DERIVATIVES NOTES
INTRODUCTION
Thirty years ago the importance of derivative securities was not as great as today.
Globalization, the need to hedge currency and interest rate risks, as well as commodity
price risks, along with significant developments in the pricing of derivative securities has
increased their significance and usage by individuals, investment firms, and corporations.
The purpose of this seminar is to provide an introduction to the different types of derivative
securities and their applications.
Topics covered include definitions, trading strategies, and pricing techniques for each of
the three main categories of derivative securities – options, futures contracts, and SWAPS.
Enjoy, and welcome to the world of “derivatives.”
I.
OPTIONS
Basic Types of Options
There are two basic types of options - call options and put options. The definitions are;
Call Option:
A call option gives the holder (option owner) the right, but not the obligation, to buy the
underlying asset for a fixed price (strike or exercise price), on or before the expiration date.
A call option, therefore, is useful when the value of the underlying asset is greater than the
exercise price. That is, exercising the option is identical to buying at the exercise price (so
you obtain the asset) and then selling the asset for its current market value. It’s the same
as “buy low and sell high.”
Put Option:
A call option gives the holder (owner) the right, but not the obligation, to sell the underlying
asset for a fixed price (strike or exercise price), on or before the expiration date. A put
option, therefore, is useful when the value of the underlying asset is less than the exercise
price. That is, exercising the option is identical to selling at the exercise price (short selling)
and then repurchasing for its current market value. It’s the same as “sell high and buy
low.” Note the “buy low and sell high” is the same thing as “sell high and buy low,” except
that the order is reversed (which is irrelevant).
1
Payoffs to a Call Option:
Since a call option involves a right, but not an obligation (hence the name), the holder can
choose the maximum of two values, its intrinsic value, or zero. Let S be the value of the
underlying asset, and X be the exercise price. The intrinsic value of the option is then,
Intrinsic Value  MaxS  X ,0
(1a)
Equation (1a) can alternatively be written as
S  X
Intrinsic Value  
 0
if S  X
if S  X
(1b)
The logic of this valuation is that, if you have an option to buy a stock for say, $20, and it’s
price is $25, you would exercise your option and make a gain of $(25-20)=$5. If the value
of the underlying asset is, say $15, you surely wouldn’t exercise the option and pay $20 for
it. In that case you would do nothing and take a value of $0. (If you really want the stock,
call your broker and buy it for $15, not $20.)
What is not yet included is the price to obtain the option in the first place. The option
holder must pay $C (paid to the holder) to obtain the right to purchase the underlying
asset. Thus, the payoff (options terminology for “profit”) is
S  X  C if S  X
Payoff  
if S  X
 C
(2)
Payoffs to a Put Option:
Since a put option involves a right, but not an obligation, the holder can choose the
maximum of two values, its intrinsic value, or zero. Again, let S be the value of the
underlying asset, and X be the exercise price. The intrinsic value of the option is then,
Intrinsic Value  MaxX  S ,0
(3a)
Equation (2a) can alternatively be written as
 X  S if S  X
Intrinsic Value  
if S  X
 0
(3b)
The holder of this option to sell, would want to do so if the price is say, $15, because it
would be possible to buy the share on the open market for $15, and sell it to the other side
of this contract for $20 for a gain of $(20-15)=$5. If the value of the underlying asset is
$25, the holder of the option would take a value of zero because it wouldn’t make any
rational sense to sell a $25 asset for $20 (you would surely lose $5).
2
As for the call option, we must note that the put option holder must pay $P for the option.
Thus, the payoffs to the holder are the intrinsic value minus what was paid for the option,
i.e.,
 X  S  P if S  X
Payoff  
if S  X
 P
(4)
The Writer’s Perspective:
What is shown above is the perspectives of call option holders and put option holders.
These are the individuals who have purchased the option (and own them). It is they who
decide when and if the option is exercised. The other side of these transactions is called
the “writer.” These individuals do not have a choice (option). If the holder wants to
exercise their option and it costs them $40, too bad. It is also a “zero-sum” game. When
the holder makes $4, they lose $4.The intrinsic values from the “writer’s” perspectives are
Call Option:
Intrinsic Value  Max S  X ,0
(4a)
 S  X  if S  X
Intrinsic Value  
0
if S  X

