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This chapter presents a brief introduction to the subject of
options. The material is extremely brief and elementary. More detailed
treatment of the theory of options can be found in Chriss[1]
(which uses minimal mathematics) and Hull[2] (which has a deeper
and more rigorous approach). The efficient market hypothesis and the
principle of no arbitrage are treated in some detail in Jacob and
Pettit[3].
Before we can discuss derivatives, we need to understand what
securities are. All the financial instruments we see around us such as
shares, bonds, futures, options, etc. are considered securities. More
precisely, a securities contract (security, for short) is a contract
issued by a government or company in order to acquire financial
capital which is used to acquire real capital, labour and management
talent. Examples of securities are common stocks which are issued by
companies to acquire capital (which make the buyer a partial owner of
the company), bonds and fixed income securities which are issued by
governments and companies which need to borrow money for both short
and long-term use, debt/equity combinations (which are basically
combinations of stocks and bonds) and third-party financial contracts
(or derivatives, which are the most important from our point of view).
The precise nature of the common stock contract is normally defined in
accordance with the laws of the country. However, almost all common
stock contracts have the following features :
- They give the owner the right to receive his share of the
remaining value of the company after all the outstanding loans, fines
etc. have been cleared.
- They give the owner the right to a share of the earnings (in
proportion to the number of shares owned by him) of the firm paid out
in the form of dividends (which normally take the form of the company
giving all shareholders a certain amount of money for each share that
they own).
- The owner has the right to sell the stock to another investor in
the secondary markets (more commonly known as stock exchanges).
While most common stock contracts have the above features, there are
several varieties of them which are distinguished by other features of
the contracts. For example they may be voting or non-voting (i.e., the
owners may or may not have the right to vote about major decisions
that affect the company), they may have restrictions on the proportion
of the stock that can be held by one investor and so on and so forth.
Bonds and fixed income security contracts are issued by governments
and companies essentially for the purpose of borrowing money. They
ordinarily specify the financial obligations the issuer has to the
owner. They generally require the issuer to pay interest and principal
in specified amounts at specified future dates. The number of such
dates (excluding the maturity) is called the number of coupons of the
bond, the terminology arising from the fact that the bond-holder has
to tear off coupons from the contract to claim the payments before the
final date.
Debt/equity combinations are financial contracts which are
combinations of debt and equity securities (in other words, a
combination of stocks and bonds). One common type of such a security
are the warrants issued with some stocks. These warrants are
securities entitling the bond-holder to purchase a specified number of
shares of the firm at a fixed price. These are similar in nature to
the options we will consider later.
Third-party financial contracts or derivatives constitute one of the
fastest growing markets in recent history and their importance today
for companies and financial institutions is difficult to
overestimate. Hence, the valuation of derivatives is an important
field of financial research.
A derivative is an instrument whose value is dependent on another
security (called the underlying security). The derivative value is
therefore a function of the value of the underlying security. The two
most commonly traded derivatives are futures and options. A futures
contract is an agreement between two parties to buy or sell an asset
(the underlying security) at a certain time in the future for a
certain price. They are traded on exchanges.
Figure 1.1:
The profit of a call option at maturity as a function of the
stock price at maturity. The strike price is taken to be $100 and the
original price of the option is assumed to be $10.
|
Figure 1.2:
The profit of a put option at maturity as a function of the
stock price at maturity. The strike price is taken to be $100 and the
original price of the option is assumed to be $10.
|
There are two basic types of options that are traded in the market. A
call option gives the holder the right to buy the underlying asset by a
certain date for a certain price. A put option gives the holder the
right to sell the underlying asset by a certain date for a certain
price. This price is called the strike price and the date is called
the exercise date or maturity of the contract.
There is a further classification of options according to when they can
be exercised. An European option can only be exercised at maturity
while an American option can be exercised at any time up to the
maturity. The value of an Asian option is dependent on the average
value of the underlying security during the term of the contract while a
Bermudan option
can only be exercised on
certain days between the present time and the maturity of the
contract.
