O-call + KI-call = European call
(10.1)
KO-put + KI-put = European put
(10.2)
As an example, consider a 100/70 % down-and-out put plus a 100/70 %down-and-in put. The down-and-out put means that a 100 % put ceases toexist whenever the 70 % barrier is breached. However, the down-and-input ensures that a new 100 % put comes into existence at the same timeand therefore a down-and-in put plus a down-and-out put is effectivelya regular European put.
10.11 BARRIER AT MATURITY ONLYThe previous sub-sections discussed barrier options where the barrierswere live continuously throughout the life of the option. However, certain
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barrier options are only live at maturity or on specific days. These barrieroptions can still be priced as a Monte Carlo process. However, barrieroptions that are only live at maturity can also be priced as a combinationof European options.Consider a trader who buys a 1 year 100/70 % down-and-in put optionon BMW where the option can only knock in at maturity. This optionis worth less than the equivalent down-and-in put option that can knockin throughout the life of the barrier option. The way to price this downand-in put option with knock in at maturity only is as follows:
r The trader buys 10 times a 1 year 70 % European put optionr The trader sells 10 times a 1 year 67 % European put optionr The trader buys one 1 year 70 % European put optionThe fact that the trader buys 10 times the 70 % put and sells 10 times the67 % put is an overhedge and therefore a conservative way to replicatethe payoff of the 100/70 % down-and-in put at maturity only. The 3 %wide put spread can be seen as a barrier shift. The payoff at maturity forthe down-and-in put (DIP) with knock in at maturity only and the payoffreplicated through a combination of European puts are both shown inFigure 10.3. Figure 10.3 clearly shows that the replication through ageared put spread plus a 70 % put is a lower estimate and thereforeresults in a more conservative (lower) price for the 100/70 % downand-in put with knock in at maturity only. Obviously, the tighter theput spread the more the replication converges to the actual price of thedown-and-in put with knock in at maturity only. The gearing of the putspread depends on the width of the put spread and the difference betweenstrike and barrier level. The gearing can be calculated by dividing thestrike/barrier differential by the width of the put spread. In other words,if one wants to replicate the 100/70 % down-and-in put with knock inat maturity only in terms of a 70 % put plus a 2 % geared put spread,i.e. a 70/68 % put spread, the gearing of the put spread would be 15(30/2 = 15).
10.12 SKEW AND BARRIER OPTIONSBarrier options tend to have large skew exposures. Luckily it is easy tosee whether a barrier option is long or short skew. Risk wise one cancompare a knock in barrier to a long option position at that barrier anda knock out barrier to a short option position at the specific barrier. This
Barrier Options
yPa
40 %
67
IPfDofat
33 %
m
R
uat
30 %
ly
f
fyopa
Percentage payoff
on
edatic
y
rit
lep
60 %
67 %
70 %BMW share price at maturity
Figure 10.3 Payoffs for the down-and-in put, knock in at maturity only, and its replicated payoff, 10 times geared 70/67 % put spread plus a 70 % put
means that the owner of a down-and-in put has a risk profile that canbe compared to being long an option at the knock-in level. Since thisis naturally a downside option, the owner of a down-and-in put optionis long skew. Another way to see that the owner of a down-and-in putoption is long skew is that, if the skew goes up, the stock is expected tobecome more volatile when it goes down and therefore the probabilityof the down-and-in put knocking in goes up. As a result the down-and-input becomes more valuable. A trader who has a short position in an upand-out call is short skew. This can be seen by comparing his positionto a call spread, where he is short a call with a strike price equal to thestrike of the up-and-out call and long an upside call with a strike equalto the barrier level of the up-and-out call. Another way to see that heis short skew is by looking at the price change of the up-and-out call ifthe skew flattens (goes down). If the skew flattens the upside impliedvolatility goes up and therefore it becomes more likely that the up-andout call knocks out, which makes the up-and-out call less valuable. Thetrader, who holds a short position in the up-and-out call, makes moneyfrom the skew flattening and is therefore short skew. Table 10.2 gives
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Table 10.2 Skew position barrier optionPosition barrier option
Skew exposurebarrier option
Direction barrier shift
Long down-and-in putLong up-and-out putLong up-and-in callLong down-and-out callShort up-and-out call
Long skewLong skewShort skewShort skewShort skew
DownwardNo shift neededUpwardNo shift neededUpward
different barrier option positions with their associated skew exposures.Obviously, if the position is opposite to the one stated in Table 10.2, theskew exposure is the opposite as well.
