, after a time interval I, of a long call option is equald21
e 2 Stto  pﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ.2 2 T  t
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AN INTRODUCTION TO OPTIONS TRADING
Of course, for a short position in either a call or a putoption the reverse holds. That is, the profit comes from and the loss from dynamic hedging. So, from the aboveone sees that when the interest rate is 0, the followingrelationship holds between  and :1 2 22  St  þ  ¼ 0
ð7:21Þ
This relationship only holds for an infinitesimally shortperiod of time I and under the assumption that the actualvolatility of the stock is equal to the implied volatility ofthe options. To see this last statement, remember thatthe option Greeks are all calculated with implied volatility as an input.
7.5
THE RELATIONSHIPBETWEEN DYNAMICHEDGING AND h WHEN THEINTEREST RATE IS STRICTLYPOSITIVE
The aim of this section is to find a relationship betweendynamic hedging and  when the interest rate is strictlypositive. In Section 7.4 it was shown that, when the interest rate is 0, the relation was that the expected profit orloss made on dynamic hedging was offset by the expectedprofit or loss made on the time value of the option, whenthe dynamic hedge was made after an infinitesimallysmall period of time I. In a way, this shows that theprice of the option is fair. Mathematically, this can
THE PROFIT OF OPTION TRADERS
89
expressed by:1 2 22  St  þ  ¼ 0
ð7:22Þ
One would expect that, when the interest rate is strictlypositive, the profit or loss made on dynamic hedging isstill offset by the profit or loss of the time value of theoption, provided the dynamic hedge is made after aninfinitesimally small period of time I. This appears tobe the case and can easily be shown from the derivationsin Sections 7.3 and 7.4.Consider an investor holding a long position in a calloption. In Section 7.4 it was shown that the profit on adynamic hedge after a small period of time I is:1 2 22  St  ¼
d212 St
epﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 2 T  t
When the interest rate is strictly positive, some additional terms need to be taken into account to get thetotal profit on the dynamic hedge. Because this investorholds a long position in the call option he not only losesmoney because of the theta but also because he could getinterest on the value of the call option. This loss is equalto the interest rate over the time period I times the priceof the option. Mathematically:7rct ¼ r St Nðd1 Þ  K erðTtÞ Nðd2 Þ
7In fact, this loss is actually equal to ðer  1Þ St Nðd1 Þ  K erðTtÞ Nðd2 Þ .But, when r is small (and this is the case since r is measured over thesmall period of time I), Taylor expansion gives that the loss is equal tor St Nðd1 Þ  K erðTtÞ Nðd2 Þ
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AN INTRODUCTION TO OPTIONS TRADING
where the unit of time of r is I, just like the unit of time of and T  t. Because this investor hedges his portfoliodynamically over the small period of time I, he has a shortposition in  ¼ Nðd1 Þ underlying stocks. This means thathe will receive interest on the value of the stock position,which is equal to:8r St ¼ rNðd1 ÞStSo, when the interest rate is strictly positive, the totalprofit on the dynamic hedge after a small period of time Iis equal to:d21
e 2 Stpﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ  r St Nðd1 Þ  K erðTtÞ Nðd2 Þ þ rNðd1 ÞSt2 2 T  t2d
1
e 2 St¼ pﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ rK erðTtÞ Nðd2 Þ ð7:23Þ2 2 T  tBecause, when the interest rate is positive, the theta of acall option after a small period of time is equal to (the unitof time for , T  t and r is I):2
ed1 2 StI ¼  pﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ  rK erðTtÞ Nðd2 Þ2 2 T  t
ð7:24Þ
the profit on a dynamic hedge is again offset by the lossbecause of theta. Of course, the same derivation can beexecuted for the long put option. The derivation for theshort call and short put option will give a reverse result.That is, the loss that is made on dynamic hedging after a8
Again, the actual profit is ðer  1ÞNðd1 ÞSt , but by Taylor expansion it can beshown that this is equal to rNðd1 ÞSt .
