Arbitrage has become associated in popular attitudes with the most ruthless and profit-driven of human impulses, but the opposite reputation might be more well-deserved. The ability to arbitrage is essential for the efficient operation of markets. An interesting application of the principle of arbitrage arose when it provided the breakthrough insight in economists’ solution to a formerly intractable problem: how to properly price the emergent financial instruments known as options.Arbitrage ; Options (Finance)

Ed Nosal

Tan Wang

English

Research Papers in Economics

every dollar that ABC’s stock is above
$50, you will receive that amount. 
Determining the payoff structure for an
option doesn’t seem that difficult. How
might one go about determining the
value or price of this option? A standard
way to think about pricing any financial
asset is to first specify the payoffs that
the asset is expected to generate in the
future; then to appropriately discount
these payoffs; and finally add up all of
the discounted payoffs. The sum of the
discounted payoffs represents the value
of the asset, and the price of the asset
should be equal to this value. 
Future payoffs for financial assets are
discounted for at least two reasons. First,
a dollar in the future is worth less than a
dollar today; so, future dollars must be
discounted to make them comparable
with dollars today. Second, if one asset’s
payoff stream is riskier than another’s,
then, holding all other things equal, the
former asset is more valuable than the
latter. Hence, riskier payoff streams
January 1, 2004
Federal Reserve Bank of Cleveland
Arbitrage: The Key to Pricing Options
by Ed Nosal and Tan Wang
ISSN 0428-1276
Arbitrage has become associated in
popular attitudes with the most ruth-
less and profit-driven of human
impulses, but the opposite reputation
might be more well-deserved. The
ability to arbitrage is essential for the
efficient operation of markets. An
interesting application of the princi-
ple of arbitrage arose when it pro-
vided the breakthrough insight in
economists’solution to a formerly
intractable problem: how to properly
price the emergent financial instru-
ments known as options. 
Arbitrage is the act of simultaneously
buying and selling assets or commodities
in an attempt to exploit a profitable
opportunity. Although the idea behind
arbitrage is fairly simple, it is quite pow-
erful because the ability to exploit such
opportunities is needed for markets to
operate efficiently. Arbitrage ensures, for
example, that buyers and sellers of for-
eign exchange can be assured that they
are getting the “correct” rates for the cur-
rencies they are buying and selling inde-
pendent of the national foreign-exchange
markets they happen to be using. 
When markets are efficient, the prices of
the objects being traded reflect their true
value. And having prices reflect true 
values is important in decentralized
economies, such as the United States,
since it is the relative prices of various
goods, services, and assets that deter-
mines how many will be produced, how
they will be allocated, and how funds will
be invested. If prices did not reflect true
value, then the resulting allocation of
goods, services, and investment would not
be, in general, economically efficient. 
This Commentary focuses on a particu-
lar episode in which the recognition of
an arbitrage “opportunity” made finan-
cial markets more efficient. It wasn’t a
chance to make a profit that got noticed,
it was the way the principles of arbitrage
could be applied to the problem of cor-
rectly pricing options. Once financial
economists figured it out, the solution
enhanced the efficiency of financial
markets because it made options useful
as hedging instruments. Such instru-
ments can be used to manage cash hold-
ings only if they are correctly priced. 
■ In Search of the Option
Pricing Formula
Options are phenomenally popular these
days, but when they were introduced, they
didn’t take off at first. No one knew how
to price them. People were using methods
based on familiar financial instruments,
like equities and bonds, and these did not
work well for pricing options.
An option is called a derivative security
because it derives both its payoff struc-
ture and its value from some (other)
underlying security. A call option on a
stock, for example, gives the holder of
the call option the right, but not obliga-
tion, to purchase a particular stock at
some date in the future. The price at
which the holder can purchase the stock
is called the strike price and the latest
possible date at which the holder can
purchase the stock—or exercise the
option—is called the expiration date. If
the holder is only allowed to exercise on
the expiration date, the option is called a
European; if the holder can exercise any
time between now and the expiration
date, the option is called an American. 