10.13 DOUBLE BARRIERSApart from options with just one single barrier, there is another commonclass of barrier options, namely the so-called double barriers. However,even with a double barrier option one has to consider each barrier separately to determine the magnitude and direction of the barrier shift.Double barrier options are typically used within structured products toachieve a specific type of payoff. Although specific examples of double barrier options and their applications within structured products aregiven in Chapter 25, it is important to devote a sub-section to double barrier options that discusses the two main types of double barrier options.Before looking at the different types of barrier options it is worth emphasising once more that the pricing of any barrier option is established byusing a Monte Carlo process. Therefore, from a trading perspective, themost interesting element of a barrier option is determining the magnitudeand direction of the barrier shift.The first type of double barrier option features a dependency betweenthe trigger of the first and second barrier. That is to say, the second barriercan only trigger if the first barrier has been triggered. Although theMonte Carlo modeling is slightly more complex because of the built incondition, it is in fact more transparent from a risk perspective than if thebarriers were completely independent. Indeed, one specifically knowsthe conditions and therefore the option characteristics under which thebarriers can be triggered. This means that one can more easily establishthe required barrier shift for each of the barriers. Consider the following
Barrier Options
69
example. A trader sells a 100 % call that knocks in at 90 % and knocksout at 120 %. However, it can only knock out after it has breached the90 % level first. For this double barrier it is relatively easy to determinethe direction of the respective barrier shifts. Since the 120 % barrier isdependent on the 90 % barrier, one should first focus on determiningthe shift for the 90 % one. Because the trader is short the double barriercall option, and the double barrier call option increases in value whenthe 90 % level is breached, the trader will want to shift the 90 % barrierlevel downward. To determine the required shift for the 120 % barrier,it is important to establish that this barrier can only trigger once thedouble barrier has already become a call option (the 90 % barrier hasbeen breached). Therefore, to determine the shift for the 120 % barrierone basically has to determine the shift for a regular 100/120 % up-andout call. The required barrier shift for a short up-and-out call is upwards.Indeed, the option goes from being an ‘in the money’ call option to notbeing a call at all and therefore goes down in value significantly. Thetrader will have hedged himself by selling shares, which he needs to buyback once the barrier level is breached. To give himself a cushion to buyback these shares, the trader wants to shift the barrier upwards at the120 % level.For the second type of double barrier option each of the two barriers can be triggered independently. In others words, when the specificbarrier level is breached, the barrier triggers regardless of whether theother barrier has been triggered. This type of double barrier option isslightly easier to model as a Monte Carlo process. However, from a riskperspective it is actually less transparent. The reason being that one hasan indefinite picture of what the option characteristics look like wheneither of the barriers breaches. This becomes clear by considering theexample above with the difference that the 120 % barrier can triggerregardless of whether the 90 % barrier has been triggered. This doublebarrier option is worth less than the conditional one in the example aboveas it can already knock out before it actually has become a call option.This also means that the barrier shift at 120 % can be less if it breachesbefore the 90 % barrier has been breached. However, if the 90 % barrier has been breached first, one requires the same barrier shift as for aregular up-and-out call. Hence, an unconditional double barrier has a bitmore ambiguity regarding the exact barrier shift. Nonetheless, by takingthe same barrier shifts as the conditional double barrier the trader justprices the unconditional double barrier slightly conservatively.