(4b)
 S  X   C if S  X
Payoff  
C
if S  X

(4c)
Intrinsic Value  Max  X  S ,0
(5a)
  X  S  if S  X
Intrinsic Value  
0
if S  X

(5b)
  X  S   P if S  X
Payoff  
P
if S  X

(5c)
Put Option:
(You might ask, how does the writer ever make any money? Note that the price of the call
and put options have not been included here. What they hope to do is collect money for an
option that never gets exercised.
3
Examples:
A call option sells for C  $8 and a put option sells for P  $3 . Both options have an
exercise price of X  $45 . Determine the payoffs for holders and writers using prices 30,
35, 40, 45, 50, 55, 60 for (1) the call option, and (2) the put option.
1. Call option
Call option “holder”
S
30
35
40
45
50
55
60
Int. Value
0
0
0
0
5
10
15
Payoff
-8
-8
-8
-8
-3
2
7
Call option “writer”
S
30
35
40
45
50
55
60
Int. Value
0
0
0
0
-5
-10
-15
Payoff
8
8
8
8
3
-2
-7
Notice that the payoffs to the holder and writer are identical but opposite in sign. That is,
when the holder earns $7, the writer loses $7. This is known as a “zero-sum game.”
Call Option
20
15
Payoffs
10
5
H
0
-5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
-10
-15
-20
S
W
4
2. Put option
Put option “holder”
S
30
35
40
45
50
55
60
Int. Value
15
10
5
0
0
0
0
Payoff
12
7
2
-3
-3
-3
-3
Put option “writer”
S
30
35
40
45
50
55
60
Int. Value
-15
-10
-5
0
0
0
0
Payoff
-12
-7
-2
3
3
3
3
Put Option
60
40
Payoff
20
H
0
-20
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
W
-40
-60
S
Trading Strategies
Straddle
Buy one call and one put on the same asset with the same exercise price. Using data from
the example above.
5
S
30
35
40
45
50
55
60
Int. Value – Call
0
0
0
0
5
10
15
Int. Value – Put
15
10
5
0
0
0
0
Payoff
4
-1
-6
-11
-6
-1
4
Straddle
40
Payoff4
30
20
10
0
-10
0
10
20
30
40
50
60
70
80
-20
S
Strap
Buy two calls and one put option on the same asset with the same exercise price.
Strip
Buy two puts and one call option on the same asset with the same exercise price.
Spreads
1. Butterfly Spread
Purchase 1 call with X 1 , sell 2 calls with X 2 , and purchase 1 call with X 3 , where
X 1  X 2  X 3 , and
X 3  X 2  X 2  X1.
The cost of this portfolio is
C1  2C 2  C3 .
2. Vertical Combination
Purchase 1 call with X 2 and purchase 1 put with X 1 , where X 1  X 2 . The cost of
this portfolio is C 2  P1 .
3. Bull Spread
6
Purchase 1 call with X 1 and sell 1 call with X 2 , where X 1  X 2 . The cost of this
portfolio is C1  C2 .
4. Bear Spread
Purchase 1 call with X 2 and sell 1 call with X 1 , where X 1  X 2 . The cost of this
portfolio is C2  C1 . This is the opposite of a “bull spread.”
Portfolio Insurance
One of the most important applications of options is for portfolio insurance. This will be
emphasized again after the presentation of the Black-Scholes option pricing model. A
simpler example is discussed here. Suppose you purchase one share of stock for $45.
There is some probability that the share will lose all of its value and your loss will be 100%.
But what if you could purchase a put option on this stock with an exercise price of $45.
The put option costs $3.
S
30
35
40
45
50
55
60
Gain/loss on share
-15
-10
-5
0
5
10
15
Int. Value - Put
15
10
5
0
0
0
0
Payoff
-3
-3
-3
-3
2
7
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Instead of suffering the full loss of value, the potential loss on this portfolio is limited to
$3, which is the price of the put option. The rationale is that as the share loses one dollar
in value, the put gains one dollar in value so that the losses on the share are offset by gains
on the put. This relationship is represented in the graph below. The reader should note
that this graph resembles a “call option” and is referred to as a “synthetic call.”
7
Share + Put Option
25
20
Payoff
15
10
5
0
-5
0
10
20
30
40
50
60
70
S
Black and Scholes:
Black and Scholes (1973) provides the first mathematical valuation of options. Skipping
the 4-5 pages of integral calculus involved, they derived the value of a call option as
C  SN d 1   Xe  rT N d 2 
(6)
where,
S
is the current market value of the underlying asset
X
is the exercise price
N d  is the cumulative normal distribution evaluated at d
r
T
d1 
is the risk-free rate of interest
is the time to expiration (expressed in years)


