price of $115 and a maturity date of June 1999
has the right to buy a share of IBM stock for $115 on or before June 18, 1999 (conventionally,
the maturity date on stock options is the third Friday of the month). If an option contract
stipulates that the buyer may only exercise the option on the day that it expires (the maturity or
expiration date), it is called a European option. If the buyer may exercise on any day during the
option’s time period, it is an American option. A buyer is free to sell the option to another party
prior to maturity regardless of whether the option is American or European. Thus, exercising an
option and selling it are two distinct activities.
This case was prepared by Susan Chaplinksy. It was written as a basis for class discussion rather than to illustrate
effective or ineffective handling of an administrative situation. Copyright © 1999 by the University of Virginia
Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail to
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spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording, or
otherwise—without the permission of the Darden School Foundation. Rev. 6/99.
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There are many positions or methods of investment that an individual may take when
investing in options contracts. The most basic of these is a long position, in which the buyer
(owner) has simply purchased an option. So, the purchaser of a call option is said to be long a
call or to hold a long call. Because of the fixed-asset purchase price they offer, call options are
attractive to individuals who believe that the price of the underlying asset is likely to rise. A
purchaser of call options on stock, therefore, may be betting on increases in stock prices.
Options contracts on stocks are generally short term, usually up to nine months.
Standardized stock-option contracts have been developed to facilitate the ready purchase and sale
of such contracts. In those contracts, the maturity date is usually given as the third Friday of the
month. Also, though the option price is listed on a per-share basis, the number of shares
contracted for is normally 100. For example, a newspaper listing of a January option at $2.35
indicates that the option expires on the third Friday in January and the price of each contract is
$2.35. But since the contract is for 100 shares, the total purchase price will be $2.35 × 100 =
$235. Note that this price (sometimes referred to as the call premium) is what you pay when you
purchase the option-it is not the exercise price you would pay for the stock if you exercise the
option. The call premium is also what a seller or writer of the call receives for granting the
option.
The actual issuer of standardized options on a stock is the Options Clearing Corporation
(OCC), and the sale or purchase of an option occurs solely between investors. In stock options,
for example, the firm whose stock underlies the options contract does not issue new shares or
receive any money in the options trade. Also, because of the way the OCC functions, the
settlement of options claims is normally a cash settlement, which is to say that there is no
physical exchange of the stock certificates. Rather, the options owner receives a cash payment
for the amount of money that` he or she could make by having the ability to exercise at the strike
price and then trading the stock on the open market.
A put option works similarly to a call option, but provides a different benefit. A put
option gives the owner the right to sell the underlying asset at a specified price over a specified
time period. For example, an investor holding a put option on IBM stock with a strike price of
$115 and a maturity date of June 1999 has the right to sell a share of IBM stock for $115 on or
before June 18, 1999. Puts provide a limit to stock losses; in this case, no matter how low the
price of IBM stock falls, the investor is still able to sell at $115. In this respect, the put acts as a
hedge or insurance for the investment; it protects the investor against the possibility of future
losses on the stock. Put options on common stock are thus attractive investments for individuals
who believe the stock price is likely to fall.
An important feature of all option contracts is that the option owner has a right, but not an
obligation to exercise it. For a call option, if the price of the underlying asset falls below the
exercise price specified in the option contract, the owner will choose not exercise the option to
buy the asset. If the option remains unexercised through the maturity date, it is said to expire
worthless. For example, consider the owner of a call on Apple Computer stock with an exercise
price of $45. If, at maturity, the stock price has dropped to $38, the owner will choose not to
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exercise the option to buy at $45 and the call will expire worthless. If the owner wanted to
purchase the stock, it would be cheaper to buy it on the open market ($38) than to pay the
exercise price specified by the options contract ($45). For this reason, the lowest payoff at
maturity to the long position is zero.
Payoff at Maturity
The payoff at maturity of an option refers to the payoff that the option yields to the owner
just before it expires on the maturity date. (Note that this does not include the amount that was
originally paid for the option.) For example, the value of a long call on common stock is greater
the more the stock price rises above the exercise price. At maturity, then, the payoff to the call
owner is calculated by subtracting the call’s exercise price from the market price of the stock
unless the call expires worthless. Because a put’s value increases as the stock price moves lower
than the exercise price, the payoff at maturity of a long put is calculated by reversing the
equation. The stock price is subtracted from the exercise price unless the put expires worthless.
It is important to recognize that for both long calls and puts, the lowest payoff at maturity
is zero. Since an option conveys the right but not the responsibility to do something, the owner
will only exercise it if it results in a gain. Thus, at expiration, the value of a long call is the
maximum of zero and the stock price minus the exercise price, and the value of the long put is
the maximum of zero and the exercise price minus the stock price.