In this thesis, I will only consider European call and put
options. For call options this is not a significant liability as
European and American call options with the same maturity and strike
prices on a non-dividend paying stock have an identical
value
. However, the value of an
American put option is always higher than that of an European put as
early exercise can be optimal under certain conditions.
From the definition of a call option, we can see that the value of an
European call option at maturity is given by
 |
(1) |
(if
then the option will not be exercised and if
, the
profit on the option will be
) where
is the value of the call
option at maturity,
is the value of the underlying security at
maturity and
is the strike price of the option. (
represents the Heaviside function defined by
if
,
and
if
).
We see that the payoff of a call option at maturity is either positive
or zero. Hence, the call option must have a positive value before
maturity, which of course is the price of the option. If we also
consider the original price of the option, the profit is then given by
 |
(2) |
where
is the profit and
is the initial price of the call
option. For the rest of this section (in fact for most of the rest of
the thesis), we will assume that the present value of the underlying
security is $100. This does not change anything as all the prices can
be rescaled by a constant factor without affecting the theory. The
profit of the call option (assuming a strike price of $100 and an
initial option price of $10) as a function of the value of the
underlying security at maturity is shown in figure 1.1.
Similarly, the payoff of a put option at maturity is given by
 |
(3) |
(if
, the option will not be exercised and if
, the profit
is
) where
is the initial price of the put option. The
profit (with the same assumed values as for the call option) as a
function of the value of the underlying security is shown
in figure 1.2.
It can be shown that if, for a given underlying security, call and put options
were available for all strike prices, then any payoff (excluding the
original price of the options) whatsoever can be duplicated. Hence, if
we can calculate the value of all possible call and put options on an
underlying security, we can calculate the value of any derivative based on that
underlying security.
Options give a person using them very high leverage for
speculation. For example, if one is convinced that the price of
Microsoft stock is going to increase over the next three months, one
can either buy the stock or call options on the stock. If we assume
that the stock's present price is $100, that the price of a call
option maturing in three months for a strike price of $100 is $10
(we assume that one call option contract gives us the right to buy 1
share) and that we have $10,000 to invest, we can either buy 100
shares or 1,000 call options on the stock. If the price rises to $120
in three months, we would make a profit of $2,000 if we had bought
the stock or a profit of $10,000 if we had bought the options. Of
course, if the price drops, the loss on the options will be greater
than the loss on the stock.
While speculators want to increase their exposure in the market,
hedgers are trying to do exactly the reverse. For example, if a fund
manager wants to insure his holding of Microsoft stock, he can sell
call options on it. If the price decreases below the strike, he will
make a profit equal to the price of the options when he sold them
while if the price increases, he incurs a loss as the options are
called. This compensates for the decrease and increase respectively in
the value of the Microsoft stock he holds. The hedger's aim is to
prevent price changes in the price of the underlying security from affecting
the value of his portfolio. Hedging can also be performed (and is, in
fact, usually done) using futures.
This principle is also called the law of one price. It states that two
equivalent goods in the same competitive market must have the same
price. Anyone who has ever gone shopping knows that this is not
exactly true. The reason this is so is that the shopping malls are
quite far from being efficient markets. Hence, before we apply this
principle, we must understand what makes markets efficient.
There are four conditions that efficient markets must satisfy. These
are
- Liquidity - this means that all players are small relative to
the size of the market and that there are no transaction costs.
- Price Continuity - this means that the price is set only by
the underlying supply-demand situation and that temporary imbalances
do not affect the price.
- Fairness - this means that there is no discrimination between
different players in the market.
- Equal access to information - this means that all market players
have equal access to all available information.
We know that at least part of the first condition is never satisfied -
there is no market where the transaction costs are zero. This does
limit the application of the no arbitrage principle as we will see
below. Most stock exchanges put in a lot of effort to ensure that the
other conditions are satisfied. The structure of the stock market
(specifically, the presence of market makers) ensures that price
continuity is more or less satisfied. Fairness is easily ensured as
only the price and time of the orders are relayed to the floor of the
exchange. While it is obviously true that some people have more
information than others, the presence of insider trading regulations
minimize the effect of the inequality in access to information.