THE PROFIT OF OPTION TRADERS
91
small period of time I is offset by the profit because oftheta. Mathematically, the foregoing reads in terms of:1 2 22  St   rct þ rSt þ  ¼ 0
ð7:25Þ
m1 2 22  St  þ rSt þ  ¼ rct
7.6
ð7:26Þ
CONCLUSION
As shown in the previous sections, the expected profit orloss from dynamic hedging is negative the profit or lossbecause of theta, provided the option portfolio is hedgeddynamically after an infinitesimally small period of time.Although this relation only holds for small periods oftime, it is clear that, approximately, this will also holdfor larger periods of time. It should be emphasized thatthis relation only holds if the stock appears to moveaccording to implied volatility. And this is where thevision of an option trader comes in. If an option traderthinks implied volatility is too low, he should buy theoption. If he is right he is likely to make a bigger profitfrom dynamic hedging (the profit formula for dynamichedging of a long option increases in volatility) than theloss he incurs because of theta. If he thinks impliedvolatility is too high, he should write an option. If heis right his loss on dynamic hedging is likely to be lessthan the profit because of theta. It is important to bear inmind that being able to say whether implied volatility istoo low or too high is no guarantee of making a profit.It really depends on after which move in the underlyingthe portfolio is made delta-neutral. This was shown in
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Section 7.1. The magnitude of the profit also depends onat which point in time the stock realizes its biggestvolatility and how close it is to ‘at the money’. Forexample, consider a long position in an at-the-moneycall option expiring in 10 days’ time. Since the gammaof an at-the-money call option increases as the optioncomes closer to maturity, it is easy to see from the profitformula of dynamic hedging – see equation (7.8) – that itis better to have a 10% move in the stock on the daybefore expiry than 10 days before expiry.In conclusion, the profit of an option trader depends ontwo aspects. First of all, he has to judge whether impliedvolatility is too low or too high and, second, he has tomake his portfolio delta-neutral after the right stock pricemovements and at the right points in time.
Chapter
8OPTION GREEKSIN PRACTICE
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W
hen trading options it is very important to understand the option Greeks, since this helps tradersto understand the risk they are facing. Although the option Greeks primarily help traders to understand theirrisk, it also helps them to determine which strategies willbenefit most from their view on a certain company ortheir view on the overall economy. The following sections will explain the option Greeks in more detail andwill give a flavour for which strategies are most appropriate to benefit from certain company or economicalevents.
8.1
INTERACTION BETWEENGAMMA AND VEGA
As mentioned in Section 6.3, gamma is large for optionswith a short time to maturity and vega is large for optionswith a long time to maturity. However, for both gammaand vega it is the case that they are largest for optionsclose to ‘at the money’. In summary, at-the-money options with a long time to maturity have large vegas andrelatively small gammas, and at-the-money options witha short time to maturity have large gammas and relatively small vegas. So, when a trader wants to decidewhether he should buy a long-dated or a short-datedoption, he is basically deciding whether he wants to doa vega or a gamma trade.In Chapter 7 it was shown that when buying an optionthe profit for the trader comes from movements in theunderlying share, so that he is able to adjust his delta.
