the share pricemoves up, the trader not only loses money on the extra shares he sold,he also continues to sell too many shares. It is these dynamics that forcea trader to mark his volatility surface per maturity with a skew that hasa downward sloping shape rather than a smile.There is one more parameter that is used to mark an implied volatilitysurface, namely the curvature. Indeed, it is the case that for very highstrikes the implied volatility does not decrease any longer but flattens out,which can be marked in by using a curvature measure. At the same timethis curvature parameter ensures that very low strikes have an even higherimplied volatility than the skew parameter indicates. This is exactly whata trader would want, as very low strikes have very little premium andtherefore sellers want to get properly compensated, which means that theskew parameter alone will not be enough and the curvature parameterwill ensure these low strikes are marked on a larger implied volatility. InFigure 5.2 the implied volatility is plotted against strike, clearly showingthe effects of both the skew parameter and the curvature parameter onthe shape of the implied volatility surface for a specific maturity againststrike price.Without confusing matters too much, even in equity there is a smallelement of a smile shape for the implied volatility per strike. However,this is almost negligible as it is only concerned with upside strikes forwhich the options have a very small premium. Obviously, nobody willsell an option for nothing and therefore a very high strike might getan incredibly large implied volatility for the simple reason that a smallpremium already translates into a large implied volatility.
Exotic Options Trading
Implied volatility
32
Curvature parameter
Skew parameter
Strike
Figure 5.2 Implied volatility plotted against strike price for a specific maturity, pointingout how the skew and the curvature parameters affect the implied volatility surface tobest fit the real world
5.4 SKEW IN FX VERSUS SKEW IN EQUITY: ‘SMILEVERSUS DOWNWARD SLOPING’In sub-section 5.3 it was shown that the implied volatility plotted againststrike has a downward sloping shape per maturity in the equity assetclass. In Foreign exchange (FX), the implied volatility plotted againststrike takes the shape of a smile for any specific maturity. In other words,the implied volatility for options on an exchange rate is higher for bothlower strikes and higher strikes than ‘at the money’. This is becausein FX either currency of the exchange rate can collapse. For example,taking the exchange rate of EUR versus USD, which is currently tradingat € 0.8 per $ 1, it is clear that if the exchange rate goes to € 0.4 per $ 1,the dollar is crashing against the euro and an exchange rate of € 1.2 per$ 1 means that the euro is crashing against the dollar. In both scenariosone can expect more uncertainty and therefore more volatility. For thisreason a trader would charge a higher implied volatility for both the 0.4strike and the 1.2 strike as it is the risk of crashing that causes traders tocharge higher implied volatilities. Figure 5.3 plots the implied volatility against the strike for the EUR/USD exchange rate, clearly showingthe ‘smile’ shape of the implied volatility surface in FX. Obviously,
33
Implied volatility
Skew
0.4
0.8
1.2Strike EUR/USD
Figure 5.3 Implied volatility plotted against strike price for the EUR/USD exchangerate, pointing out the ‘smile’ shape of the volatility surface in FX
the extent of ‘skew’ in either direction depends on which currency isconsidered to be more stable. For example, both the euro and the dollarare considered stable currencies and therefore the ‘skew’ for both lowerand higher strikes is very similar. However, the magnitude of ‘skew’ onan exchange rate between a stable currency and an unstable currencycan be very different for downside strikes than for upside strikes. Forexample, consider the exchange rate between the euro and the Brazilianreal (BRL), which is currently trading at BRL 2.6 per € 1. In this caseit is very unlikely to see a crash of the euro versus the BRL, whereas acrash of the BRL versus the euro is highly possible. In other words, atrader would charge a much higher implied volatility for an FX optionwith a strike of BRL 3.6 per € 1 than for an FX option of BRL 1.6 per€ 1. The reason being that an exchange rate of BRL 3.6 per € 1 meansa large devaluation of the Brazilian real, which will be accompaniedwith large volatility, whereas an exchange rate of BRL 1.6 per € 1 is alarge devaluation of the euro and will most likely be accompanied with adecrease in volatility as it means that the BRL is becoming more stable.Figure 5.4 plots the implied volatility against strike for the BRL/EURexchange rate, and clearly shows that the skew is much more prominentfor higher strikes, i.e. a devaluation of the BRL, than for lower strikes. It
Exotic Options Trading
Implied volatility
34
1.6
2.6
3.6Strike BRL/EUR
Figure 5.4 Implied volatility plotted against strike price for the BRL/EUR exchangerate, pointing out the biased ‘smile’ shape of the implied volatility surface between astable and an unstable currency
could even be argued that the implied volatility should initially go downfor slightly lower strikes than ‘at the money’, after which it picks up forvery low strikes.
5.5 PRICING OPTIONS USING THE SKEW CURVEWhen pricing options it is important to take the skew curve into account.That is to say, pricing an option means pricing it off the full volatilitysurface. Therefore, if one prices, for example, a downside option, onewould look at the volatility curve and price it on a higher implied volatility than an ‘at the money’ option. For a plain vanilla downside optionthe effect of skew on the price is straightforward. However, for a largeclass of exotic options the effect of skew on the price is not that obvious.Hence, each section discusses separately the effect of skew on the priceof the exotic option in question.
6Simple Option Strategies1Before really delving into everything that has to do with exotic options,it is good to be familiar with some simple but commonly used optionstrategies. It is also a prerequisite to know the jargon of the differentoption strategies when dealing with exotic options. Especially becausesimple option strategies are often used to hedge the risks of an exoticoption.The most popular option strategies amongst investors usually involvemore than just buying or selling an option outright. In this chapter severaldifferent option strategies are explained. Why investors would executethese option strategies is discussed from a break-even point of view.As a bonus, some of the following option strategies have an embeddedskew position and can therefore serve as a simple introduction to morecomplex skew positions.
6.1 CALL SPREADOne of the most popular strategies is the call spread. A call spreadinvolves nothing more than two calls, one with a low strike and anotherone with a higher strike. An investor is said to be buying the call spreadif he buys the lower strike and sells the higher strike. Selling the callspread means selling the lower strike and buying the higher one. Buyingthe call spread is called a bullish strategy because the investor benefitsif the underlying increases in value. However, the investor’s profits arecapped because he has sold another call with a higher strike to fundhis bullish view on the stock. As an example consider an investor whobuys an at the money call on BMW and partially funds this by sellinga 120 % call on BMW. In this case the investor profits from an increasein BMW’s share price up to the point where the share price has reached1Parts of this chapter have been previously published in de Weert, F. (2006) An Introduction to OptionsTrading, John Wiley & Sons Ltd, Chichester. Reproduced with permission.
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Exotic Options Trading
Payoff
20 %
0BMW Price
ATM
120 %
Figure 6.1 Payoff profile at maturity for an ATM/120 % call spread on BMW
120 % of its initial value. Every percentage gain over 20 % does notmake the investor any money, hence his profits are capped. When thetwo payoff profiles of a long ATM call option and a short 120 % calloption are combined, one gets the payoff graph in Figure 6.1.From a skew perspective it is interesting to look at the call spreadposition from the trader’s point of view. The trader, who is selling the100 % call and buying the 120 % call, is short skew as he sells a lowerstrike option with a higher implied volatility than the higher strike optionwhich he buys. The trader would therefore benefit if the skew goes down.To put it differently, the trader would make money if the skew becomesmore shallow. This can happen when
r The implied volatility of the 100 % call goes down and the impliedvolatility of the 120 % call stays the same. This results in a smallerdifference in implied volatility between the 100 % call and the 120 %call and hence the skew is more shallow.r The implied volatility of the 120 % goes up and the implied volatility ofthe 100 % call stays the same. This means that the implied volatilityof the 120 %, which the trader is long, moves closer to the implied
Simple Option Strategies
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volatility of the 100 % call. The balance is that the skew becomesmore shallow and the trader makes money as a result.r The implied volatility of the 100 % call goes down and the impliedvolatility of the 120 % call goes up. Again, the result is that the skewis less steep, which results in a profit to the trader.The above shows the possibility for the trader to make money as aresult of skew becoming less steep. Obviously, one could think of evenmore scenarios in which the trader benefits from a more shallow skew.For example, when both the implied volatility of the 100 % call andthe implied volatility of the 120 % go down, but the implied volatilityof the 100 % call goes down more than the implied volatility of the120 % call. However, this scenario is not as clear cut as the above threescenarios. For example, if the vega of the 120 % call is larger than thevega of the 100 % call, a parallel downward shift in implied volatilityfor all strikes has a larger negative effect on the trader’s long 120 % callposition than the positive counter effect on the trader’s short 100 % callposition. This means that, even if the implied volatility of the 100 %call goes down more than the implied volatility of the 120 % call itmight not be enough to offset the negative effect because of the longvega loss.