The ability to exercise—or not—gives
options different characteristics from the
underlying securities on which they’re
based, and that fact makes them difficult
to price. Consider a European call option
on ABC stock with a strike price equal to
$50 and an expiration date in one year. If
you own this option, what will be your
possible payoffs one year from now?
Clearly, if ABC’s stock price is below
$50 one year from now, you will not
exercise because you would be paying
$50 for something that is worth less than
that; in these circumstances your payoff
will be zero. If the stock price is above
$50, you will exercise the option, and forshould be discounted more heavily than
less risky payoff streams.
It might seem as if calculating the value
of the option should be fairly straightfor-
ward. We know when the payoffs to the
option will be received and what possible
values they may take. All we have to do
is to discount these expected future pay-
offs to determine the option’s value. 
But here’s the rub: What discount factor
do we use? A discount factor should
reflect the underlying risk of the asset.
Since the option’s value is related to
movements in the stock price of the ABC
company, it might seem reasonable to
discount the option’s future expected
payoffs with ABC’s discount factor. The
problem here is although the movement
in the option’s payoff perfectly tracks the
stock price movements when they are
above $50, it does not at all track the
movement of the stock price when the
price is below $50. 
This implies that the risk characteristics
of the option are quite different from
those of the underlying stock. And
herein lies the obstacle that prevented
generations of researchers from solving
the option pricing formula: No appro-
priate discount factor could be found.
■ Arbitrage Basics 
The big breakthrough came when two
economists recognized that arbitrage
was the secret to unlocking the pricing
formula. The first step to grasping their
discovery is to understand the implica-
tions arbitrage has for the pricing of
anything in general.
To see these, consider a simple example.
Imagine that you visit a street, Main
Street, which is lined with apple ven-
dors. On the south side of the street all
the vendors sell apples for 10 cents a
piece, and on the north side they sell
them for 20 cents a piece. Whatever the
reason apples are trading at two differ-
ent prices, it is possible for you to profit
from the apparent “mispricing.” 
Suppose you go to the south side of the
street and ask a vendor if he would be
willing to “lend” you an apple, which
you would repay with an apple momen-
tarily. You in turn sell the borrowed
apple on the opposite side of the street
for 20 cents. (If apples were stocks,
what we have just described is “short-
ing” a stock.) With 20 cents in hand you
run across the street, purchase an apple
for 10 cents, and return this apple to the
south-side vendor from whom you bor-
rowed it. When all is said and done, you
have earned 10 cents without using any
of your own wealth: This corresponds to
an infinite rate of return! But why stop at
only 10 cents? If you borrowed 2 apples
from the vendor, you could have made
20 cents; if you borrowed 100 apples
you could have made $10. It appears that
you will be able to generate an indefinite
amount of money by simply selling
apples on the one side of the street and
buying them on the other. 
Unfortunately (for you), the forces of
supply and demand will take hold and
limit how much you can make. By
attempting to sell large amounts of
apples on the north side, you will bid the
north-side price down, and by attempting
to buy large amounts on the south side,
you will bid the south-side price up. In
the end, the price of apples on both sides
of the street will be the same, and your
infinite-rate-of-return investment will
disappear.
The apple parable helps explain why, for
example, the U.S./Canadian dollar
exchange rate in the New York foreign-
exchange market will be the same as in
the Tokyo foreign-exchange markets. If
there was a discrepancy between these
rates, financial institutions would have
an opportunity to make an infinite return
by selling the currency at the “over-
valued exchange” and buying currency
at the “undervalued exchange.” The
forces of supply and demand will ulti-
mately equate the two exchange rates. 
The key lesson here is that objects that are
the same should trade at the same price; if
they do not, then an arbitrage opportunity
exists. Attempts to cash in on the opportu-
nity will actually eliminate it and, in the
end, prices will be the same.
■ Forget the Discount Factor,
It’s All About Arbitrage
Applying the lesson to options tells us
that if there are two options written on
the same stock, with the same strike
price and with the same expiration date,
then the prices of these two options must
be the same. But it does not tell us what
that price should be. Or does it?