ln S X   r   2 2 T
 T
d 2  d1   T

is the instantaneous yearly variance
Black and Scholes then used the put-call parity relationship [Stoll, 1969] to find the value
of a put option as
P  Xe  rT 1  N d 2   S 1  N d 1 
(7)
8
There are five variables that affect the value of put and call options: Their relationships
are;
C
0
S
C
0
X
C
0
 2
C
0
T
C
0
r
P
0
S
P
0
X
P
0
 2
P
0
T
P
0
r
Example:
A stock has a current price of $47, and its variance is   0.2 . The time to expiration is
T  6 mos. , the exercise price is X  50 , and the risk-free rate is r  6% . Find the price
of a call and put option.
1. Call option
d1 


ln 47 50  .06  .2 2 .5
2
.2 .5
 .1557
d 2  .1557  .2 .5  .2969
From the cumulative normal distribution, N  .1557  .4381 and N  .2969  .3833 .
Substituting these values into (6)
C  47.4381  50e .06.5 .3833  1.99
2. Put option
And substituting these values into (7)
P  50e .06.5 1  .3833  471  .4381  3.52
Hedge Ratios and the Black-Scholes OPM
In an earlier discussion it was mentioned that options could be used to “insure” portfolios.
The technical aspect of this is more, but only a little, complicated than discussed. What is
needed is to compute “hedge ratios” for the call and put options. These are denoted by
“H.”
Call option:
H C  N d1 
Put option:
H P  1  N d1 
Example:
Suppose you are the manager of a portfolio that is worth $100m. The risk-free rate of
return is 5%,   .25 , and the investment horizon is 4 years. How could put options be
used to limit the loss on this portfolio?
9
First, compute the hedge ratio for the put option,
d1 


