Profit and Loss Analysis© Dallas Brozik, Marshall University

All financial contracts are just combinations of the three basic contracts, spot,

forward, and option. The problem with understanding financial contracts is that they can

contain many of these basic building blocks, so it can be confusing to understand what is

happening with the entire contract. One way to understand any financial contract is to use

a graphical technique called profit and loss analysis. The profit and loss of a complex

financial contract is merely the sum of the components of that contract.

Spot ContractThis one is really easy. There is no profit and loss diagram for a spot contract. A

spot contract is executed immediately, and then it is over. The price that was paid

becomes a matter of record. Profit and loss diagrams are only useful when there is a time

dimension in the contract. Most financial contracts include a time dimension. This is not

to say that there are no spot contracts in finance. The purchase or sale of an asset for

immediate delivery is a spot contract. The spot transaction often creates the other

positions, but in and of itself the spot contract has no future.

Forward ContractA forward contract is the most common type of contract used. This contract can

have one side, a long position, or two sides, long and short positions. The easiest way to

understand the profit and loss of such contracts is to consider simple cases. Assume that

a person owns some "thing". This "thing" could be an automobile, a Picasso painting, or

a share of stock. The profit or loss that is realized on this investment is dependent on what

happens to the value of the "thing" in the market. This value can include factors like wear-

and-tear, rarity, or even the state of the overall economy. If the original price of the “thing”

was $100 and the investor purchased the “thing” (this could be a spot transaction), the

investor has control of the "thing". This is called a long position, the profit and loss diagram

would look like this.

If the market price of the "thing" increases from $100 to $10, the owner has made

a $10 profit, but this profit will not be realized unless the "thing" is sold. If the market price

declines to $90, the owner has a loss of $10, again realized only if the "thing" is sold. No

one can control what the market price of a "thing" will be since that price is a function of a

lot of factors and how they affect other market participants. But if a person is shrewd and

can estimate what the market will do, he or she can buy a "thing" for a low price and wait

until the market price goes up. Then the "thing" can be sold for a profit. This is the idea

behind the old saying "Buy low and sell high."

Note that in order to make a profit, the market price of the "thing" must go up. In the

case of some "things", a person can be pretty sure that the price will go up, eventually. A

piece of timberland has value as land and because of the crop of growing trees, but the

owner may have to wait a while to make a profit before selling either the timber, the land,

or both. Some "things" might be quite speculative in nature, like fine art. If the market

wants to pay in excess of $40 million for a painting of a bunch of flowers on a given day,

there is no guarantee that the same painting will hold that value over the years, especially

if the Surgeon General reports that simulated flowers cause simulated hay fever.

Items that are considered collectibles are especially subject to the vagaries of the

market. If Ellen purchases a delivery truck and uses it her business, the truck will help

generate cash flows by allowing her to serve a wider clientele. If Ellen uses her money to

purchase a classic Edsel, she will have to rely on someone out there wanting that vehicle

badly enough to pay her more than she paid for it. The pricing of collectibles can be

regarded as the "Greater Fool Theory". In order to make money on collectibles, there has

to be someone more foolish than the owner who wants the "thing" more badly. Regardless

of whether a person's fortune is based on production or personal preference, the profit and

loss diagrams are the same.

Certain types of forward contracts have two sides, a long and a short position. The

short position obligates the holder to deliver the item specified in the contract while the long

position obligates its holder to receive the item specified in the contract. Short positions

are used by individuals or firms who produce "things" since the contract guarantees the

producer a price for a specified delivery. Long positions are used by individuals or firms

that need the "things" being produced, either as an intermediate or end product. The long

position guarantees the buyer a delivery of the "things" at a specified time and place.

The most common of this type of forward contract is a commodity futures contract,

though other types exist. A farmer may choose to sell his corn crop forward, that is, he

promises to deliver a certain number of bushels of corn for a certain price, say $3 a bushel.

Another party, like a big cereal mill, might choose to purchase the corn for delivery after it

is harvested. Note that by making this contract both parties have eliminated price risk;

each side knows the exact price of the corn that is subject to this contract. If the market

price of corn changes, it makes no difference for the parties in the contract. The obligation

of completing the contract at the price specified in this futures (forward) contract will not

change. This is called a hedge. Both sides are locked in to the contract.

