C. Códigos Matlab 105
C.2. Lenguas de Mathieu
Consider again the definition of a call option given in Chapter 3. A call option gives the holder the right but not the obligation to buy a given quantity of stock (100 shares) at a given price (the exercise price) on or before a given date (the expiry date). Call options are not
like stock in that they are not issued by a corporation but are sold short or written by other market participants. Every time someone buys an exchange traded stock call option someone must sell one. For every long position there is a short position.
What does it feel like being short a call option? The key phrase in the definition of the option is "the right but not the obligation". The holder (i.e. the party who bought the option) has the right but not the obligation. The individual who is long the option can choose, on or before expiry, if he wants to take up the stock or not.
This crucial aspect of a call option gives rise to the special kinked expiry price profile shown in Figure 3.1.
Imagine now the situation of being short such a call option (exercise price =
$100). If you are short this instrument you have given someone else (the holder) the option to choose between calling stock from you at $100 or not. You have no say in the decision but will have an idea as to what the holder will do. If the stock price is above $100 on expiry he will issue a call notice and you have to deliver stock in return for payment of $100 per share. If you do not already own stock you have to go into the market and buy some. If the price has risen significantly this can involve large losses. Say the stock price on expiry is $120. then the call notice will entail you buying stock at $120 and selling on to the option holder at $100, a loss of $20 per share. Since there is no upper limit to share prices one can see that shorting call options can be a very dangerous activity. In practice one does not usually need to go to the bother of buying the stock: it is often easier to just buv hack the short option and suffer the same loss. But what if the stock price is below
$100 on expiry? Obviously the holder would choose not to exercise his right to buy stock at $100 since it is cheaper in the market. In this situation, the holder would let the option expire worthless. In this situation the individual who shorted the option would he essentially buying it back at zero. This is the best outcome tor naked writers of call options. The term "naked" here refers to short call positions that arc not hedged in any way. Naked sellers take in option premiums and hope that worthless expiry results. Recall the • nn.' vcar call option referred to throughout Chapters 3 and 4. With the ^lin.k price at $99 and one year to go, the option is priced at $5.46. A n.ikfij call writer selling one option would receive
$546. If the instrument expires out-of-the-money and worthless he would be essentially !^i.i\inr. the option back at $0 and have made a profit of $546. This is ';- most the seller can make and that is the situation of all option
sellers; the very best they can do is to keep the initial option premium and there is no limit on the very worse that can happen.
Option sellers and option buyers are always in opposite situations. Every dollar made by one party is a dollar lost by the other. Option sellers have a limited profit potential and an unlimited loss potential whereas option buyers have a limited loss potential and an unlimited profit potential. Expiring profit and loss profiles are therefore also opposite but in a graphical sense. Opposite in a graphical sense here means all diagrams of short option positions are mirror images of long option positions. Figure 5.1 gives the expiring "value" of the long and short call option in question.
Figure 5.1 (a) shows the value of the long call option as either positive or zero and Figure 5.1(b) the short call option as negative or zero. With the stock price at
$120 the long option is worth $20 per share or $2.000 and this is the sum realised by the holder on closing out (i.e. selling) his long position. At this stock price the short option is shown as worth —$2,000 meaning that this is the sum the writer has to pay up to close out (i.e. buy back) his short position. The negative sign is used to illustrate the fact that at this stock price the short position is in debt to the tune of —$2,000. At higher stock prices the debt to the short position increases and this is clearly illustrated by the negatively sloping line. Above the exercise price the expiring profile has a slope of -$100. Every increase in $1 causes a loss of $100 to the option seller. Above the exercise price the stock exposure is short 100 units.