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Every time the trader adjusts his delta, he is basicallybuying the underlying share at a low level and selling it ata high level. The reason that movements in the underlying share cause movements in delta is because of thegamma. So, if a trader thinks that a share will go througha volatile period in the next 3 months, after which theshare will become less volatile, the best strategy is to buya short-dated option. This will provide the trader with alarge gamma, so that he will benefit most from the movements in the underlying share. This is a very popularstrategy during the reporting season, since this is thetime when companies give their results and outlook forthe year, which can cause large movements in the underlying share.Another strategy is to buy a long-dated option. As mentioned, a long-dated option has a small gamma and a largevega. So, in contrast to a short-dated option, when a traderbuys a long-dated option he is not so much concernedwith the realized volatility but more with the impliedvolatility. That is to say, when a trader buys a 3-yearoption he wants the 3-year implied volatility to pickup, after which he can sell the option at a better level.The reason that he is not concerned with the realizedvolatility in the near future is because a 3-year option hasa tiny gamma, so he is not going to benefit much fromadjusting the delta of this option. However, when therealized volatility picks up, the implied volatility of a3-year option is likely to pick up as well, only not asmuch as the implied volatility of a 3-month option.This illustrates once more that, when a trader expectsvolatility to pick up in the near future, he is better off
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buying a short-dated option than a long-dated. From thisit is clear that the only justification for buying a longdated option is that the trader expects a big event in themedium to longer term. This event can be a war, an election, the oil price, a merger or virtually anything that islikely to move the share price.The above makes it clear that understanding the optionGreeks not only enables traders to manage their risk, butit also enables them to understand what strategy bestsuits their view on the economy. However, the abovediscusses two very simple strategies for two differenteconomical events. In reality, traders will try to dospreads to give their view on the economy some gearing.For example, just before a reporting season one spreadcould be to buy a 1-month option and sell a 3-monthoption. This strategy will give some gearing to the viewthat the annual report will cause the share price to have abig one-off move, after which the share price will settledown and hardly move at all. Equally, to give gearing tothe view that the share price will hardly move for thecoming year after which a rise in oil prices will cause theshare to go through a volatile year, the best strategy is tobuy a 2-year option and sell a 1-year option.This section only discussed the aspects of gamma andvega strategies under the assumption that the underlyingshare price will be close to the chosen strikes of theoptions throughout the whole term of the options. Thisis because, if the share price moves too far away from thestrike price of an option, the option will have neither
OPTION GREEKS IN PRACTICE
97
gamma nor vega. This issue will be addressed in Section8.2
8.2
THE IMPORTANCE OFTHE DIRECTION OF THEUNDERLYING SHARE TO THEOPTION GREEKS
The problem traders face when buying a long-dated optionis that they do not know where the underlying share willbe trading in, say, 2 years’ time. This section will showwhy it is so important that a trader also needs to have aview on the direction of the stock when trading longdated options rather than just having a view on thevolatility of this stock.Suppose an option trader buys an at-the-money optionwith an implied volatility of 25 expiring in 2 years’ time.As shown in the previous section, this is a typical vegatrade, meaning that the trader expects a pickup in impliedvolatility in the near future after which he can sell theoption at a higher implied volatility level. Since the traderwill have hedged his exposure to the underlying stockthis strategy will make him a profit provided he is rightabout the pickup in implied volatility. However, let’ssuppose that this particular option is very illiquid, so itis highly unlikely that the trader will be able to sell theoption on in the market. This means that the trader willbe stuck with this particular option position for the life ofthe option. One would expect that, if the underlying sharerealizes a volatility of 25 over the life of the option, the
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trader will neither make nor lose money. Unfortunately,it is not as simple as that. Consider a scenario where theshare price drifts away from the strike within 2 years in alow-volatile manner, after which it becomes very volatilecausing the volatility over the life of the option to beexactly 25. In this scenario the trader will most definitelylose money. This is because when the share moves awayfrom the strike the option will have little gamma left. Aswas shown in Chapter 7, a small gamma will cause thetrader to adjust his delta relatively infrequently, in whichcase he will not make enough money from delta-hedgingto earn back the premium paid for the option.
8.3
PIN RISK FOR SHORT-DATEDOPTIONS
It might seem counterintuitive that one of the most riskyoptions appears to be very short-dated at-the-money options. The problem with, say, a 1-day at-the-money optionis that it will be hard to assess whether this option willexpire in or out of the money. The direct result is thatwhen an option trader decides to sell such a 1-day at-themoney call option he does not know what hedge to putagainst this option. Of course, the trader could look at thedelta of this option and hedge accordingly, but becausethe option has such a large gamma the delta almostbecomes meaningless. Inevitably, the trader will haveto take a view on the underlying share price at expiration.In the case of an at-the-money call option the traderwould have to assess whether the underlying share pricewill be higher than the strike at expiration, in which case
OPTION GREEKS IN PRACTICE
99
he will hedge the call option with a one delta (buy thesame number of shares as options), or lower than thestrike, in which case he should hedge with a zero delta(do not buy any shares). Obviously, this strategy can bevery costly if the trader is proved to be wrong. Even if thetrader puts the trade on with a 50% delta (buy half thenumber of shares as options) it could still be a very costlystrategy because of the large gamma. The reason thegamma is so large is because the delta will change from0 to 1 or vice versa every time the underlying share pricegoes through the strike. This is what traders call ‘pinrisk’, the underlying share price being exactly at the strikewhen the option almost expires.