The cereal company is on the side of the contract that obligates it to buy corn for $3

a bushel. If there is a drought and the market price of corn rises to $4 per bushel, the

contract has become profitable to the cereal company. The contract plus $3 per bushel

gets corn worth $4 per bushel. The contract itself is worth $1 per bushel. If there is a

bumper crop of corn and the market price falls to $2 per bushel, the cereal mill is still

obligated to pay $3 per bushel, so the contract costs the firm $1 per bushel.

The value of the forward contract depends on the price of the underlying commodity,

and the only reason the forward contract has value is because it can be traded separately

from the commodity. When the forward contract and the underlying commodity are

considered as a single unit, there is no price risk for that complete unit. Inside the unit, the

prices of the two components might change, but they change in opposite directions by

equal amounts. The profit and loss diagram for the cereal company holding the long

futures (forward) contract is

The farmer is on the opposite side of the contract from the cereal mill. Since the

farmer must deliver the corn, this is called a short position. The farmer is obligated to sell

corn at $3 per bushel, regardless of market price. If the price of corn goes to $4 per

bushel, the farmer will have lost a dollar of market value since he must sell for $3 per

bushel. If the price of corn drops to $2 per bushel, the farmer will make $1 per bushel

above market value solely because of the contract. The profit and loss diagram for this

short position in the futures (forward) contract is

The long and short forward contracts are opposites. The profit and loss diagrams

show that the gain of one party is exactly matched by the loss of the other party. This is

an important aspect of forward contracts. The gain from one party is exactly the opposite

of the other party. The way to see that the contracts are opposite is to add them together,

point by point. For each value along the x-axis of the graph, add the value of both the long

and short positions together. The process of adding together profit and loss diagrams is

very important in understanding how complex financial contracts really work. In this case,

the sum of this long forward contract and this short forward contract would be:

+

1 In all types of mathematical processes, some numbers may have opposite effects whencombined by a specific operation. For example, +100 and -100 are opposites underaddition since their combination equals 0, the additive identity. Under multiplication thesetwo numbers are not opposites since the result of that operation does not produce themultiplicative identity of 1. When performing subtraction, +100 and -100 are not oppositessince the result of the operation is nonzero, and under division they are not opposites sincethe result of the operation is -1, which is the negative of the multiplicative identity (close butno banana). In none of the possible mathematical operations would the values "cancel"out. Stamps and appointments are canceled, not numbers. Numbers offset.

The resulting profit and loss line for the sum of the two contracts is the x-axis which

represents no gain and no loss. For each market value of the underlying commodity, the

profit on one contract is exactly offset by the profit on the other contract. This is an

important usage of the word opposite. Contracts that are opposites have offsetting effects.

Anyone who held both the long and short side of the same forward contract would have no

net gain or loss when the contracts matured. Another, much poorer, way to say this is that

opposite contracts "cancel each other out" when held together1.

It might seem strange that people would willingly involve themselves in contracts of

this type. Forward contracts can be used to eliminate risk, in this case the variability of the

price of corn. Sometimes the participants in the corn market, both producers and end

users, feel that risk reduction is worth the possibility of losing a little potential profit. The

farmer has a guaranteed price for his corn that will lock in his profit, and the cereal mill has

a guaranteed cost for the corn which allows the mill to plan production and costs. Both

sides can benefit from this type of contract since it acts as insurance for both parties.

The concept of risk in forward contracts deserves closer examination. If a farmer

has corn to sell, he is "long" in corn. Notice that this long position occurs naturally since

the farmer is in the business of growing corn. Any fluctuation in market price will cause the

farmer to have a profit or loss in corn relative to his starting position. If the market price of

corn today is $3.00 a bushel, any change in market price will generate a profit or loss for

the farmer relative to his position today. The forward contract locks in today's position and

removes risk, but for the contract to be able to do this, it must be an "opposite" of the corn.

For a farmer who has sold his corn forward, the combination of the corn (a long position)

and the forward contract (a short position) is riskless since the combination of the two

positions produces the same results regardless of market activity. But if the corn is

separated from the forward contract, both still have risk since market activity will change

the value of both positions. Each increase in the price of corn is offset by a decrease in the

value of the forward contract.

When used for hedging purposes, forward contracts remove the risk from "natural"

positions by creating a contract opposite to the "natural" position. A farmer who is

"naturally" long in corn, can remove the risk from that position by selling the corn forward

and creating a short position. A cereal mill that is "naturally" short in corn, can remove that

risk by purchasing corn forward with a long contract. The long and short contracts are

opposites and have offsetting effects for both parties. If a party without a natural position

holds a forward contract, this party could bear the risk of an unmatched long or short

position. This is what happens to speculators in the futures markets (futures contracts are

special types of forward contracts). These individuals take forward positions even though

they do not have a "natural" position, and they bear the risk of these contracts.

Option ContractThe profit and loss diagrams for option contracts are actually pieces of the profit and

loss diagrams for forward contracts. An option contract allows the participants to have only

the up or down side of either the long or short positions, so there are four profit and loss

diagrams for option contracts.

Put Options

Assume there is a farmer would like to have a contract for selling his corn after

harvest, but he would like to have the best deal possible. If the price of corn drops, he

would like to have a guaranteed minimum price, but if the price of corn increased, he would

like to have the right to sell his corn on the market. Under an ordinary short futures

(forward) contract, the farmer would be protected if the market price of corn fell, but if the

price increased he would be obligated to sell at a below-market price. What this farmer

really wants is a contract with a profit and loss diagram that looks like the left half of the

profit and loss diagram for the short forward contract. This way he has a contract that

removes risk if the market moves against him but does not remove risk if the market moves

in his favor. This would be the best of all possible worlds for someone in a "naturally" long

position. The farmer really wants a contract that looks like this:

This ideal contract is called a long put option because the holder of the contract, the

"long" position, controls the right to invoke the contract or not. This is a slightly different

meaning of "long" than was used in forward contracts because it refers to control rather

than taking delivery. It is called a put option because the holder of the contract has the

right to "put" the item in contract, in this case corn, in the hands of a buyer under conditions

of the contract should the seller choose to do so. The price at which the corn can be sold

is called the exercise price. The holder of the long put is protected from bad things and

gets to keep good things.

For this type of contract to exist, there must be some other party willing to take the

other side. That party would face a profit and loss diagram exactly opposite of the one

above, and it would look like:

This "ideal" short put option is an absolutely miserable contract to hold. At best, the

holder of the contract makes no profit, and if the contract is invoked losses will be suffered.

The holder of this contract does not even have control. This is still a put option, since the

holder can have the "thing" put to him, but it is a short position since the holder of the

contract has no control. This is roughly equivalent to standing in the middle of a railroad

track just to see if a train might come along. In this form, this is a really bad contract.

As might be expected, no rational players want anything to do with this type of option

contract. This could mean that there would be no put options available, but there are still

people like the farmer who want to have the best of both worlds, guaranteed prices and

market prices, whichever is higher. In order to get someone to take the short position of

a put contract, the person who wants to have a long position must offer an incentive (bribe).

If someone is going to stand in the middle of a railroad track and take the chance of being

hit by an adverse market price movement, it has to be worth his while. But even this is not

enough. The person who is being coaxed onto the track knows that a train will come along

eventually, and that this is still a losing proposition. So a time limit is put on the contract.

The seller of the contract takes some money from the buyer and only has to stand on the

track for a limited amount of time.

The buyer of the contract spends money to get the contract and is in the long

position because he controls whether the contract will be invoked or not. The seller of the

contract takes money to stand in the middle of the railroad track for a specified amount of

time. The amount of money that is required for the contract depends on how likely the

seller thinks it is that a train will come along in the specified time. This train may be

powered by engines like the weather, the economy, or the performance of some other

contract. The seller of the contract will be influential in deciding how much money is

necessary for him to climb onto the tracks; if the buyer does not offer enough, there will be

no contract.

The effect of this exchange of money between the buyer and the seller of the put

option is a spot contract and has the effect of shifting the profit and loss diagrams for both

the buyer and seller. Regardless of what happens, the buyer paid money to the seller. The

seller gets to keep this money whether or not the train rolls over him.