In 1973, economists Fisher Black and
Myron Scholes showed how the notion
of arbitrage can be used to price an
option. Their big insight was that the
payoff structure of an option can be
replicated by a portfolio of market-
traded assets. Since the cash payoffs to
the portfolio and the option are identi-
cal, it must be the case that the price of
the option equals the value of the portfo-
lio; otherwise, an arbitrage opportunity
would exist. A simple example might
help explain their approach.
Suppose that today’s price for ABC stock
is $40. In one period from now the price
of ABC stock will either rise to $60 or
fall to $20. There is a European option on
the ABC stock that has an expiration date
in one period, with a strike price at $50.
The borrowing and lending interest rate
is 25 percent per period. 
This is all the information we require to
price the option. Note that the payoff to
the option in one period will be either
$10 if the stock price rises or zero if it
falls. The Black-Scholes insight was to
construct a portfolio of existing assets
that can replicate a payoff of $10 when
the stock price goes up and zero when it
goes down. 
The portfolio in this example would
consist of 
1/4 share of ABC stock and $4,
which has been borrowed. The value of
this portfolio today is $6—the stock is
worth $10, and $4 must be repaid. 
The payoff in one period matches that of
the option: Your holdings of ABC stock,
if the price rises, will be worth 
1/4 x $60,
or $15. The $4 borrowed will be repaid
with $5 since the interest rate is 25 per-
cent, so the value of the entire portfolio
will be $10. If the stock price falls, your
holdings of ABC will be worth 
1/4 x $20
or $5. In this case, the portfolio’s value
one period from now is zero: Your hold-
ings of ABC stock just offset what you
owe. In summary, the portfolio pays $10
when the stock price is high, and zero
when it is low.
Since the payoff to your portfolio is
identical to the payoff of the option, it
stands to reason that the price of the
option should be $6, the value of your
portfolio today. And it is an arbitrage
argument that demonstrates that this
must be so. 
To see this, suppose that the option was
selling at $7. Just as in the apple exam-
ple, you can make an infinite rate of
return by selling the “expensive good”
(the option) and buying the “cheap
good” (the portfolio). If you sell the
option, you will receive $7 but you will
be required to pay $10 in one period
from now if the stock price turns out to
be high. After selling an option, you canpurchase the portfolio described above
for $6. That leaves you with $1 in your
pocket. The portfolio will pay off $10
one period from now when the stock
price is high, which is exactly what you
need to pay off your option obligation.
Hence, you have made $1 for sure with-
out using any of your wealth. But you
will continue as long as the profit
opportunity exists, and the forces of
supply and demand will ultimately
equate the value of the portfolio with
the price of the option.
So far, all this seems pretty simple, but
that is because the example is simple. In
reality when someone purchases an
option, the expiration date is many days
into the future and, as a result, the stock
price has the opportunity to move “up
and down” many times. So the relevant
question is how to price an option when
the price of the underlying asset can
move around many times before the
expiration date. Black and Scholes real-
ized that the arbitrage-portfolio replica-
tion argument described above needed
to be used repeatedly. To illustrate, let’s
continue with our simple example.
Suppose that the option on the ABC
stock, with strike price of $50, has an
expiration date three periods from now.
Today, date 0, the price of ABC is $40;
at each date the price can either increase
or decrease $20. If, however, the stock
price ever is zero, it remains at zero for-
ever. Figure 1, which is a “stock price
tree,” describes the set of possible stock
prices for ABC over the next three peri-
ods. We will use it to calculate the price
of the option today, at date 0.
Suppose that at date 2, the stock price is
$40. In this situation, the option will
either pay $10 or $0 at date 3. From the
example above, we know that the price
of the option will be $6, and that a port-
folio consisting of $4 borrowed and 
1/4
stock of ABC can replicate the payoff of
the option. Similarly, if the stock price
at date 2 is $80, then the option will pay
off either $50 ($100 – $50) or $10
($60 – $50), depending upon whether
the stock goes up or down at date 3. A
portfolio consisting of borrowing $40
and buying one unit of ABC stock can
replicate the payoffs of the option. This
portfolio, and hence the option, has a
value equal to $40. Finally, at date 2, if
the stock price is zero, the payoff to the
option will be zero at date 3, and a port-
folio consisting of “nothing,” which
costs nothing, is able to replicate the
option’s payoff. 