11Forward Starting OptionsA forward starting option is an option which does not start until theso-called forward start date. This also means that the strike price ofthis option is dependent on the underlying share price at this forwardstart date. The price of a forward starting option can in most cases beestimated with the equivalent regular option which has the same term.Obviously, to exactly price this forward starting option one would use aMonte Carlo process that simulates all the different paths according tothe stochastic variables, calculates the price by adding the payoffs foreach path and subsequently divides this sum by the number of generatedpaths. This would give the exact price of the forward starting option,assuming that the stochastic variable inputs are correct and enough pathsare generated. In other words, pricing the forward starting option is easyonce one understands how to input the right stochastic variable for theMonte Carlo process. This is exactly what this section will focus on.Moreover, this chapter will show that skew is the variable to be mostwary of when pricing a forward starting option.
11.1 FORWARD STARTING AND REGULAROPTIONS COMPAREDConsider a forward starting 100 % put option on BMW that starts in onemonth and then has a maturity of 1 year. The fact that the forward startingoption has a strike of 100 % means that, in one month, the strike of theforward starting option is set to the prevailing stock price on that date.Suppose that BMW stock is trading at € 40. One can estimate the price ofthis forward starting option on BMW by pricing a 1 year ‘at the money’put on BMW. This price will be almost the same as the 100 % one monthforward starting option which has a maturity of 1 year from the date ofthe forward start date. Obviously, there is a slight difference in pricebecause of the fact that the buyer of a forward starting option only holdsan option in one month but has to pay for it up-front. This means thatone has to discount the price of the regular 1 year option by one monthto get the price of the 1 month forward starting option. Apart from this
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Exotic Options Trading
interest rate differential there are two other potential pitfalls in derivingthe price of a forward starting option from the equivalent regular option.The first pitfall is that there might be a dividend in the 1 month up tothe forward start date and not the year after. This means that the forwardstarting option has one dividend less than the regular option. Therefore,to derive the price of the forward starting option from the regular option,one has to price the regular option without the dividend that occurs in themonth up to the forward start date. Otherwise, the price of the forwardstarting put option would be overestimated. The second pitfall is that,depending on the strike level in one month, one might not be using thecorrect implied volatility to price the forward starting option. Indeed,it was shown in Chapter 5 that the lower the strike price the higher theimplied volatility. This means that if BMW’s share price goes down inthe month up to the forward start date, the regular option should havebeen priced at a higher implied volatility in order to derive the price ofthe forward starting option from the equivalent regular option. However,if the stock price were to go up during that month, the regular optionshould be priced at a lower implied volatility to derive the correct pricefor the forward starting option. The problem of skew can be solvedby going long delta if one buys the forward starting option and goingshort delta if one sells the forward starting option. The next sub-sectionshows how to hedge the skew risk during the forward start period for theforward starting option. The last sub-section deals with another distinctskew feature that strongly affects the price of a forward starting option,namely the term structure of skew.
11.2 HEDGING THE SKEW DELTA OF THE FORWARDSTART OPTIONThe previous sub-section showed that the price of any forward startingoption is dependent on the underlying share price at the forward startdate, as the strike determines what implied volatility each option shouldbe priced at. However, the strike price only becomes known on the forward start date. Therefore, the implied volatility is always an unknownin pricing the forward starting option. Whether a trader prices the forward start option with a Monte Carlo process or by deriving it from theequivalent regular option, he always uses the current share price as areference for the implied volatility of the forward starting option. Thismeans that, if the trader buys a forward starting option, he loses money ifthe share price goes up and, if he sells a forward starting option, he loses
Forward Starting Options
73
money when the share price goes down. The trader can easily hedgethis exposure by buying or selling the underlying shares. The followingexample shows how this is done in practice.Consider the 100 % forward starting put option on BMW, where thestrike price is set in 1 month to then have a maturity of 1 year. SupposeBMW is trading at € 40 and the ATM implied volatility is 20 %. Furthermore the vega of this forward starting option is 0.30 % and the traderbuys 1 million of these forward starting puts. Also, because of the effectof skew the ‘at the money’ implied volatility goes down by 0.15 % fora 1 % (€ 0.40) move up in BMW’s share price. Since the trader’s bestguess for the ATM implied volatility in one month’s time is the currentimplied volatility of 20 %, the trader exposes himself to an increase inBMW’s share price. Luckily the trader can easily hedge himself to thissensitivity in share price by buying shares. The number of shares thetrader needs to buy can be calculated as follows. For every € 0.40 increase in BMW’s share price the trader loses 1 million multiplied by€ 40 multiplied by 0.30 % multiplied by 0.15 %, which equals a loss of€ 180 per € 0.4 increase in share price. This can be hedged by buying180/0.4 = 450 BMW shares.