Assume that the price of the contract is $.50 per bushel and that the price at which

the contract can be invoked is $3 per bushel. This price at which the contract can be

invoked is called the exercise price or the strike price. The diagrams now look like:

If the market price of corn has risen to above $3 per bushel by the end of the

contract, the farmer will choose to let the option contract lapse and sell the corn on the

open market. In this case, the farmer is able to sell his corn in the market for more than the

price specified in the option contract, but the farmer (long) has lost $.50 per bushel and the

seller (short) has made $.50 per bushel on the option contract. The train did not hit the guy

standing on the track. There may have been a train coming, and it may even have been

visible, but the seller got to step off the track before the train got there due to the time limit

in the contract.

If the market price of corn drops to $2 per bushel, the farmer (the long put holder)

will choose to invoke the contract at the exercise price of $3 per bushel. The seller just got

hit by the train. The seller of the contract (the short put position) must now buy corn for $3

per bushel, which represents a loss of $1 per bushel relative to the market price. But the

seller was previously paid $.50 per bushel for the contract, so the net loss to the seller is

only $.50. The farmer makes $1 per bushel relative to the market, but since he paid $.50

per bushel for the contract, his net profit on the contract is only $.50 per bushel.

Had the market price only dropped to $2.50 per bushel, the contract would have still

been invoked, but neither party would have made or lost any money. The farmer's gain of

$.50 per bushel on the sale of the corn would have just offset the price paid for the contract.

Similarly, the seller of the contract would pay the excess price for the corn with the money

originally given him by the farmer in exchange for the contract.

The long and short positions of the put option are opposites. If the profit and loss

diagrams from the two sides are summed, the resultant profit and loss diagram would have

a value of zero for every point on the x-axis. Even the existence of the price paid to the

seller from the buyer makes no difference in the summation since one position is exactly

offset by another.

Comparison of the profit and loss diagrams for the long and short positions of the

forward contract and the long and short positions of the put option shows that something

is missing. The shape of the profit and loss diagram for a long put option is similar to the

left half of the profit and loss diagram for the short forward contract. The profit and loss

diagram for the short put contract is similar to the left half of the long put contract. The put

options also have an offset because there was a cost involved in making the seller stand

on the railroad track, but the 45o portions of the diagrams are the same. The put options

are really the left portion of the forward contracts, broken at the exercise price.

Call Options

The right portions of the forward contracts also represent valid option contracts, the

call options. Suppose the purchasing manager for a cereal manufacturer has the job to

make sure that enough corn is available to make the corn flakes that the company sells to

grocery stores. Once he knows how much corn is needed, he could enter into enough

futures (forward) contracts to assure delivery at the specified times at specified prices. But

his performance will be evaluated not only by how well he does in getting the corn but also

by how cheaply he can get the corn.

The ideal contract for the purchasing manager would be one that allowed him to take

delivery of corn at a fixed price, unless the price of the corn dropped, in which case he

would like to pay the lower price. Since he wants to have the ability to "call" the corn away

from the party on the other side of the contract, he would be want to have a long call option.

The call gives him the right to take corn away from someone else, and the long position

gives him the right to control whether or not the contract is invoked. An ideal long call

option, from his point of view, would have a profit and loss diagram like this:

This long call contract would give him the right to purchase corn at $3 per bushel.

If the market price of corn rose to $4 a bushel, he would invoke the contract and save

(make a profit relative to the market) $1 per bushel. If the market price dropped below $3

a bushel, he would let the contract lapse and purchase the corn on the open market. Of

course, for the purchasing manager to have a long position with a contract, someone else

must have a short position, and for this perfect contract, the profit and loss diagram for the

short position would look like:

Just like with the put contract, the person on the short side of a perfect call contract

would never make a profit. And just like with a put contract, if someone is going to be

induced to stand on the railroad track, an inducement must be provided by giving him

money and setting a time limit for the contract.