Calls and puts are often referred to as being in-the-money, at-the-money, or out-of-themoney. An option is in-the-money if there is a positive payoff to the owner upon exercise, that is,
if the owner would reap a gain if the option were exercised immediately. An option is out-of-themoney if the payoff to the owner upon exercise is negative, meaning that an immediate exercise
would result in a loss. (Remember, however, that in this case, the owner of a long option would
simply not exercise, so the minimum option payoff is zero.) A call option on common stock,
therefore, is in-the-money when the underlying stock’s price is higher than the call’s exercise
price and out-of-the-money when the price of the stock is lower than the exercise price. A put
option on common stock is just the opposite. It is in-the-money when the stock price is lower
than the exercise price and out-of-the-money when the stock price is higher than the exercise
price. Both types of calls are at-the-money when the underlying asset price and the option’s
exercise price are equal.
Written Call and Put Options
For the writer, or seller, of a call or put option, the exposure and potential payoff of the
options contract are the mirror image of the purchaser’s position. For instance, the writer of a call
option contract on common stock guarantees to sell the stock at the exercise price if the
purchaser decides to exercise the call. The writer of an option is said to hold a written option.
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Because it is like a short stock position in this respect (providing the opposite return of a long
position), a written call or put is sometimes referred to as a short call or a short put.
Looking at the payoff to a written call, the higher the stock price at maturity is over the
exercise price, the more the writer loses. On the other hand, if at maturity the stock price is lower
than the exercise price, the option will expire worthless and the writer loses nothing. So for the
writer of the call, the payoff at maturity is the lower of zero and the exercise price minus the
stock price. For example, consider a written call on IBM stock with a strike price of $110, if the
market price of the stock is below $110, say $105, the payoff will be zero; the call holder will
not exercise the option, so the call writer keeps the stock and loses nothing. On the other hand, if
at maturity the stock price is greater than $110, say $118, the call owner exercises the call. This
forces the call writer to sell shares for an amount below the market price, resulting in a negative
payoff (an $8 loss in this example), as the call writer gets only $110 for stock worth $118.
In reviewing the example just given, you will notice that options payoffs are a zero sum
game. In the example just given, when at maturity the stock price is below the exercise price
both the owner and the writer receive a payoff of $0 because the option expires worthless. When
the stock price is higher than the exercise price, the payoff to the call owner is +$8, while the
payoff to the writer is −$8. This is a demonstration of why options are viewed as side bets—
regardless of the movement of the stock price, the long and the short positions always sum to
zero.
A written put works in the opposite direction of a written call. For a written put, the
writer contracts to buy the stock at the exercise price if the put owner chooses to exercise. At
maturity, if the stock price is higher than the exercise price, the put expires worthless and the
writer is relieved of the obligation, thus neither gaining nor losing. If the stock price is lower
than the exercise price, the put is exercised and the writer must buy at a higher than market price,
thus losing the difference. For the writer of a put, the payoff at maturity is the minimum of zero
and the stock price minus the exercise price.
To sum up, there are no negative payoffs for long calls and puts, and there are no positive
payoffs for written calls and puts. Remember that these payoffs do not include the money
(premium) originally paid by the owner and received by the seller for the option.
Combining Positions
There are many ways that stock and option positions may be combined to produce a
particular kind of payoff. For example, purchasing a put with an exercise price of $50 on a stock
an investor already owns guarantees a minimum value of $50 for the portfolio. If the stock price
rises above the exercise price, the put will expire worthless but the investor still owns the stock.
For instance, if the stock price goes to $60, the total portfolio payoff equals $60 (stock) + $0
(put) = $60. If the stock price drops to $40, the total portfolio payoff to the put equals $40 (stock)
+ $10 (put) = $50.
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Value of Options Prior to Maturity
The previous examples have considered the payoffs to option positions at maturity. As
we have seen, the payoff depends on only two factors: the underlying stock price and the
exercise price. When an investor buys a call or put, the price paid or the option premium is the
premium paid for the trading rights the option carries. This option price, as we noted, is not the
exercise price. What then determines the value of an option prior to maturity? The premium or
value of an option depends partly on the underlying asset price and the exercise price just
discussed. It also depends on four other factors: the volatility of the underlying asset, the
maturity of the options contract, the risk-free rate of interest, and the dividends paid on the stock.
The value of a call option prior to maturity is determined by the Black-Scholes option-pricing
model. This model will be covered in upcoming classes.