11.3 THE FORWARD START OPTION AND THE SKEWTERM STRUCTUREIt is important to recognise that the term structure of skew stronglyimpacts the price of a forward starting option. This has been overlookedby numerous investment banks for a prolonged period and has thereforeresulted in large losses on their side. The following example clearlyshows the risks of the skew term structure in pricing a forward startingoption.Consider a trader who is selling a three year 70 % put option on BMWthat is forward starting in 2 years. If the trader blindly prices this optionoff his implied volatility curve he will look at the 3 year point on hisvolatility surface, which has a much lower skew than the skew for the1 year maturity. However, a three year option that is forward starting in2 years is effectively a 1 year option and should therefore be priced withthe skew of the 1 year maturity. To make this more intuitive, suppose thatthe 3 year ‘at the money’ implied volatility is 24, the 1 year skew is 1.5 %per 10 % and the 3 year skew is 0.5 % per 10 % change in strike price.This means that the volatility surface prescribes an implied volatility of25.5 % for the 70 % 3 year put forward starting in 2 years. However,
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Exotic Options Trading
since the option is forward starting in 2 years, the forward starting 70 %put should be priced with an implied volatility using the 1 year skew,which takes the implied volatility to 28.5 %. In other words, the impliedvolatility one should use to price a forward starting option is that derivedfrom the ‘at the money’1 implied volatility point at expiry of the forwardstarting option and, in case of a downside strike, one should adjust thisvolatility by the prescribed skew of the maturity of the real term of theforward starting option, i.e. the time between expiration and the forwardstart date.
11.4 ANALYTICALLY SHORT SKEW BUTDYNAMICALLY NO SKEW EXPOSUREThe paradox with a forward starting option is that a short position inthe 70 % put on BMW, expiring in 3 years and forward starting in 2years, is analytically short skew but has no skew exposure dynamicallyduring the forward starting period. Dynamically short skew means that,if the share price goes down, the position gets shorter vega and if theshare price goes up, the position gets longer vega. A short position ina regular 3 year 70 % put option on BMW is dynamically short skewas this position gets shorter vega with the share price moving down.The reason that a forward starting option is not short skew dynamicallyduring the forward starting period is that, if the share price moves downthe strike moves down as well. Thus, for a regular short downside option,a downward moving share price means that the share price moves closerto the strike price and therefore the position gets shorter vega. For a shortforward starting option, a downward moving share price does not changeanything for the vega position as the percentage strike stays the same.However, both a short position in the regular and the forward startingoption are short skew analytically. Analytically short skew means that ifthe skew increases, i.e. the difference in implied volatility increases fora 10 % difference in strike, the position loses money, i.e. the downsideoption becomes more valuable and therefore being short this downsideoption costs money. This is obviously the case for a short position ineither the regular option or the forward starting option.The fact that a forward starting option is analytically short skew butnot dynamically makes it extremely difficult to hedge one’s exposure to1One should really derive it from the ‘at the money forward’ volatility. This is the implied volatility at theforward level rather than the current ‘at the money’ level.