Let us now consider an imaginary market where all the above conditions
are satisfied. In this market, participants, acting in their own
self-interest, use available information to attempt to secure a higher
return for their investments. In doing so they collectively ensure that
price movements in response to new information are instantaneous and
unbiased and will fully reflect all relevant information. Competition
will drive security prices from one equilibrium level to another so
that the change in price in response to new information will be
independent of the prior change in price. In other words, the price of
a security follows a Markov process which is only dependent on the
latest information released to the market. This is an extremely
important conclusion which we shall use extensively later. Most option
pricing models assume that the stock prices follow certain specific
Markov processes.
Empirical tests have shown that real markets satisfy most, though not
all, of the conditions necessary for an efficient market. Please refer
to Jacob and Pettit[3] for a summary of these tests.
The no arbitrage principle can now be easily seen to hold in an
efficient market. For, if we assume that two equivalent securities A
and B have different prices (say A has a lower price than B) then a
market participant can make a risk-less profit by buying A, changing it
to B (since A and B are equivalent, this is possible) at no cost and
then selling B. (If one is uncomfortable with changing security A to
B, one can just buy A and sell B. Since A and B are equivalent, the
final payoffs are the same and cancel each other leaving one with the
initial profit of the difference in the prices.) Hence, an upward
pressure on the price of A is created (as A is being bought by the
participant) as is a downward pressure on the price of B. This will
continue until the prices of A and B become equal. In an efficient
market, this will be a very short period of time as all the
participants are aware of arbitrage opportunities when they
arise. Note that the absence of transaction costs is of fundamental
importance in this discussion. If we assume that a single transaction
has a fixed cost
, then the price of A and B can have a
difference of up to
(since two transactions are required to
benefit from the arbitrage opportunity) as the transaction costs will
wipe out the arbitrage profit.
The principle of no arbitrage is actually more general than the
efficient market hypothesis. This can easily be seen by considering a
market where only some of the people have all the available
information. These people will still try to exploit arbitrage
opportunities as they arise and will exert pressure on the market to
satisfy the principle of no arbitrage.
Hence, in an ideal market, arbitrage opportunities will not exist. We
can use this fact for option valuation. This is done by replicating an
option using a portfolio consisting of zero-coupon bonds and the
underlying security. If we can make this portfolio replicate the
payoff of the option, then the principle of no arbitrage assures us
that the option will have the same value as the portfolio.
We can use the principle of
no arbitrage to show that the value of an European call option is the
same as the value of an American call option as long as no dividends
are paid. To do so, we consider the following portfolios :
Portfolio A : one American call option plus an amount of cash equal
to
(
is the strike price,
is the risk-free
interest rate,
is the maturity time and
is the present
time)
.
Portfolio B : one share.
The value of the cash in portfolio A at expiration is
. At some
earlier time
, it is
. If the call option is
exercised at time
, the value of portfolio A is
which is obviously less than
when
since
. Hence,
portfolio A is always worth less than portfolio B if the call option
is exercised prior to maturity. If the call option is held to
expiration, the value of portfolio A at time
is
while that of portfolio B is
. Hence, at maturity, portfolio A is
greater than or equal in value to B. Therefore, the value of portfolio
A is less than that of portfolio B if the call is exercised early
while it is at least as much as the value of portfolio B if the call
is only exercised at maturity. Hence, early exercise of the call
option is never optimal and the value of the American and European
call options are the same.
We can use the principle of no arbitrage to derive an important
relationship between European put and call prices with the same strike
price. To do so, we consider the following portfolios :
Portfolio A : one European call option plus an amount of cash equal
to
Portfolio B : one European put option plus
one share.
Both are worth
at expiration of the options. Hence, the
portfolios must have the same value at the present time. Therefore,
 |
(4) |
where
is the value of the European call option and
is the
value of the European put option.
Hence, calculating the value of the call option also gives us the
value of the put option with the same strike price. Therefore, in the
rest of this thesis, we will only consider how to calculate the value
of the call options.
Next: Stochastic Processes : An
Up: thesis
Previous: Overview
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Marakani Srikant
2000-08-15