COMBINATION #1: THE VERTICAL CALL SPREAD

Probably the simplest option strategy is the vertical call spread (or sometimes simply referred to as the call spread). This strategy involves buying one call option and simultaneously selling another call option with the same expiry date but a different exercise price. The example we will consider is the $100/$110 one-year call spread on the stock that we have used throughout this book. Say we buy the

$100 strike option and sell the $110 strike option. If you buy the lower strike and sell the higher strike (with call options), market terminology dictates that you have bought the spread. It should be obvious that the lower strike call option will always be more expensive than the higher strike option and so the combination will have a positive price, i.e. some investment is required. Leaving aside the question of the initial call spread cost, consider the value of the combination on expiry. Table 7.1 and Figure 7.1 show the expiring values and prices of the individual components and the total spread. We use the usual notation employed for short option positions, i.e. that of negative value. A negative value represents the sum of cash required to liquidate the position.

With the stock price below $100. both options expire worthless and so the spread also expires worthless. With the stock price above $100 bill below $110, the long call option gives exposure to 100 long units •>f stock but the short call option gives zero exposure. With the stock l-iricr iihove $110. the higher strike call gives exposure of 100 short 'mils ol stock and this completely cancels out the long exposure given 1^\ the lower call. So at any stock price above $110, the call spread ' ;ihu is capped at $1,000 and the call spread price will be capped at

Figure 7.1 Expiring call spread value

$10 per share. The expiring price or value profile below $110 is very much like that of a simple long call position. The addition of the short call position has the effect of removing any further profits above the $110 level, capping the unlimited profit potential of the long option.

It is interesting to consider the spread as one compound investment vehicle. Up until this stage all the financial instruments we have considered have had a price discontinuity or kink at expiry. Treated as

Table 7.1 Expiring call spread value (long one $100 strike call and short one $110 strike call)

Table 7.2 Call spread price one year from expiry

Stock Long call price Short call price Call spread price (strike - $100) (strike = $110) price

(a) (b) (c) = (a) + (b)

80 0.40 -0.08 0.32

85 0.98 -0.25 0.73

90 2.02 -0.63 1.39

95 3.67 -1.33 2.34

100 5.98 -2.50 3.48

105 8.95 -4.23 4.72

110 12.50 -6.58 5.92

115 16.52 -9.52 7.00

120 20.89 -13.00 7.89

125 25.50 -16.92 8.58

130 30.28 -21.20 9.08

one entity, the call spread is different in that it has two kinks and this makes the price profile before expiry doubly interesting. The price of the call spread prior to expiry can similarly be obtained from the prices of the two separate components.

Table 7.2 and Figure 7.2 give the prices of the individual components and the spread for various stock prices one year from expiry.

Call spreads are very popular with outright investors and specula-

FiHure 7.2 Call spread price before expiry

tors because they are cheaper than outright call options and require less of a stock price move to break even. If one believes that the underlying stock price is going to rise significantly in the future one would simply take a long position in the $100 strike call option. However, if one believes that the price rise is not going to be that large, say only 10%, then it may make sense to buy the call spread. The call spread will be cheaper than the outright call option and if the final stock price is not much higher than $110, the final portfolio values will be similar. Since the spread is cheaper, the percentage return will be higher and speculators are usually very concerned about percentage returns. As an example, if the stock price is $100, the outright $100 strike call costs $5.98 but the spread only costs $3.48. If the stock price rises to $110 by expiry, the outright call will be worth $10 producing a profit of 67%, whereas the call spread being worth the same $10 will have produced a profit of 187%. To break even with the outright call option, the underlying stock price must rise to at least $105.98 by expiry, whereas the rise only has to be to

$103.48 for the spread. Buying a call spread is less risky than buying an outright call but this is offset by the limited upside potential.

The price of the call spread prior to expiry is curved. At lower stock price levels the price curve sits above the expiry profile and at higher levels the curve sits below the expiry boundary. With one year to expiry the slope of the price curve is very low and in this situation never gets above 0.20. A more detailed study of the various sensitivities of the call spread can be obtained by looking at the price profile at different times to expiry and these are given in Figure 7.3.