6.2 PUT SPREADA put spread is opposite to a call spread in the sense that, when an investorbuys a put spread he is executing a bearish strategy, whereas buying acall spread is a bullish strategy. Buying a put spread means buying ahigher strike put and selling a lower strike put. Since a put with a higherstrike is always more expensive than a put with a lower strike and thesame maturity, the investor has to pay a premium to put this strategy on.When an investor buys a put spread he partially funds his bearish view onthe underlying stock by selling a lower strike put. However, by partiallyfunding his bearish view he is also limiting the profits of his strategy,because the investor only benefits from the stock going down, up to thestrike of the downside put. As an example consider an investor who buysan at the money put on Volkswagen (VW) and partially funds this byselling an 80 % put on VW. In this case the investor does profit from adecline in VW’s share price up to the point where it has reached 80 % ofits initial value. If the share price drops below 80 % of its initial value,
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Payoff
20 %
0VW Price
80 %
ATM
Figure 6.2 Payoff profile at maturity for an ATM/80 % put spread on Volkswagen
the investor does not benefit any longer. Again, when the two payoffprofiles of a long at the money put and a short 80 % put are combined,one gets the payoff graph in Figure 6.2. It is important to understandthat the payoff in Figure 6.2 is not the investor’s profit because he wouldhave paid an initial premium to put this strategy on. So, if this strategycosts him 2 % of the initial share price, the investor only starts to makemoney if the share price drops below 98 % of the initial share price.However, had he just bought an at the money put it would have costhim considerably more, for example 5 %, and therefore the share pricewould have to drop below 95 % of the initial share price before theinvestor would start to make money. Obviously, the put outright wouldnot limit his upside potential to 18 % but in theory to 95 % if the shareprice went to zero.Consider the trader who is selling the 100 %/80 % put spread. In thiscase the trader is long the 80 % put and short the 100 % put. Since heis long the lower strike option and short the higher strike option, thetrader is long skew. This means that the trader would benefit from asteepening skew. In other words, if the difference in implied volatility
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for a 10 % change in strike goes up, the trader, who is long skew, makes aprofit.2
6.3 COLLARThe collar or the risk reversal is used to give gearing to a certain view onthe underlying rather than limiting the upside potential of that strategy,as for the call or put spread. A collar consists of a call with a higherstrike and a put with a lower strike. If an investor is very bullish on astock and he thinks the downside for the stock is very limited, he couldbuy an upside call and sell a downside put to partially fund this strategy.However, the investor might not be limiting his upside potential when thestock goes up, but by selling the put he does expose himself to the stockgoing down. So the investor is said to be gearing his view because hecan actually lose more than his investment when the stock ends up lowerthan the strike of the put. If an investor is bearish on a stock he could buya downside put and sell an upside call to fund his view partially or in full,but he therefore exposes himself to the stock price going up. Whether aninvestor is paying or receiving money to put on a risk reversal strategydepends on the forward and the skew. It is therefore ambiguous to talkabout buying or selling a risk reversal.Consider an investor who is very bullish on BMW’s share price andtherefore wants to buy a 1 year 110 % call. He also thinks that if the shareprice goes down it will not go down more than 10 % in the coming year.Therefore he wants to fund his 110 % call by selling a 90 % put withthe same maturity. By combining the payoff profile of a long 110 % callwith a short 90 % put one would get the graph in Figure 6.3. Again, it isimportant to remember that Figure 6.3 does not show the profit profileof the investor. The profit depends on how much the investor received orpaid for putting this 90 %/110 % collar on in the first place. If BMW’sdividend yield is higher than the risk free interest rate, the investor willmost definitely have received money for this strategy. This is because,if the dividend yield is higher than the risk free interest rate, the 1 yearforward is less than 100 % of the initial share price, which makes the2Depending on the net vega position of the put spread, a steepening in skew does not always result in a profitto the trader. This was discussed in sub-section 6.1. A steepening of the skew as a result of the 80 % impliedvolatility going up and the 100 % staying the same or of the 100 % going down and the 80 % implied volatilityremaining equal or lastly of the 80 % implied volatility going up and the 100 % volatility going down alwaysmakes the trader a profit.