Other Commonly Used Derivatives
Warrants
Though in some ways like call options, warrants are a separate type of financial
derivative. A warrant is an option to purchase a specified number of shares of common stock at a
specified price over a specified period of time. Warrants are most often issued in connection with
bonds and are usually detachable, meaning they may be detached from the bonds and traded
separately. When issued with debt, warrants are usually used as a “sweetener” to make the deal
easier to sell or to lower the interest rate paid.
In some ways, warrants are like call options, but with a couple of differences. One is that
most warrants are long term in nature and may be exercised over several years before they
expire. Another distinct difference is that while call options are transactions solely between
investors, when a warrant is exchanged, the company of the underlying stock actually issues new
shares of the stock and receives the warrant holder’s money.
Convertible securities
A convertible security is one that can be converted at the option of the holder into another
security but of a different type. For example, a convertible bond is a bond that may be converted
at the owner’s choice to common stock. The bond contract stipulates the exchange ratio (for
example: 50 shares of stock in return for a $1,000 face-value bond). Convertible bonds are often
used by a firm as a means of obtaining a lower interest rate, reaching otherwise unattainable
outside capital, or selling stock at a premium over current market rates. Convertible securities
often carry provisions whereby the issuer may, at its discretion, call the issue for redemption,
thus increasing the firm’s control over when the owner converts. Once a security is converted,
the process cannot be reversed.
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Futures and forwards
A futures contract is a contract to buy or sell a stated asset at a specified price at a
specified time. Unlike an option, a futures contract requires the holder to act on the maturity
date; the holder does not have an option. Just as in the case of options, futures contracts are sold
on many different types of underlying assets—wheat, cocoa, and sugar, for example. A financial
futures contract is one in which the underlying asset is some type of financial security, such as
U.S. Treasury bonds, foreign currencies, and stock market indexes. For example, a long futures
contract on Spanish pesetas requires the owner to buy a specified amount of pesetas at a
specified price on a specified day.
A forward contract is the same as a futures contract except for two features. Futures
contracts are standardized and tradable on secondary markets, whereas forward contracts are
usually customized and not tradable in secondary markets. Additionally, while forward contracts
are settled at maturity, futures contracts are marked to market daily, meaning that there is a daily
cash settlement and the owner of the future takes his or her part of the gain or loss each day. For
the futures owner, it is as if the contract were settled at the end of the day and a new one opened
the next morning.
Like options contracts, futures and forward contracts provide a way to hedge against
uncertain events. A U.S. company with French franc−denominated debts coming due, for
example, may purchase a futures contract to buy French francs at a fixed exchange rate (the
dollar price of francs in the futures market) if the firm predicts a declining dollar. This futuresmarket transaction locks in the dollar cost to repay the debt and hence eliminates the exchange
rate risk. Unlike options, futures and forward contracts have no limits on their payoffs. If the
investor holds a purchase contract and the asset price rises above the exercise price, the investor
gains. If the asset price drops below the exercise price, he or she loses. Because the holder of a
futures or forward contract is obligated to exercise, the contract cannot expire worthless and the
potential loss or gain is unlimited.
Let’s look at an example of a forward contract. Suppose that an investor purchases a
three-day forward contract on deutsche marks in which she agrees to buy (German deutsche
mark) DEM1000 for $250. This is equivalent to an exchange rate of DEM4 to the U.S. dollar.
During the contract period the market exchange rate fluctuates, and at the end of the third day
(the time the contract expires), deutsche marks are trading at DEM3.8 to the dollar, raising the
market price of DEM1000 to $263.16. The investor fulfills the contract, and since the market
value of the asset has risen, he or she receives a positive payoff of $263.16 − $250 = $13.16.
That is, the investor pays $250 to purchase deutsche marks worth $263.16.
Now consider the same situation using a futures contract. Like the forward, the future is a
three-day contract for the purchase of DEM1000 for $250. But unlike the forward, the future is
marked to the market price each day. Suppose at the end of day one, the exchange rate is listed at
DEM5 per dollar. The price of the underlying asset of DEM1000 is now worth only $200, so the
futures contract is marked to market and the payoff to the investor is $200 − $250 = −$50, for a
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first day loss. At the end of day two, the dollar has declined in value and the exchange rate is
posted at DEM4.25/$. The futures contract is again marked to market and the investor receives a
payoff of $35.29 ($235.29 − $200). When the contract expires on the third day, the dollar has
again declined in value to DEM3.8/$, and the investor receives a final payoff of $263.16 −
$235.29 = $27.87. Ignoring the time value of money, the total payoff of the futures contract to
the investor is −$50 + $35.29 + $27.87, or $13.16, which is the same as the payoff to the
forward.