Forward Starting Options
75
forward starting options. This is because one can only hedge with regularoptions, which are both analytically as well as dynamically short skew.This means that if one hedges a short 70 % forward starting option bybuying regular options, one needs to role up the strike if the share priceis going up and role down the strike if the share price is going down.To make matters worse, sub-section 11.3 showed that a three year 70 %put forward starting in 2 years is analytically mainly exposed to theone year skew. That would mean that one has to hedge a short positionin this option by buying a 1 year 70 % put. However, as time passesone would need to role forward this hedge to continue to be hedged onthe 1 year skew. This means that it is extremely difficult to analyticallyhedge forward starting options against skew. First of all, one wouldneed to hedge with options having a maturity equal to the term of theforward starting period, i.e. the maturity minus the forward start date,and continue to role this maturity forward as time lapses. On top of that,one would need to role the strike up if the share price moved up and rolethe strike down if the share price moved down.
11.5 FORWARD STARTING GREEKSThe Greeks of a forward starting option are similar to the equivalentoption with a time to maturity equal to the period between forward startdate and expiration. However, delta,2 gamma and theta will only kick inonce the forward starting option has actually struck, i.e. on the forwardstart date. Only then will delta, gamma and theta get a value. For vegathis is not the case and it does have a value regardless of whether theforward starting option has struck. In other words, even if the forwardstarting option has not struck yet it is already exposed to changes inimplied volatility. The vega of a forward starting option is similar toan option with a time to maturity equal to the period between forwardstart date and expiration of the forward starting option. The only maindifference is that the level of volatility used to determine this vega shouldequal the implied volatility of the actual expiration of the forward startingoption rather than the level associated with the actual term of the forwardstarting option.
2
This does not take the skew delta into account.
12Ladder OptionsLadder options are designed for investors who want to get exposure tothe upside of a stock while at the same time locking in the performanceof the stock if it ever goes above certain levels. This type of option isparticularly popular among retail investors and is typically structuredas a capital guaranteed note with unlimited upside participation and theadded advantage that a certain performance is guaranteed once the stockgoes above a certain level. The following sub-section gives an exampleof a ladder option.
12.1 EXAMPLE: LADDER OPTIONConsider an investor who wants to go long BMW stock but wants tomake sure that if the stock ever goes above 105 % or 110 % he receivesat least 5 % or 10 % respectively. The investor decides to buy a 1 yearladder option which has full participation in the upside of BMW stockand assures a payoff of the greater between the performance of BMW and5 % if the stock ever goes above 105 % and the greater of the performanceof BMW and 10 % if the stock ever goes above 110 %. Mathematicallythe payoff at maturity looks like:BMWT − BMW0LadderCT,= max X,BMW0where⎧0 % If BMW never > 105 % during the one year term of the⎪⎪⎪ladder option⎪⎪⎪⎨5 % If BMW ever > 105 % during the one year term of theX=ladder option⎪⎪⎪⎪⎪⎪⎩ 10 % If BMW ever > 110 % during the one year term of theladder optionAlthough the payoff of the ladder option looks complicated, it is surprisingly easy to price with the knowledge of barrier options with rebates.
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Exotic Options Trading
12.2 PRICING THE LADDER OPTIONThe ladder option can be priced as a series of higher strike knockout calls with rebates, where the strike of the following knock-out callequals the barrier level of the previous knock-out call and the rebates areequal to the ‘additional’ performance that gets locked in. Obviously, thehighest strike call does not have a knock-out level. The ladder option insub-section 12.1 can be priced by the following set of options:
r Investor is long a 100 % call which knocks out at 105 % to then pay arebate of 5 %.
r Investor is long a 105 % call which knocks out at 110 % to then pay arebate of 5 %.
r Investor is long a 110 % call without any knock-out level.Since, at the barrier level, the rebate of the knock-out call is equal tothe intrinsic value,1 the discontinuity in option price is minimal over thebarrier. Because the difference in option price is so small over the barrierthe trader barely needs to apply any barrier shift. