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Lecture 12 Introduction to Optio

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					                          Lecture 12
                       Introduction to Options


                                AIM OF LECTURE 12
                           Learn the terminology of options
                           Learn how to use payoff diagrams
                           Introduce to Put-Call Parity


                            12.1 WHAT IS AN OPTION?
An option is a contract (a financial contract) between two parties. We refer to the two parties
as holder and writer. There are two types of options: call and put. There are two categories:
European and American. (We will only cover European options in this course, but you should
be aware of the difference between them).

A call option gives the holder the right (but not the obligation) to purchase a pre-specified
financial asset at a future date at a pre-specified price. We refer to the pre-specified asset as
the underlying asset, and the pre-specified price as the exercise price. If the holder uses
his/her right to purchase we say that the option is being exercised.
     The writer of a call option is obliged to sell the underlying asset to the holder of the
option if the holder exercises the option.

A put option gives the holder the right (but not the obligation) to sell the underlying asset at
a future date at the exercise price.
     The writer of a put option is obliged to buy the underlying asset from the holder, if the
holder chooses to exercise.

We did not say anything about the date when exercise can take place. If there is only one
date at which exercise can take place we refer to this option as a European option. That date
is referred to as the exercise date. The exercise date is stated in the contract.

On the other hand, if the holder can choose to exercise at any date before a pre-specified date
we refer to the option as American. Under some circumstances it is never optimal for the
holder to exercise before the last day, then the American is equivalent to the European.

We shall from now on only cover European options.
                                                             Financial Markets: Theories & Evidence 2


The following table summarises the definitions:


                 Party:         Call                                Put
                 Holder         Right to purchase                   Right to sell
                 Writer         Obligation to sell                  Obligation to buy
                                (if option exercised)               (if option exercised)


We say that the holder has a long position in the option, and the writer has a short position
in the option. So we can refer to long call, long put, short call, and short put, as the basic
option investment strategies.

You may now suspect that the holder has an advantageous position relative to the writer: the
holder can choose but not the writer. Because of the option to choose the option has got a
value. For the writer to enter into this contract, there must be a compensation payed to the
writer. This compensation is the price of the option. We would like to know what the price
would be and what it would depend on. This is what option pricing is all about.

In practice the option contracts are standardised. They are only issued for some (standard)
underlying assets, and for some (standard) maturities (exercise dates), and for some (standard)
exercise prices. This is to make sure that the market is liquid enough. Options are usually
issued by a clearing house, and investors take long and short positions with the clearing
house.


                                       12.2 PAYOFF DIAGRAMS
It is often useful to think of the different options in terms of payoff diagrams. They tell how
the net value of an investor’s position at the exercise date1 depends on the value of the
underlying asset. By position we mean long call, short call, long put, short put.


First some notation:
     C = price of European call option
     P = price of European put option
     S = current share price (underlying asset)
     X = exercise price
and some terminology:
     if S > X the call option is in-the-money and the put option is out-of-the-money
     if S = X the call and the put options are at-the-money
     if S < X the call option is out-of-the-money and the put option is in-the-money



1
    The Black and Scholes formula can give us the value of an option at any date, not just the exercise date.
                                                  Financial Markets: Theories & Evidence 3


We shall look at payoffs at the exercise date. We may denote with an asterisk (*) the values
at the exercise date:
     C* = price of European call option at the exercise date
     P* = price of European put option at the exercise date
     S* = share price at the exercise date

Clearly, if the share price is greater than the exercise price the call option will be exercised.
The value of such and option (which is in-the-money) is S*-X. On the other hand, if the price
of the share is lower than the exercise price, the option has no value. To summarise




Similarly for a put option, it will never be exercised if the share price is greater than the
exercise price (because the holder of the option would never sell the share to the option writer
to the price of X when (s)he can sell the share on the stock market to the price of S*). If the
share price is lower than the exercise price the option will be exercised, and the value is X-S*.