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Exotic Options Trading
Payoff
0BMW price
90 %
110 %
Figure 6.3 Payoff profile at maturity for a 90 %/110 % collar on BMW
1 year 90 % put more valuable than the 1 year 110 % call. On top of thatthe skew makes the 90 % put even more valuable and the 110 % call lessvaluable.Another feature of the collar is that it has a relatively high delta,because, if a trader buys a put, he needs to buy shares to hedge himselfand if the trader also sells a call, he needs to buy even more shares inorder to be delta hedged. Because of this high delta the collar can alsobe used to express a dividend view. If a trader thinks BMW is likelyto increase its dividends he quite happily takes the other side of theinvestor’s strategy to buy the put and sell the call. Because, if BMWdoes increase its dividends, the forward should have been lower than theforward with the current dividends and the put will therefore increase invalue and the call will decrease in value. The trader is also exposed tochanges in skew as he is long the lower strike option and short the higherstrike option. He is therefore long skew and benefits if the skew goes up.
6.4 STRADDLEThe straddle is a very popular strategy for investors who think theunderlying stock will move away from its current level but do not knowwhether it will be up or down. A straddle consists of a call option and
Simple Option Strategies
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a put option with the same strike and same maturity. Since buying astraddle means buying both a call and a put option, the underlying needsto move away from the strike price considerably for the strategy to makemoney. For that very same reason selling a one year ATM straddle is avery attractive strategy for investors who think the underlying will movesideways for the coming year. The investor will take in the premiumof the straddle and will most likely have to pay very little at maturitybecause the stock will not have moved away much from current levels.So, on either the put or the call, the payout will be nothing and on theother one it will be very little.As an example, consider an investor who knows that TomTom, themaker of car navigation systems, is due to come out with a statementwhich will be a real share mover. However, the investor does not knowwhether it will affect the share price positively or negatively. For thatreason he decides to buy a 6 months at the money straddle on TomTom.Combining the two payoff profiles of a long at the money call and along at the money put, one will get Figure 6.4. To show the differencebetween the payoff of buying an ATM straddle on TomTom and theprofit of buying this straddle, suppose the investor paid € 3 for the ATMstraddle on TomTom and the share price is currently trading at € 20. The
Payoff
0TomTom price
ATM
Figure 6.4 Payoff profile at maturity for a long ATM straddle on TomTom
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Exotic Options Trading
Profit
0TomTom price–3
17
20
23
Figure 6.5 Payoff profile at maturity for a long ATM straddle on TomTom
profit profile at maturity of buying an ATM straddle on TomTom wouldlook like Figure 6.5.The straddle is the perfect tool to play the volatility. Since a straddleconsists of both a call and a put, it has a very large vega exposure. On topof that it has very little dividend exposure as the deltas of the call and theput offset each other. Therefore, a trader who believes that the volatilitywill go up in the near future should buy a straddle and subsequently deltahedge it. A trader who believes that the volatility will go down shouldsell a straddle and delta hedge it rigorously.