Assume that someone is willing to stand in front of the train for a cash payment of

$.75 per bushel for a contract with the exercise price of $3 per bushel. The profit and loss

diagrams for the long and short call contracts look like:

The profitability analysis is along the same lines as with a put option. If the market

price for corn at the end of the contract is $4 per bushel, the holder of the long position will

call the corn away from the holder of the short position for $3 per bushel. But since the

seller of the call (the short position) had already received $.75 per bushel, his overall loss

was only $.25 on a bushel of corn worth $4. The holder of the call paid a net price of $3.75

a bushel for the corn ($3.00 for the corn and $.75 for the contract that set the price at $3.00

per bushel) which is worth $4 a bushel. The holder saved (made a profit) of $.25 per

bushel of corn.

Any time the market price of corn is above the exercise price of $3 per bushel, the

contract will be invoked and the holder of the long call will require the holder of the short

call to deliver the corn at $3 per bushel. The net price for this corn is actually $3.75, and

the price of corn must rise above $3.75 per bushel before the buyer of the corn profits. In

the event that the market price for corn is below $3 per bushel, the contract will be allowed

to lapse. The seller of the contract winds up richer by $.75 per bushel, but the buyer of the

contract had a type of "insurance policy" against the price of corn going too high.

The long call and the short call are opposite contracts. There cannot be a long call

without a short call, and the sum total of the profit and loss diagrams for both contracts is

zero, just as it was with the put option contracts and with forward contracts. The long call

option contract is the equivalent of the right side of the long forward contract (with an offset

due to the price paid the seller), and the short call option contract is the equivalent of the

right side of the short forward contract (with an offset due to the money received from the

buyer).

The risk characteristics of options differ from those of forward contracts. Forward

contracts are able to remove risk from "natural" positions by creating a contract with

offsetting effects. Since an option contract is only a partial forward contract, some of the

offsetting effects are missing. The holders of the options, the long positions, have paid

money to have the privilege to bear that portion of the market risk which will be beneficial

to them. The long option holders have purchased the right to the "upside" risk. The short

option positions effectively sell their upside risk, but they are still exposed to "downside"

risk.

The short option holders are no fools, though. If they are going to stay in the game,

they must win at least as much as they lose. The price paid to the sellers of options, the

short positions, must cover the losses realized when options are exercised. Since the short

position has the potential for limited gain but unlimited loss, this means that the short option

positions must win the bet with the train most of the time. If they did not, pretty soon there

would be no one to stand on the tracks because they would all be broke, dead broke. The

sellers of the options are paid to bear the "downside" risk associated with the contracts, and

they are paid enough to cover their losses.

Puts vs. Calls

It is extremely important to note that put contracts and call contracts are not

opposites. The profit and loss diagram for a long call is the mirror image of a long put if

and only if both contracts have the same exercise price and the same amount of money

is paid to make the contract worthwhile to the sellers of the contracts. More importantly,

a put contract and a call contract do not arise simultaneously. The buying of a long put

means that someone else is accepting a short put, not that someone else is creating a call.

Put options and call options are totally different contracts; they may look like mirror images

under certain circumstances, but they are not opposites.

A proof that the put and call contracts are not opposites can be seen by examining

the sum of both contracts. Suppose that in the market there are offered for sale both put

options and call options for corn. Further assume that the put option has an exercise price

of $3 per bushel at a cost of $.50 per bushel and that the call option has an exercise price

of $3 per bushel at a cost of $.75 per bushel.

If a put option were the opposite of a call option, someone who bought both a call

and a put should find himself in a neutral position. Since this individual would hold a long

call option and a long put option, his overall profit and loss diagram would simply be the

sum of both of component profit and loss diagrams. The sum of the contracts can be found

by graphing the profit and loss diagrams for both contracts and adding them together point

by point. The resultant profit and loss diagram from a sum of the two would look like:

+

The resultant profit and loss diagram is definitely not zero at all points; the put option

and call option are definitely not opposites. In fact, in this combination they created a

contract with financial characteristics entirely different than either the put or call option.

This resultant contract shows how complex financial contracts can be analyzed. All

complex contracts are combinations of basic contracts. The combination of a long put

option and a long call option at the same strike price is called a long straddle, and for the

put and call contracts described above, the profit and loss diagram of the combined

contracts is the profit and loss diagram for a long straddle. All long straddle profit and loss

diagrams will have this same basic "V" shape though the numeric values at which the "V"

crosses the x-axis will be determined by the exercise prices of the contracts and the price

paid for each.