Let’s look at the net-payoffs (at exercise date) from entering an options contract. Figure 7.1
shows the payoff of buying a call option at 5 with exercise price 140.


Figure 12.1




                        [Consult Figure 8.3 in Copeland and Weston]




The payoff of writing a call option is the mirror picture of buying (Figure 12.2).
                                                  Financial Markets: Theories & Evidence 4


Figure 12.2




                        [Consult Figure 8.3 in Copeland and Weston]




The payoff of buying a put option at price 7 with exercise price 90 is illustrated in Figure
12.3.

Figure 12.3




                        [Consult Figure 8.3 in Copeland and Weston]




Finally, issuing a put option is illustrated in Figure 12.4.

Figure 12.4



                        [Consult Figure 8.3 in Copeland and Weston]
                                                 Financial Markets: Theories & Evidence 5


As you can see from Figure 12.2, writing a call option may be potentially dangerous since
the loss may be unlimited. To cover for such a loss a writer of a call may be required to own
the share, or paying margin payments to the clearing house.


                            12.3 COMBINING OPTIONS
Options can be combined to create virtually any position. We will show two such positions
here.


12.3.1 Straddle

Consider buying one call option and one put option with same exercise price and same
maturity date. (I the exercise price is the same, as well as the maturity date, the price of the
call must be greater than the price of the put, we will see this in section 12.4).

The net position is shown by the dotted line below in figure 12.5


Figure 12.5


                        [Consult Figure 8.6 in Copeland and Weston]
                                                  Financial Markets: Theories & Evidence 6


12.3.2 Replication of Short Selling

Consider buying one put option and issuing one call option with same exercise price.

The net position is shown by the dotted line below in figure 12.6. Notice that this position
exactly replicates shortselling!


Figure 12.6




                                     [Figure not available]




                           12.4 THE PUT-CALL PARITY
As hinted above, there is a relation between the price of a call and the price of a put (if they
have the same exercise price and the same maturity date). This relation is based on a no-
arbitrage argument. The reason is that one can form portfolios with puts and calls which give
the same payoff diagrams at maturity. Then the portfolios must cost the same (otherwise there
would be arbitrage opportunities.

To form the portfolios riskless lending is needed. Let Rf denote the risk-free interest rate for
the period up until the maturity date.

Consider the following two investment strategies (portfolios):
(1) buying one call option with exercise price X and investing X/(1+Rf) in treasury bills.
(2) buying one put option with exercise price X and buying one share (the underlying asset).

Consider first strategy (1). In the event that the call finishes in the money, the call should be
exercised, the value of the call is then S-X. At the same time the value of the treasury bills
have grown to X and therefore the net position is S-X+X = S. If the call finishes out-of-the-
money, the call should not be exercised, and its value is zero. But the bond has still grown
to X, so the net value is X.
                                                  Financial Markets: Theories & Evidence 7


So strategy (1) offers either S or X, whichever is greater:

                                           max{S,X}

Consider strategy (2). In the event S>X (the put is out-of-the-money) the put is not exercised
and the value of the net position is S. If the put finishes in-the-money so that S<X, the put
will be exercised, and its value is X-S. The value of the net position is value of put plus value
of share, thus X-S+S = X. So strategy (2) also offers either S or X, whichever is greater:

                                           max{S,X}

Since (1) and (2) offer the same pay-off structure they must have the same value, i.e.

                                     C + X/(1+Rf) = P + S

                                 This is the Put-Call Parity!


Notice that if X=S then C = P + XRf/(1+Rf) > P, so the call has a higher price than the put.
(Obviously if X is "close" to S, C>P holds).


                                    12.5 NEXT TIME
Next time we will derive a formula for valuing options, and practice the formula with
numerical examples.


                                       REFERENCES
Copeland, Thomas E., and J. Fred Weston, Financial Theory and Corporate Policy, Addison-
Wesley, Chapter 8, (parts A-C).

The following reference is optional:

Hull, John, (1997), Options, Futures, and Other Derivatives, Prentice-Hall, chapters (6)-7.

				
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