ln 100 100  .05  .25 2 4
2
.25 4
 .65
N .65  .7422
H P  1  .7422  .2578
With this figure we can do one of two things – we can (i) hold 100m in the portfolio and
buy put options on it, or (ii) put 25.78% of the portfolio in riskless T-Bills and 74.22% left
in the portfolio. Now suppose that the portfolio loses 1% of its value.
(i)
Loss on portfolio = (.01)(100m) = 1,000,000
Gain on puts = (.2578)(1,000,000) = 257,800
Total loss = 742,200
(ii)
Loss on portfolio = (.01)(74,220,000) = 742,200
Loss on T-Bills = 0
Total loss = 742,200
II.
FUTURES CONTRACTS
What Is a Futures Contract?
A futures contract is a legally binding standardized agreement between two parties to buy
or sell a predetermined amount of a commodity, such as corn, during a specified month in
the future (the delivery month) at a price (the future price) which is determined at the time
the contract is established. This summary is for general illustration purposes only.
Please contact a commodity broker of your choice for information specific to your
operation.
Where Are Futures Contracts Traded?
Futures contracts (as well as options on futures contracts) are traded at 11 different
commodity exchanges in the U.S. as well as abroad. Futures contracts on the major
domestic agricultural crops are traded at the Chicago Board of Trade (CBOT), the Kansas
City Board of Trade, the Minneapolis Grain Exchange, the New York Cotton Exchange
and the Coffee, Sugar and Cocoa Exchange.
10
What Benefits Are Gained From Buying and Selling Futures Contracts?
One of the most important benefits gained from trading in the futures market is that traders
can assume any of a wide range of commodities or other assets with a relatively small
initial investment. The initial investment includes a commission of approximately $50 per
contract and a margin. A margin is a good faith deposit. When a trader assumes a futures
position, he or she locks in a price for future delivery for the underlying commodity. This
fixed price is the future price at which the contract is bought or sold. Subsequently, as the
price of the actual commodity rises or falls, the futures price follows suit, making or losing
money. A benefit derived from selling futures contracts is that it enables the trader to
establish, in advance, an approximate price for crops he or she intends to harvest and
market at some future time. This provides protection against dangerous price swings, and
enables speculators to profit from market fluctuations. Speculators are market participants
who have absolutely no interest in owning or selling a physical commodity, but have the
money to take on risk -- buying and selling futures contracts in hopes of making a profit.
What Is a Hedge?
A hedge is the buying or selling of a futures contract for protection against the possibility
of a price change in the physical commodity that the trader is planning to buy or sell. There
are two types of hedges: a long hedge and a short hedge. A short hedge is the selling of a
futures contract to protect the sale price of a commodity the trader is planning to sell. A
long hedge is the buying of a futures contract to protect the purchase price of a commodity
the trader is planning to buy. Most traders are able to liquidate or offset contracts prior to
delivery. The long trader can offset a futures contract by subsequently purchasing a
contract with the same delivery month. While most contracts entered into do not result in
delivery, the threat of delivery still tends to serve the purpose of keeping the prices of
futures contracts and their underlying cash market in reasonable alignment with one
another. The cash market is where physical commodities are bought and sold.
When Should Traders Hedge?
In order to successfully hedge, producers must first determine what target price they need
to cover cost of production to make a reasonable profit. Using cost of production figures
and devising a reasonable profit margin, producers can establish their target price range.
11
The target price range should be viewed as a goal that may or may not be obtained during
the market year.
An Example of the Short Hedging Process
A producer decides to explore a variety of marketing alternatives, including futures, rather
than settle for whatever the local elevator is willing to pay at harvest time. The producer's
ultimate goal is to improve his or her bottom line. The producer estimates it will cost $2 to
produce one bushel of corn. Once the producer sees corn prices in a range where a profit
can be made, he or she decides to hedge a portion of the crop by selling 1 CBOT December
corn futures contract. The standard contract size for 1 CBOT corn futures contract
equals 5,000 bushels.
By early May, CBOT December futures hit $2.60 a bushel. To lock in a selling price of
$2.60, the producer sells 1 CBOT December corn futures contract. As it turns out, the
Midwest experienced a bumper crop year. Corn yields were above normal, causing prices
to drop. By harvest, corn prices fell to $1.90 a bushel. The producer offsets his/hers futures
position by purchasing 1 CBOT December corn futures contract. The result of the
producer's hedging activities were:
CASH MARKET
FUTURES MARKET
May: Plans to sell 5,000 bushels of corn; sells Sells 1 CBOT December corn futures
1 CBOT December corn
contract @ $2.60/bu
October: Sells 5,000 bushels of corn in the Buys 1 CBOT December corn futures
cash market @ $ 1.90/bu
contract @ $ 1.90/bu
Sales price of corn
$1.90/bu
Plus futures gain ($2.60 - $1.90)
$0.70/bu
Net sale price
$2.60/bu
By using CBOT corn futures, the producer increased his or her final sales price from $1.90
to $2.60 a bushel. The producer accomplished his or her goal. Better yet, the final sale was
60 cents a bushel higher than his or her production expenses.
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Futures Pricing (An “Arbitrage” Approach)
One of the basic principles of finance is that two identical assets should have the same rate
of return, and the same price. To this extent, suppose we have the situation shown below;
-
A 77-day T-Bill that yields 14%
-
A 167-day T-Bill that yields 12%
-
A 77-day futures contract that delivers a 90-day T-Bill with a yield of 8%.
There are two ways to have a “T-Bill” investment that lasts for 167 days. The first is to
buy a 167-day T-Bill. The second way is to buy a 77-day T-Bill and a futures contract that
delivers a 90-day T-Bill, 77 days from now. According to theory, the yields on these two
approaches should be the same.
Currently:
?
1  .12360 1  .14360 1  .08360
167
77
90