It is important to recognize that with these contracts, the investor has no limits to her
exposure to gain or loss. If at maturity the market price of deutsche marks had been higher, say
DEM2.5/$, the total payoff to the investor would have been even greater ($400 − $250 = $150).
Similarly, if the market price of deutsche marks had fallen below the contracted price, to
DEM6/$ for example, the investor would have to buy at a higher rate than market and the payoff
would result in a loss ($166.67 − $250 = −$83.33). Remember: there are no exposure limits on
forwards and futures.
Swaps
A swap is an agreement between two parties to exchange a series of cash flows in the
future. In essence, a swap is just a series (a portfolio) of forward contracts. Like other derivative
contracts, swaps contain a specified transaction and maturity. One type of swap is a simple
currency swap, in which two firms agree to exchange different currencies over time at a fixed
price. For example, consider a U.S. company that needs British pounds sterling (GBP) for endof-year payments over the next three years and a firm in the U.K. needing dollars for similar
purposes during the same time period. To protect against exchange rate risk, both firms would
like to lock in the price they will pay for their currency needs. Therefore, they enter into a threeyear contract in which they agree to swap a specified amount of the currencies at the end of each
of the three periods. The schedule might be something like this: year 1, $1000/GBP750; year 2,
$1000/GBP730; year 3, $1000/GBP735.
In that example, the currency exchange is the purchase price and the yearly settlement is
the maturity date for each forward in the series. There are many types of swaps available,
including more complex currency transactions, interest rate swaps, and bond swaps.
Put-Call Parity
You have seen that the payoff at maturity gives the value of an option at the time it
expires. However, we can use a particular combination of positions in options and an underlying
asset here to demonstrate put-call parity, the fundamental relationship between the price of a call
and the price of a put. Suppose an investor takes the following special position: purchase a share
of stock, purchase a put on the stock and write a call on the stock, with the exercise price on the
put and the call being equal. The payoff at maturity to this combination is constant; it equals the
exercise price. To see this, consider the following payoff in Table1.
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Table 1. Payoff at maturity.
Stock
Price
5
10
15
20
25
30
35
+
+
+
+
+
+
+
Payoff to Long Put,
Exercise Price = 20
15
10
5
0
0
0
0
+
+
+
+
+
+
+
Payoff to Short Call,
Exercise Price = 20
0
0
0
0
−5
−10
−15
=
=
=
=
=
=
=
Total Payoff
at Maturity
20
20
20
20
20
20
20
Since the payoff at maturity to this combination is guaranteed, the position is riskless.
And if the position is riskless, it must earn the risk-free rate of return. Therefore, the total price
you would have to pay today for this combination of a long stock, a long put, and a short call
must always equal what you would have to pay for another means of obtaining the same payoff.
In that case, the other method of obtaining a guaranteed (riskless) payoff would be to use the
U.S. government bond market, where we can view U.S. government interest rates as risk-free
rates. The price today (in the government bond market) of obtaining a future payoff (the exercise
price) is the future payoff discounted at the risk-free rate of return. This relationship can be
expressed as follows in Equation 1:
Stock price + Put price − Call price = Exercise price/(1 + Risk-free rate)Time to maturity
(1)
Equation 1 could also be rearranged as:
Stock price + Put price = Exercise price/(1 + Risk-free rate)Time to maturity + Call price
(2)
Equation 2 is called put-call parity. The value of the two sides must always be equal
because of the no-arbitrage condition that a risk-free investment must earn the risk-free rate of
return. Otherwise an investor could earn a riskless profit (arbitrage profit) by buying and selling
asset positions that had no risk but offered different returns. The market, however, does not allow
this type of profit opportunity to persist. Put and call prices adjust to ensure that there are no
long-term arbitrage opportunities. As long as there are no major transaction costs preventing the
arbitrage opportunities, these market forces lead to put-call parity for European options.
The put-call parity relationship is helpful because given the value of any three of its
components, we can easily find the fourth value. Thus, if we can calculate the call value and
observe the stock price and risk-free rate, we can arrive at the value of a put. For example,
consider a share of stock with a current market price of $25. If a 90-day call option on the stock
with an exercise price of $30 is priced at $1.35 and the risk-free rate of return is 7%, the value of
a 90-day put on the stock with an exercise price of $30 is:
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$25 + Put price − $1.35 = $30/(1.07)90/360
or
Put price = $30/(1.07)90/360 + $1.35 − $25 = $5.85
To reemphasize, if the put price were different from $5.85, an investor would be able to earn
arbitrage profits. But since we know that if this price were observed, the market would correct
for it we can be assured that $5.85 is an appropriate price for the put.
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