More complex financial contracts might require a few more steps of adding

component profit and loss diagrams, but if taken in a stepwise fashion the overall profit and

loss diagram for even the messiest financial contract can be determined. Once the profit

and loss diagram is known, the investor has information that can be valuable in deciding

whether or not a certain investment is appropriate. For example, if the investor does not

think that corn will rise to above $4.25 per bushel or drop to below $1.75 per bushel, a

straddle of this type will only result in a loss for the buyer of the straddle.

To illustrate the analytical power of profit and loss diagrams, consider the following

situation. Farmer Jones, who is naturally long in corn, would like to protect himself from

possible declines in the price of corn, and he can do so by buying a call option. But he

would like to get this protection at a reduced cost, which means he must sell some contract

in order to make back some of the money he spent on the call option. The only way he can

make money is to sell an option, but because Farmer Jones does not want to have to buy

any corn (have corn put to him), he must sell a call option. The deal can be made to work

since options with different exercise prices sell for different prices.

Assume that in the market corn is currently selling for $3.00 per bushel. Further

assume that a call option with an exercise price of $3.00 per bushel sells for $.50 per

bushel and that a call option with an exercise price of $4.00 per bushel sells for $.25 per

bushel. The reason that the call option with the $4.00 exercise price sells for less than the

call option with the $3.00 exercise price is because the market price of corn is already

$3.00 a bushel. The price of corn will be more likely to exceed $3.00 than $4.00 within the

period of the contract since the price of corn is already at $3.00. The train (the market price

of the corn) is further away from $4.00 a bushel than it is from $3.00 a bushel, so there is

less risk for the seller of the contract. Since there is less risk for the seller, it should cost

less to get him to climb onto the tracks.

There could also be call options with exercise prices lower than the current market

price, and these would have prices that reflect that they currently have value. For example,

if a call option on corn had an exercise price of $2.00 per bushel while the market price of

corn was $3.00 per bushel, the contract would have an exercise price of at least $1.00 per

bushel. This contract has a $1.00 profit already “built in” since the corn can be bought for

$2.00 per bushel and resold immediately for $3.00 per bushel. No rational player would

sell the this contract for less than $1.00 per bushel, and the price would be higher if there

was any time left in the contract during which the price could change even further. Since

Farmer Jones is seeking protection at a low price, these contracts would probably not be

of interest to him.

Assume there are also put options on corn available in the market, and the put

option with an exercise price of $3.00 per bushel sells for $.75 per bushel, and the put

option with an exercise price of $2.00 per bushel sells for $.10 per bushel. There could

also be put options available at other exercise prices, and these prices would reflect the

exercise price, the current price of corn, the time to expiration of the contract, and the

market's expectations of what will happen to the price of corn during the period of the

contract.

Back to Farmer Jones. His big fear is that the price of corn will drop below $3.00 a

bushel, but he thinks there is a chance that the price could go up. He would like to protect

himself against a drop in market price by buying a put option, but he must decide which

option to buy. He really wants to guarantee a price of $3.00 per bushel, so he has to buy

a put option with an exercise price of $3.00, and this contract will cost him $.75 per bushel.

With the put option, Farmer Jones has control of a contract that will allow him to force

another party to pay $3.00 per bushel for the corn should Farmer Jones choose to invoke

the contract. If the price of corn falls, he has a guaranteed $3.00 per bushel price. If corn

is selling for more than $3.00 per bushel when the contract expires, Farmer Jones will not

invoke the contract and sell his corn on the market for the higher price.

If the price of this put option cuts into Farmer Jones' profit too much, he can sell a

call option order to recover some of the money. By selling the call option he makes a

contract for someone else to take his corn from him at the price specified in the option

contract. If he feels that it is unlikely that the price of corn will go above $4.00 per bushel,

Farmer Jones also feels it is unlikely that anyone would call the corn away from him at

$4.00 per bushel. If he sells a call option with an exercise price of $4.00 per bushel,

Farmer Jones really expects to be able to keep the money and still have his corn after the

option expires.