1.05398  1.04840
Since the left-hand-side has a higher yield, an individual would “buy” the 167-day T-Bill.
The money to do this would be obtained by selling the right-hand-side.
The price of the 167-day T-Bill is
P0 
1m
1  .12
167
360
 $948,786.08
Thus, sell $948,786.08 worth of 77-day T-Bills, and, simultaneously, take a short position
in a futures contract that calls for delivery of a 90-day T-Bill 77 days from now. At time
77, the price of a 90-day T-Bill is
P0 
1m
1  .12
90
360
 $980,943.65
Also, at time 77, the value of the 77-day T-Bills originally sold short is
P0  $948,786.081  .14 360  $975,752.39
77
13
Cash flows at time “0”
Short 1 77-day T-Bill in an amount of
$948,786.08
Use proceeds to but 1 167-day T-Bill for
$948,786.08
Total
$0
Cash flows at time “77”
Deliver 1 90-day T-Bill worth
$975,752.39
Receive payment for 90-day T-Bill
$980,943.65
Total
$5,191.26
To prevent this arbitrage, the futures contract on the T-Bill must have a yield of
1  .12 360  1  .14 360 1  F  360
167
77
90
F  .103056
Thus, the price of the futures contract must be
P0 
1m
1  .103056
90
360
 $975,752.39
Futures and Hedging
Futures contracts can be used for hedging purposes in much the same manner as options.
Example:
This is an example of using Interest Rate Futures to create a “short hedge.”
Charlotte Insurance Co. plans to satisfy cash needs in six months by selling its Treasury
bond holdings for $5m at that time. It is concerned that interest rates might increase over
the next three months, which would reduce the market value of the bonds by the time they
are sold. To hedge against this possibility this possibility, Charlotte plans to sell Treasury
bond futures. It sells 50 contracts ($100,000 per contract) for 98-16 (or 98 16/32 of par
value).
Now suppose that the actual price of the futures contract declined to 94-16 because of an
increase in interest rates. Charlotte can close out its short futures position by purchasing
contracts identical to those it has sold. If it purchases 50 contracts at the prevailing price,
its profit per contract would be
Selling price
$98,500
(98.5% of $100,000)
Purchase price
$94,500
(94.5% of $100,000)
Profit
$4,000
14
The gain on the futures contracts will help offset the losses on its bond holdings.
III.
SWAPS
Interest Rate SWAPS
A SWAP arrangement obligates two counterparties to exchange cash flows at one or more
future dates. For example, a foreign exchange SWAP might call for one party to exchange
$1.6m for £1m in each of the next five years. An interest rate SWAP with notational
principal of $1m might call for one party to exchange a variable cash flow equal to $1m
times LIBOR for $1m times a fixed rate of 8%.
Consider, for example, a firm that has issued long-term bonds with total par value of $10m
at a fixed interest rate of 8%. The firm is obligated to make interest payments of $800,000
per year. However, it can change the nature of its interest rate obligations from fixed to
floating by entering a SWAP agreement.
A SWAP with notional principal of $10m that exchanges LIBOR for an 8% fixed rate will
bring the firm fixed cash inflows of $800,000 per year and obligate it to pay instead $10m
x rLIBOR . The receipt of the fixed payments from the SWAP agreement offsets the firm’s
interest rate obligations on the outstanding bond issue, leaving it with a net obligation to
make floating rate payments.
Suppose that the SWAP is for three years and the LIBOR rates turn out to be 7%, 8%, and
9%.
Floating rate payments are;
Year 1
$700,000
Year 2
$800,000
Year 3
$900,000
Fixed payments are $800,000 for each year. Thus, in the first year the fixed rate payer
would remit $100,000 to the floating rate payer. In the second year there would be no
transfer, and in the third year the floating rate payer would remit $100,000 to the fixed rate
payer.
On the other hand, the firm could have retired its fixed-rate debt and reissued floating rate
debt. But, this is more expensive.
15
Currency SWAPS
Consider a SWAP agreement to exchange dollars for pounds for one period only. Next
year, for example, one might exchange $1m for £.6m. This is no more than a simple
forward contract in foreign exchange. The dollar paying party is contracting to buy British
pounds in one year’s time, for a number of dollars agreed upon today. The forward
exchange rate is F1  $1.67 / pound . We know from the interest rate parity theorem that
this price should be related to the spot exchange rate, E0 , by the following formula
F1  E0 1  Rus 1  RUK 
This relationship allows us to determine a fair SWAP rate.