8.4
THE RISKIEST OPTIONS TOGO SHORT
In a low-volatility environment implied volatilitiesusually trade at a premium to realized volatility. Theresult is that it looks attractive to go short options.However, there is quite a good reason for it to look soattractive. The premium one receives for shorting anoption is to cover the event risk. To make this moreclear, consider the following example. An option traderis considering to short an at-the-money 6-month optionon Ahold. Ahold is the Dutch retailer that came intotrouble after accounting scandals at some of its subsidiaries. After a large drop in the share price, causing thevolatility of the stock to be extremely high, the volatilitysettled down. Suppose the 6-month historical volatilityis currently 32% annualized (approximately 2% a day).
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Although the 6-month historic volatility is 32% it ishighly unlikely that anybody would want to short a 6month option on Ahold for an implied volatility of 32%.The reason is that the chance of other irregularities atAhold is just too high, which makes the chance of anotherbig move in the near future quite considerable. Because ofthis high event risk the probability of the stock realizingmore than 32% is just too high. Even if the stock were torealize less than 32% for most of the coming 6 months,and after one announcement of more irregularities itmoves 10% taking the volatility over 6 months exactlyto 32% annualized, the option trader would still losemoney. The reason for this is that the losses ongamma-hedging are squarely related to the move in theunderlying share – see formula (7.16). This means thattwice as big a move results in four times as big a loss. Inthe case of Ahold moving 10% on 1 day, this would resultin 25 times as big a loss because the average move is 2%a day.However, if a trader has good fundamental reasons tobelieve that there will be no further irregularities atAhold, the above phenomenon can also give him a goodopportunity to make money. If he shorts the option andthere are no further scandals he will have been paid apremium in terms of volatility that will not be realized.
Chapter
9SKEW
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W
hen trading options one needs to be familiar withthe phenomenon of ‘skew’. At first, skew is something that is difficult to grasp, but it is perfectly explicable. This chapter gives the definition of skew and givesthe reasons for it.
9.1
WHAT IS SKEW?
Skew is very much related to implied volatility, which isexplained in Chapter 4. Chapter 4 explained that, giventhe price of a specific option in the options market, onecan calculate the implied volatility of this option. Itappears to be the case that options with lower strikeprices have higher implied volatilities. That is to say,that the ’market’ prices options with lower strike pricesrelatively more expensive than options with higher strikeprices. The phenomenon where options with lower strikeprices have higher implied volatilities is called ‘skew’.So, when two 3-month options on Royal Dutch/Shell arecompared, one with a strike price of $40 and one with astrike price of $32, the one with the strike price of$32 will most definitely have a higher implied volatility.This sounds odd considering that both options are expiring in 3 months’ time and, as mentioned in Chapter 4, theimplied volatility is the market’s expectation of the realized volatility of Royal Dutch/Shell in the 3 monthsahead on which the strike price has no influence. Thereasons for skew will be explained in Section 9.2.
