This is a book on option theory and many “how to” books are available giving very full descriptions of trading strategies using combinations of options. There is no point repeating all that stuff here. However, even the most theoretical reader needs a knowledge of how the more common combinations work, and why they are used; also, some useful intuitive pointers to the nature of time values are examined, before being more rigorously developed in later chapters.
Most of the comments will be confined to combinations of European options.
(i)Call Spread (bull spread, capped call):This is the simplest modification of the call option.
The payoff is similar to that of a call option except that it only increases to a certain level and then stops. It is used because option writers are often unwilling to accept the unlimited liability incurred in writing straight calls. The payoff diagram is shown in the first graph of Figure 2.6.
It is important to understand that a European call spread (and indeed any of the combinations described below) can be created by combining simple options. The second graph of Figure 2.6 shows how a call spread is merely a combination of a long call (strike X1) with a short call (strike X2). The third graph is the payoff diagram of a short call spread; it is just the mirror image in thex-axis of the long call spread.
Call Spread Long and Short Calls Short Call Spread
Payoff
Payoff Payoff
ST
X1 X2
ST
X1 X2
ST
X1 X2
Figure 2.6 Call spreads
(ii)Put Spread (bear spread, capped put):This is completely analogous to the call spread just described. The corresponding diagrams are displayed in Figure 2.7.
Put Spread Long and Short Puts Short Put Spread
Payoff Payoff
Payoff
ST
X1 X2
ST
X1 X2
ST
X1 X2
Figure 2.7 Put spreads
(iii) In glancing over the last two sets of graphs, the reader will notice that the short call spread and the put spread are very similar in form; so are the call spread and short put spread. How are they related?
2.5 COMBINATIONS OF OPTIONS
All the payoff diagrams used so far have been graphs plotting the value of the option position at maturity against the price of the underlying stock or commodity. But the holder of an option would have had to pay a premium for this position (the price of the option). To get a “total profits” diagram, we need to subtract the future value (at maturity) of the option premium from the payoff value, i.e. the previous payoff diagrams have to be shifted down through thex-axis by the future value of the premium. Similarly, short positions would be shifted up through the x-axis.
Call Spread Short Put Spread Box Spread
X1 X2
r T
1 2
(C - C ) e
1 2
(X X ) X1
X2
r T
1 2
(P - P ) e
X1 X
- 1 2
Figure 2.8 Equivalent spreads
The effects of including the initial premium on the final profits diagram of a call spread and a short put spread are shown in the first two graphs of Figure 2.8. The notationC1,C2,P1,P2 is used for the prices of call and put options with strikesX1,X2.
The diagonal put and call payoffs are 45◦ lines, so that the distance from base to cap must be X2−X1 as shown. Recall the put–call parity relationship for European options C+Xe−r T =P+S, from which
(C1−C2) er t+(P2−P1) er t =X2−X1
It follows immediately that these two final profit diagrams are identical. All of these payoffs could be generated using just puts or just calls, and the costs would be the same. This theme is developed further below. Although it is possible to create spreads with American options, re- member that the put–call parityequalityno longer holds; American puts and calls are therefore not interchangeable as are their European counterparts.
(iv)Box Spread:The third graph of Figure 2.8 shows an interesting application of the concepts just discussed. By definition, a put spread is perfectly hedged by ashortput spread; but we have just seen that a European short put spread is identical to a European call spread. Thus a put spread is exactly hedged by a call spread. The combination of the two is called a box spread.
Suppose we buy a call spread forC1−C2and a put spread forP1−P2; the put–call parity equality of the last paragraph shows that this will cost (X1−X2) e−r T.
Since a box spread is completely hedged, this structure will yield precisely X1−X2 at maturity. In other words, a combination of puts and calls with individually stochastic prices yields precisely the interest rate.