So, we have determined the price of the
option at date 2 for all possible out-
comes. By using the same kind of rea-
soning we will be able to determine the
price of the option at period 1 when the
stock price is $60 and when it is $20 and
finally at period 0.
Figure 2, which is an “option price tree,”
describes the price of the option for the
various realizations of the stock price.
At date 1, if the stock price is $60, then
a holder of the option is holding an asset
that will be worth either $40 or $6 at
date 2, depending upon whether the
price of the stock goes up or down at
date 2. A portfolio consisting of borrow-
ing $22.40 and buying .85 of ABC stock
can replicate these date-2 payoffs. This
portfolio is worth $28.60; hence, the
value of the option at date 1 when the
stock price is $60, denoted by node a in
figure 2, must be equal to $28.60. Simi-
larly, if the stock price is $20 at date 1,
a portfolio consisting of .15 of ABC
stock will replicate the date-2 payoff of
the option. This portfolio costs $3;
hence, the value of the option at node b
in figure 2 is $3. The price of the option
at date 0 can be determined by working
backward in the same fashion: It will be
given by the value of a portfolio that
pays off $28.60 if the stock price
increases to $60 at date 1 and $3 if the
stock price decreases to $20. A portfo-
lio that consists of borrowing $7.84 and
buying .64 of ABC stock will give such
payoffs, and it costs $17.76 to buy such
a portfolio. Therefore, the value of the
option at date 0 is $17.76.
Black and Scholes used the method of
repeated, or “dynamic,” portfolio repli-
cation, along with an arbitrage argu-
ment, to determine the price of the
option. At each date, the value of the
portfolio is precisely equal to what is
needed to buy a new portfolio that can
replicate the subsequent period’s pay-
offs. In the end, the date-3 payoffs to
the option can be replicated. If, in our
example, the price of the option is not
equal to $17.76, say it is $20, an infi-
nite rate of return can be made; you sell
the option and buy the date-0 portfolio.
The returns to the portfolio with
dynamic replication will precisely pay
off your option obligations, you pocket
$2.24, and the desire to capture more of
this “free money” will imply that the
forces of supply and demand will
equate the value of the replicating port-
folio and the option.
■ Arbitrage and Economic
Efficiency
Arbitrageurs help make markets effi-
cient. When, for some reason, prices
get out of line with one another, arbi-
trage will get the prices back in line. As
a result, prices will reflect the “true val-
ues” of the traded objects. The notion
of arbitrage has improved the efficiency
of markets in another, and perhaps,
unexpected way. The ideas behind arbi-
trage helped financial market partici-
pants price derivative assets, such as
options. These derivative assets can be
used to manage risk and, when cor-
rectly priced, will enhance the effi-
ciency of markets and firms and, as a
result, society. 
FIGURE 1 STOCK PRICE TREE
40
60
20
80
40
0
100
60
20
0
date 0 date 1 date 2 date 3
FIGURE 2 OPTION PRICE TREE
40
6
0
50
10
0
0
a
date 0 date 1 date 2 date 3
bFederal Reserve Bank of Cleveland
Research Department
P.O. Box 6387
Cleveland, OH 44101
Return Service Requested:
Please send corrected mailing label to
the above address.
Material may be reprinted if the source is
credited. Please send copies of reprinted
material to the editor.
Ed Nosal is a senior economic advisor at the
Federal Reserve Bank of Cleveland, and
Tan Wang is the Peter Lusztig Associate
professor of Finance at the University of
British Columbia.
The views expressed here are those of the
authors and not necessarily those of the Fed-
eral Reserve Bank of Cleveland, the Board of 
Governors of the Federal Reserve System, or
its staff. 
Economic Commentary is published by the
Research Department of the Federal Reserve
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Arbitrage: the key to pricing options

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