Notice that this combination of options does not create a conflict for Farmer Jones

and his corn. If the price of corn does fall below $3.00 per bushel, the party holding the call

against the corn will not make Jones sell his corn for $4.00 per bushel (though Jones might

wish this would happen). Since there is no claim on the corn, Jones is free to use the put

option which he controls to force someone else to pay him $3.00 per bushel. If the price

of corn goes above $4.00 per bushel, Farmer Jones will be forced to sell his corn for $4.00

per bushel, but he would not want to use the put option to sell the corn for $3.00 per bushel

anyway. These two contracts with this set of strike prices will not cause any problems for

selling the corn, but if option contracts are used inappropriately conflicts could occur.

The purchase of the put option sets a floor price for Farmer Jones' corn at $3.00 per

bushel. The sale of the call option sets a ceiling price for the corn of $4.00 per bushel. If

the price of corn is between $3.00 and $4.00 per bushel when the contracts expire, neither

will be invoked, and Farmer Jones will be able to sell his corn at the market price.

Farmer Jones really has three positions in the corn market.

1) He has corn which can be sold at the market price.

2) He controls a put option which allows him to sell his corn for $3.00 per bushel.

3) He has sold a call option which can force him to sell his corn for $4.00 per bushel.

This combination of corn and options might seem confusing, but graphical analysis

provides an easy way to see what the net result of all contracts and market positions will

be. The first step in the analysis is to construct the profit and loss diagram for each of the

market positions relative to the price of corn today, which happens to be $3.00 per bushel.

Farmer Jones has corn that currently sells for $3.00 per bushel. Any change in the

market price of corn will create either a profit or loss on this corn. This is a long forward

position, and the profit and loss diagram looks like this.

Farmer Jones also controls a put option (he is long the put option) that will allow him

to sell his corn for $3.00 per bushel, if he chooses to exercise his option. He paid $.75 per

bushel for this option, and his profit and loss diagram for this position looks like this.

Farmer Jones sold a call option (he is short the call option) which will allow another

party to take his corn away from him for $4.00 per bushel. He was paid $.25 per bushel

to take this position, so his profit and loss diagram is

Farmer Jones' overall position in the market is the sum total of these three

component positions. This position is found by adding the three components together

point-by-point as was done in the previous example. What this means is that all the profit

and loss diagrams must have the same scale on the x-axis and then aligned vertically. For

each point on the overall position graph, simply add the values of the component graphs

at that point. It is useful to note that since the component graphs are composed of straight

lines, the overall position graph will also be composed of straight lines. It is also worth note

that the only time the overall position graph can only change direction at a point where one

of the component graphs changes direction. The overall position profit and loss diagram

would look like this:

Long

Forward

Position

Long

Put

Option

Position

Short

Call

Option

Position

Overall

Position

When corn is selling for $2.00 per bushel, Farmer Jones has lost $1.00 per bushel

on the corn but made $.25 bushel on both the put and the call options. His net position is

a loss of $.50 per bushel relative to $3.00. Another way to analyze what happens at this

price is to consider what will actually happen to the corn. Since Farmer Jones is long the

put option, he will force the other party to the contract to buy the corn for $3.00 per bushel.

This put option cost Farmer Jones $.75 per bushel, so his net position is so far a loss of

$.75 per bushel. But Jones had sold a call option which would allow someone else to buy

his corn for $4.00 per bushel. Since corn is selling for only half that price, the other party

will not exercise the option; if he really wants corn he will buy it on the open market.

Farmer Jones gets to keep the $.25 per bushel he charged for selling the call option, so his

net position for his corn and both option contracts is a loss of $.50 per bushel relative to

$3.00. This situation will hold for all prices up to $3.00 per bushel since Jones will exercise

his put option and the call option will not be exercised against him. Once he sells the corn,

he will net $2.50 per bushel.

When corn is selling for $3.00 per bushel, neither option will be exercised. The

holder of the call option will not pay Farmer Jones $4.00 per bushel for corn that is only

worth $3.00 per bushel. Farmer Jones will not force the other party to buy his corn for

$3.00 per bushel because it is simpler just to sell the corn on the open market than to

arrange for the sale and delivery of the corn through the option contract. Jones will sell the

corn outright for $3.00 per bushel, but this profit will be offset somewhat by the cost of the

put option ($.75 per bushel) and the income from the sale of the call option ($.25 per

bushel). Once he sells the corn, he will net $2.50 per bushel.