There are two purposes for which this structure is used. First, if one (or more) of the four options, bought in the market to make the box spread, is mispriced, the yields on the cash investment may be considerably more than the interest rate. This is quite a neat way of squeezing the value out of mispriced options. Second, gains on options sometimes receive different tax treatment from interest income, so that this technique has been used for converting between capital gains and normal income.
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2 Option Basics
(v)Straddle:This is another popular combination of options with the payoff shown in the first graphs of Figure 2.9 . This consists of a put and a call with the same strike price. People invest in this instrument when they think the price of the underlying stock or commodity will move sharply, but they are not sure in which direction. Clearly, this is tantamount to betting on the future volatility of the stock.
Payoff Payoff
Straddle Strangle
ST
X
ST
X1 X2 Figure 2.9 Straddle and strangle
Strangle:A slightly modified version of the straddle is shown in the second graph of Figure 2.9.
A straddle is quite an expensive instrument, but by separating the strike prices of the put and the call, the cost can be reduced.
(vi)Collar:One of the most important uses of an option is as a hedge against movement in the underlying price. Typically, the owner of a commodity can buy an at-the-money put option;
for each $1 drop in the commodity price, there is a $1 gain in the payoff of the put. The put option acts as an insurance policy on the price of the commodity.
If an insurance premium is too expensive, it can be reduced by introducing an “excess”
or “deductible”. For example, the owner of the commodity bears the first $5 of loss and the insurance covers any further loss. This would be achieved by buying a put whose strike price is $5 below the current market price.
Another way in which the insurance cost can be decreased is by means of a collar. In addition to buying a put, the commodity holder sells a call with strike somewhere above the current commodity price. The first graph of Figure 2.10 shows the payoff for a collar. BelowX1, the
Collar Long Commodity Net Exposure
Payoff Payoff
Payoff
ST
X1
X2 ST X1 ST
X2 0
S0
Figure 2.10 Collars
2.5 COMBINATIONS OF OPTIONS
commodity holder receives $1 from the put for each drop of $1 in the price. AboveX2, he pays away $1 under the call option for each $1 rise in the price. If the option positions are combined with his position in the underlying commodity (second graph), the result is his net exposure to the commodity (third graph). BetweenX1andX2he is exposed to movements in the price;
outside these limits he is completely hedged. A particularly popular variety is thezero-cost collarwhere the strike prices are arranged so that the receipt from the call exactly equals the cost of the put.
(vii)Butterfly: As with simple put and call options, the writer of a straddle accepts unlimited liability. This can be avoided by using a butterfly, which is just a put spread plus a call spread with the upper strike of the first equal to the lower strike of the second. Like the straddle, this instrument is basically a volatility play, but the upside potential for profit has been capped.
The payoff diagram is given in the first graph of Figure 2.11.
Payoff
Condor Payoff
Butterfly
ST
X
ST
X1 X2 X3 X4 Figure 2.11 Butterfly and condor
Condor:This instrument is very similar to the butterfly, but has different strike prices for the put spread and call spread.
(viii) In Section 2.5(iii) it was shown how a European spread could be constructed either from puts or from calls; this is equally true of the butterfly and condor. One is occasionally confronted with a very complicated payoff diagram which needs to be resolved into its underlying puts and calls. A condor provides a good example of how to proceed.
r
Starting from the left end, move towards the right along the payoff diagram for the condor.r
The first direction change is atX1, where the line bends down 45◦. This is achieved with a short call (−C1).r
Moving on toX2, the line bends up 45◦: long call (−C2).r
AtX3, the line bends up 45◦: long call (+C3).r
AtX4, the line bends down 45◦: short call (−C4).The condor could therefore be constructed as−C1+C2+C3−C4.
On the other hand, moving from the right to the left of the diagram, this sort of reasoning would yield a combination−P4+P3+P2−P1. As a further variation, we could conceptually break the condor in two, constructing one half out of puts and the other half out of calls.
These complex payoffs are therefore ambiguous, in that they can be constructed in several different ways. But for European structures, put–call parity always assures that the cost is the same, whatever elements are used to build them.
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