SKEW
9.2
103
REASONS FOR SKEW
There are two main reasons for skew which have to benamed together, because one or the other still does notexplain skew. First of all, as long as stock markets existthe experience is that when stock markets are goingdown they are more volatile than when stock marketsgo up. This still does not explain why an option with alower strike price should have a higher implied volatilitythan an option with a higher strike price, since for bothoptions the realized volatility goes up by the sameamount. Apparently, this downward movement followedby a pickup in volatility has a bigger impact on the optionwith the lower strike price than on the one with thehigher strike price. The reason for this is that when thestock is going down it moves away from the higher strikeprice and closer to the lower strike price which, as wasshown in Section 6.1, causes the option with the lowerstrike price to get a larger gamma. According to Chapter 7this, in turn, means that the (low-strike) option delta hasto be re-hedged more frequently resulting in a larger lossfor the investor who sold the low-strike option. Naturally, the seller of a low-strike option wants to get compensated for this phenomenon, which will cause theinvestor to ’charge’ a higher implied volatility for thelower strike option such that he gets a relatively higherpremium for a lower strike option than for a higher strikeoption.
104
9.3
AN INTRODUCTION TO OPTIONS TRADING
REASONS FOR HIGHERVOLATILITIES INFALLING MARKETS
As stated in Section 9.2, it is the case that when shareprices are going down they become more volatile, whichultimately causes skew. The reason for this is that whenshare prices are going down investors are getting moreuncertain about the future and tend to panic. In thisperspective it is good to compare volatility with uncertainty; when there is a lot of uncertainty share prices tendto be more volatile, whereas when there is a lot of certainty share prices move in a very unvolatile manner.Another less scientific and distinct reason, which playspossibly a less significant role, is that although a shareprice is going down investors still think in absolutemovements, which, of course, causes higher volatilityas this is measured in percentages.
Chapter
10SEVERAL OPTIONSTRATEGIES
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T
he most popular option strategies amongst investorsusually involve more than just buying or selling anoption outright. In this chapter several different optionstrategies will be explained. Why investors would executethese option strategies is discussed from a break-evenpoint of view.
10.1
CALL SPREAD
One of the most popular strategies is the call spread. Acall spread involves nothing more than two calls, onewith a low strike and another with a higher strike. Aninvestor is said to be ‘buying the call spread’ if he buys thelower strike and sells the higher strike. ‘Selling the callspread’ means selling the lower strike and buying thehigher one. Buying the call spread is called a ‘bullishstrategy’ because the investor benefits if the underlyingincreases in value. However, the investor’s profits arecapped because he has sold another call with a higherstrike to fund his bullish view on the stock. As an example, consider an investor who buys an at-the-money callon BMW and partially funds this by selling a 120% call onBMW. In this case the investor will profit from an increase in BMW’s share price up to the point where theshare price has reached 120% of its initial value. Everypercentage gain over 20% will not make the investor anymoney, hence his profits are capped. When the two payoffprofiles of a long at-the-money call option and a short120% call option are combined, one gets the payoff graphin Figure 10.1.
107
Payoff
SEVERAL OPTION STRATEGIES
20%
0BMW price
At the money
120%
Figure 10.1 Payoff profile at maturity for an at-the-money/120%call spread on BMW.
10.2
PUT SPREAD
A put spread is the opposite of a call spread in that whenan investor buys a put spread he is executing a ‘bearishstrategy’, whereas buying a call spread is a bullish strategy. Buying a put spread means buying a higher strike putand selling a lower strike put. Since a put with a higherstrike is always more expensive than a put with a lowerstrike and the same maturity, the investor will have topay a premium to put this strategy on. When an investorbuys a put spread he partially funds his bearish view onthe underlying stock by selling a lower strike put. However, by partially funding his bearish view he is alsolimiting the profits of his strategy, because the investoronly benefits from the stock going down, up to the strike
AN INTRODUCTION TO OPTIONS TRADINGPayoff
108
20%
0VW price
80%
At the money
Figure 10.2 Payoff profile at maturity for an at-the-money/80% putspread on Volkswagen.