If the price of corn is between $3.00 and $4.00 per bushel, neither option will be

exercised. Farmer Jones will not use the put option to sell his corn at a below market price,

and the holder of the call option will not pay Farmer Jones a premium for the corn. Jones

will sell his corn for the market price, and his profit will be offset by the price of the put

option and the income from the call option. Since these two options have a net cost of

$.50, Farmer Jones' overall net profit will always be $.50 less than the market price of the

corn.

If corn should sell for $4.00 per bushel or more, Farmer Jones will have the corn

called away at that price by the holder of the call option. Jones will not even think about

exercising the put option since he would not want to sell the corn for half of its value and,

besides, he will not have the corn since it has been called away from him. He will be paid

$4.00 per bushel for the corn, but this profit will be reduced by the $.50 per bushel net cost

of both option contracts. His net profit relative to the starting price of $3.00 per bushel will

be $.50 per bushel. Once he sells the corn, he will net $3.50 per bushel.

It is important to note that the previous analysis was done relative to the market

price of the corn when Farmer Jones entered into the contracts. This was a valid reference

point since it allowed Jones to understand what would happen to his profits relative to the

point where he was at the time of the deals. It is also possible to structure these diagrams

based on the absolute value of the corn. This only effects the Long Forward Position by

scaling the y-axis to be the price of corn, as shown below.

This approach has no effect on the profit and loss curves for the option contracts

since they already had an absolute zero reference point on the y-axis. The overall curve

will shift, however, to reflect the change in the Long Forward Position and will look like this:

It makes no difference whether the relative or absolute scaling is used, both

techniques provide the same information. In the example above, Farmer Jones was able

to use a put option to establish a floor price $2.50 per bushel for his corn (net of all

expenses), and his use of a call option to reduce his overall expense created a net ceiling

price of $3.50 per bushel for the corn. The use of the profit and loss diagrams allows the

combined effect of all market positions to be seen. Had this overall position been

unsatisfactory, Farmer Jones could have sought other contracts and combinations that

would have been more to his liking.

The value of the profit and loss diagrams is that they unambiguously show Farmer

Jones' position for all market values of his corn. He will have no surprises when it comes

time to sell the corn. Note that since there are different levels of possible outcome Farmer

Jones is still at risk. The probabilities of the various outcomes have not been specified,

however, so it is not possible to quantify the amount of risk Farmer Jones will bear. In real

applications, it might be possible to develop probability estimates for the various outcomes

so that a more complete analysis of risk and return could be conducted.

Limitations of Profit and Loss AnalysisThough profit and loss diagrams and their analysis is a powerful tool, the technique

is not appropriate for certain applications. The measure of profit and loss used in the

diagrams does not consider the time value of money. This means that profit and loss

analysis cannot be used for valuation calculations. Money does have different values at

different points in time. Any analytical techniques that ignores time value of money cannot

be used to find the value of any given security.

Profit and loss analysis is really a tool of decomposition and recomposition.

Financial contracts can be very complex, and when taken as a whole these contracts can

be difficult to understand. All complex financial contracts are really just combinations of the

three basic contracts, spot, forward, and option, and profit and loss analysis provides the

technique by which these basic components can be considered as parts of the whole.

Profit and loss analysis allows the complex contract to be broken down into their

components. These components can then be arranged so that their combined effect can

be determined. Profit and loss analysis allows the investor to "look under the hood" of a

financial contract and understand what is really happening inside the complete financial

package.

SummaryProfit and loss diagrams are valuable tools in the analysis of financial contracts.

Even the most complex financial contract is merely a combination of spot, forward, option

contracts. Spot contracts do not have any profit and loss associated with them since the

contract is completed immediately. Forward contracts have two possible positions, long

and short, and option contracts have four positions, long and short call and long and short

put. The six profit and loss diagrams associated with forward and option contracts are the

building blocks for developing the profit and loss diagrams for more complex contracts.

Graphical analysis permits the development of overall profit and loss diagrams that

combine the effects of several different individual contracts.