6.5 STRANGLEThe strangle is very similar to the straddle with the difference that thecall and the put do not have the same strike. So, a strangle consists of acall with a higher strike and a put with a lower strike. Therefore, like thestraddle, the strangle is also for investors who think that the underlyingshare will move away from current levels. However, an investor mightprefer a 90 %/110 % strangle to an ATM straddle because he thinks thatthe share price will move much more than 10 % from current levels buthe wants to minimize his loss in case the share price does not move at
Simple Option Strategies
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Payoff
0TomTom price
90 %
ATM
110 %
Figure 6.6 Profit profile at maturity for a long 90 %/110 % strangle on TomTom
all. By using the strangle to support his view, his initial investment ismuch less than when buying the ATM straddle. If the share does movemuch more than 10 % from current levels he profits almost as much aswith an ATM straddle, but he has paid much less than in the case of thestraddle, which makes his strategy less risky. As an example, consideran investor who thinks that TomTom’s share price will either move awayfrom its current € 20 level by at least 20 % in the next six months or notdo anything at all. Instead of buying an at the money straddle where theinitial investment is large and in case of a big move the profit at maturityis large as well, the investor decides to buy a 6 months 90 %/110 %strangle where the initial investment is small and in case of a big movethe profit is almost as large as for the straddle. The payoff profile atmaturity for this strategy is shown in Figure 6.6.To see that this strategy is less risky than buying a six months ATMstraddle, suppose the investor paid € 1.50 for the strangle and as per theexample in section 6.4 he would have to pay € 3 for the ATM straddle.If the share price were to move down by 20 %, TomTom’s share pricewould be € 16. The strangle would therefore make a profit of € 0.50(the strike of the 90 % put is € 18 and therefore makes € 2 on this put)and the straddle would make a profit of € 1. The two different profit
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StraddleProfit016.5
23.5
Strangle
–1.5TomTom price
–317
18
20
22
23
Figure 6.7 Profit profiles at maturity for a long 90 %/110 % strangle and an ATMstraddle on TomTom
profiles of the straddle and strangle are shown in Figure 6.7. Figure 6.7clearly shows that the strangle is a less risky strategy but the potentialprofit is less as well. However, for very big moves the difference inprofit between the straddle and the strangle is very small in percentageterms.
7Monte Carlo ProcessesMost exotic options are ultimately priced with the aid of a Monte Carloprocess. However, before using a Monte Carlo process one first needsto establish the inputs for this Monte Carlo process. These inputs canonly be determined when understanding the exact risks and behaviourof the Greeks. Therefore, when pricing exotic options it is much moreimportant to be aware of the risks than to fully understand the actualpricing of it through a Monte Carlo process. Nonetheless, it is importantto have an understanding and intuition of the workings of a Monte Carloprocess. This section is looking to provide exactly this understandingwithout discussing the different mathematical models that are concernedwith generating the paths of a Monte Carlo process.
7.1 MONTE CARLO PROCESS PRINCIPLEThe principle of a Monte Carlo process is to generate a large but finitenumber of paths, aggregate the option payoffs associated with each pathand subsequently divide the sum by the number of simulated paths. Thesepaths are generated according to the process by which the stock or otherunderlying is assumed to move. For example, under Black–Scholes,stock price movements are assumed to behave like a Brownian motion.This means that the paths would be generated by a computer programusing a standard normal distribution, where at any point in time the nextmove is distributed as a standard normal distribution with the impliedvolatility as standard deviation of this normal distribution. Armed witha large set of paths one can estimate the price of an option by addingthe option payoffs corresponding to each respective path and dividingit by the number of simulated paths. In other words, let ωi be the pathcorresponding to the i-th simulation and let f (ωi ) be the correspondingoption payoff, the option price obtained through Monte Carlo simulationisN1 f (ωi ).N i=1
(7.1)
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Although formula 7.1 comes across as a very rough estimate of thespecific option price, the Central Limit Theorem says that the error inthis estimation converges to zero with a speed of1√ .N
(7.2)
This means that a large enough number of paths ensures that an optionprice obtained with Monte Carlo simulation is very close to the fair valueof this option.
7.2 BINOMIAL TREE VERSUS MONTECARLO PROCESSA binomial tree is another popular iterative method to calculate the priceof an option. The difference between a binomial tree and a Monte Carloprocess is that the binomial tree requires a discrete (not continuous intime) probability measure and it is therefore a less accurate approximation of the real movement of the underlying. On the other hand, becauseit has a finite number of observations, a real expectation can be calculated as the probabilities do add up to 1, whereas a Monte Carlo processonly converges for a large enough number of paths to the expected valueand therefore the price of the option. Since, for a continuous process,the probability of one specific path occurring is zero, it is impossiblefor a Monte Carlo process to calculate an expectation directly. However,for a large enough number of paths, one can expect the proportion ofpaths within a certain range to be equal to the probability of that stockprice falling in that range prescribed by the process according to whichthe stock is supposed to move. Although the binomial tree and a MonteCarlo process bear some significant differences, the reason that the binomial tree is discussed in this section is because it is the easiest way toshow how a Monte Carlo process derives the value of, for example, anAmerican option where the possibility of early exercising should alsobe taken into account in the price.