of the downside put. As an example, consider an investorwho buys an at-the-money put on Volkswagen (VW) andpartially funds this by selling an 80% put on VW. In thiscase the investor will profit from a decline in VW’s shareprice up to the point where it has reached 80% of itsinitial value. If the share price drops below 80% of itsinitial value the investor will not benefit any longer.Again, when the two payoff profiles of a long at-themoney put and a short 80% put are combined, one getsthe payoff graph in Figure 10.2. It is important to understand that the payoff in Figure 10.2 is not the investor’sprofit because he would have paid an initial premium toput this strategy on. So, if this strategy cost him 2% of theinitial share price the investor will only start makingmoney if the share price drops below 98% of the initial
SEVERAL OPTION STRATEGIES
109
share price. However, had he just bought an at-the-moneyput it would have cost him considerably more – forexample, 5% – and, therefore, the share price would haveto drop below 95% of the initial share price before theinvestor would start making money. Obviously, the putoutright would not limit his upside potential to 18% butin theory to 95% if the share price goes to 0.
10.3
COLLAR
The collar or ‘risk reversal’ is used to give gearing to acertain view on the underlying rather than limiting theupside potential of that strategy, as for the call or putspread. A collar consists of a call with a higher strike anda put with a lower strike. If an investor is very bullish on astock and he thinks the downside for the stock is verylimited he could buy an upside call and sell a downsideput to partially fund this strategy. However, the investormight not be limiting his upside potential when the stockgoes up, but by selling the put he does expose himself tothe stock going down. So, the investor is said to be ‘gearing’ his view because he can actually lose more than hisinvestment when the stock ends up lower than the strikeof the put. If an investor is bearish on a stock he could buya downside put and sell an upside call to fund his viewpartially or in full, but therefore exposes himself to thestock price going up. Whether an investor is paying orreceiving money to put on a risk reversal strategy dependson the forward and the skew. It is therefore ambiguous totalk about buying or selling a risk reversal.
AN INTRODUCTION TO OPTIONS TRADING
Payoff
110
0BMW price
90%
Figure 10.3BMW.
110%
Payoff profile at maturity for a 90%/110% collar on
Consider an investor who is very bullish on BMW’s shareprice and, therefore, wants to buy a 1-year 110% call. Healso thinks that if the share price goes down it will not godown more than 10% in the coming year. Therefore, hewants to fund his 110% call by selling a 90% put with thesame maturity. By combining the payoff profile of a long110% call with a short 90% put, one would get the graphin Figure 10.3. Again, it is important to remember thatFigure 10.3 does not show the profit profile of the investor. The profit depends on how much the investorreceived or paid for putting this 90%/110% collar on inthe first place. If BMW’s dividend yield is higher than therisk-free interest rate, the investor will most definitelyhave received money for this strategy. This is because, ifthe dividend yield is higher than the risk-free interest
SEVERAL OPTION STRATEGIES
111
rate, the 1-year forward is less than 100% of the initialshare price which will make the 1-year 90% put morevaluable than the 1-year 110% call. On top of that, theskew will make the 90% put even more valuable and the110% call less valuable.Another feature of the collar is that it has a relatively highdelta, because if a trader buys a put he needs to buy sharesto hedge himself and if the trader also sells a call he needsto buy even more shares in order to be delta-hedged. Because of this high delta the collar can also be used toexpress a dividend view. If a trader thinks BMW is likelyto increase its dividends, he will quite happily take theother side of the investor’s strategy to buy the put and sellthe call. Because, if BMW does increase its dividends, theforward should have been lower than the forward withthe current dividends and the put will therefore increasein value and the call decrease in value.
10.4
STRADDLE
The straddle is a very popular strategy for investors whothink the underlying stock will move away from itscurrent level but do not know whether it will be up ordown. A straddle consists of a call option and a put optionwith the same strike and same maturity. Since buying astraddle means buying both a call and a put option, theunderlying needs to move away from the strike priceconsiderably for the strategy to make money. For thatvery same reason, selling a 1-year at-the-money straddleis a very attractive strategy for investors who think the
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AN INTRODUCTION TO OPTIONS TRADING
underlying will move sideways for the coming year. Sincethe investor will take in the premium of the straddle, hewill most likely have to pay very little at maturity because the stock will not have moved away much fromcurrent levels. So, on either the put or the call the payoutwill be nothing and on the other one it will be very little.As an example, consider an investor who knows that