The two discontinuities on expiry cause the call spread to have very unusual features, the most interesting one being time decay effects. So far we have discussed portfolios that either suffer from time decay or benefit from time decay.

This compound instrument does both. It is clear from Figure 7.3 that below a special stock price ($104 in this case), the passage of time reduces the instrument's value but that above the special price, time passing has the effect of increasing the instrument value. And at one specific price, the instrument is completely independent of time. At first this behaviour seems most peculiar. When you buy a single put or call option, the moment you have bought it, time begins to erode its value. When you sell short a single option, the moment you sell, time begins to act in your favour. With this instrument, the effect depends on the price of the underlying. The reason for this behaviour is of course easy to understand—it is due to

Figure 7.3 Time decay effects on call spread

the fact that the portfolio is simultaneously long and short options with different exercise prices. The net effect of time decay is the sum of the effects of time decay on the individual components. At stock prices lower than $104, the negative time decay effects of the lower call option outweigh the positive time decay effects of the upper call option. At higher prices the reverse is true and at one specific price the two effects match.

As with individual options, the passage of time causes the price curve to move towards the expiry boundary and a study of the way in which the curve moves can shed light on the other option sensitivities. In this case the expiry boundary is more complex and comprises three straight lines with corresponding implications for the sensitivities. The discussion can be reduced to considering the two regions above and below the special stock price of $104.

Lower Region

1 he price profiles in the lower region are similar to those of a long call option.

Along any given curve the deltas are positive and increase \\ith increasing stock price. Time passing has the similar effect of i educing the deltas at lower prices and increasing the deltas at higher puces. The price curvature or gamma is positive and this increases as expiry approaches.

Upper Region

The price profiles in the upper region are similar in shape to those of a short put option. Along any given curve the deltas decrease with increasing stock price.

Time passing has the similar effect of increasing the deltas at the lower prices and decreasing the deltas at higher prices. The price curvature is negative and this becomes more pronounced as expiry approaches.

The delta of the combination is always positive or zero and so the appropriate hedge would be a short position in the underlying. Initially the maximum delta is only 0.2 and so the size of the hedge would be quite small. This is due to the fact that a long way from expiry, the two opposing stock exposures almost cancel out.

However, the fact that the curvature is positive in the lower region and negative in the upper region, makes things complicated for the volatility player attempting to stay delta neutral. In the lower region, the gamma of the position is positive and so rehedging would entail selling stock on the way up and buying on the way down—

the combination is a long volatility position. In the upper region the gamma of the position is negative and so rehedging would entail buying stock on the way up and selling on the way down—the combination is a short volatility position. There is a point on the boundary between the two regions where the curvature or gamma switches from positive to negative, i.e. a point where the gamma is zero and no rehedging would be required. This corresponds to the point with no time decay. At this point the portfolio is neither long nor short volatility and if hedged with the correct amount of short stock, neither long nor short the market.

It may seem strange that someone would want to be in the above position. Until now we have assumed that the individual player, before entering the market, makes a decision about being long or short volatility and so the above combination would not suit. There are situations, however, when the individual may decide that if the market rises, volatility will fall and if the market falls, then volatility will rise. The above combination would perfectly suit such a view. The player would automatically become short volatility on the way up and long volatility on the way down.

Another situation that may make such a combination attractive, is one in which the individual viewed the upper option expensive compared to the lower option.

Here the individual has no view on volatility or the direction of the market price but wishes to arbitrage out the

difference in option values. In order to do this he must short the expensive option and buy the cheap one. The net combination would be long of the underlying and in order to remove market risk, the appropriate hedge would be used. The portfolio would be run either to expiry or until the apparent anomalous price difference disappeared.

Whatever the reason for entering such a portfolio, it is clear that managing the market hedge and monitoring the effects of time and volatility changes is non-trivial—and this is a relatively simple combination.