7.3 BINOMIAL TREE EXAMPLEConsider the binomial tree in Figure 7.1 on BMW’s share price.Figure 7.1 depicts a 2 year binomial tree where with a 50 % chancethe stock price is multiplied by 1.1 the following year and with another
Monte Carlo Processes
119
50 %
110
50 %
100
50 %
47
50 %
98
50 %
91.91
50 %
80.64Figure 7.1 Binomial tree with 50 % chance of stock price being multiplied by 1.1 and150 % chance of being multiplied by 1.1and a dividend of € 2 in the second year
50 % chance the stock price is divided by 1.1 the following year. There isone catch in this binomial tree because there is a € 2 dividend in the second year and therefore the stock prices in the second year are € 2 lowerthan expected. Knowing the 2 year stock movement according to the binomial tree in Figure 7.1 it is easy to price a 2 year € 97 European call.Indeed, there are only two paths where the € 97 European call has a payout. Namely when BMW’s stock price is € 98 after two years and whenit is € 119, both of which have a 25 % chance of occurring. Assumingthat the interest rate is zero, the price of the 2 year € 97 European call is0.25 × (98 − 97) + 0.25 × (119 − 97) = € 5.75When pricing a two year € 97 American call, one also has to checkat every point in the tree whether it is optimal to early exercise theAmerican call option. For the tree in Figure 7.1, the only point thatneeds to be checked is the € 110 stock price after 1 year. In other words,one has to look at the part of the tree in Figure 7.2 to determine whetherthe option of early exercising has an impact on the price of the Americanoption.The payout on the € 97 American call is € 13 if one were toexercise after 1 year. Assuming that the interest rate is zero and thatBMW’s share price is trading at € 110 after one year, the present valueof the expected value of the € 97 American call in the second year is0.5 × (119 − 97) + 0.5 × (100 − 97) = € 12.5
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119110
50 %50 %
98Figure 7.2 Second year of the binomial tree from Figure 7.1 zoomed in to the ‘up’ path
This means that it is optimal to exercise the 2 year € 97 American callon BMW after the first year if BMW is then trading at € 110. The priceof the American call can therefore be established as0.5 × (110 − 97) + 0.5 × 0 = € 6.5.
7.4 THE WORKINGS OF THE MONTECARLO PROCESSKnowing that, for a large enough number of paths, the percentage ofpaths falling within a certain range converges to the probability of thatrange occurring according to the process by which the stock is supposedto move, a Monte Carlo simulation can effectively be seen as an expectation. Therefore, determining the fair option value through a MonteCarlo process is ultimately the same idea as calculating the expectedvalue of an option from a binomial tree. For a Monte Carlo process, acomputer program randomly generates paths for which the associatedoption payoffs are aggregated and subsequently divide by the simulatednumber of paths to get the fair option value. On the other hand, a binomial tree reduces the stock price movements to a finite number of pathsand hence compromises the accuracy of the stock price movements, butthe expected value is real as the probabilities do add up to 1. In otherwords, increasing the number of paths of a Monte Carlo process meansthat the outcome resembles an expected value more closely and therefore the Monte Carlo estimation is closer to the fair value of the option,whereas increasing the number of paths of a binomial tree means that thebinomial tree more closely resembles the actual stock price movementsand hence the expected value of the option is closer to the fair value ofthe option.
8Chooser OptionBefore proceeding with discussing real exotic options, the chooser option is an interesting one to discuss. The chooser option is often characterised as an exotic option because it comes across as an option thatis difficult to price. However, closer investigation of the chooser optionshows that it can actually be priced as a series of regular options.The buyer of a chooser option has the right to decide, up to a certaindate, whether it should be a call option or a put option. The strikes ofeither the put or the call can be the same but need not necessarily be thesame. When the chooser option specifies that the strikes are the same,the chooser option is referred to as a simple chooser. When the strikesor even the expiries are not the same the chooser is referred to as acomplex chooser. The following sub-section discusses an example ofhow a simple chooser can be priced.
8.1 PRICING EXAMPLE: SIMPLE CHOOSER OPTIONConsider a European chooser option on BMW with a maturity of 2 years,a strike price of € 40 and the buyer has 2 months to choose whether hewants the chooser to be a € 40 call or a € 40 put, see Table 8.1. Inother words, the fact that the cho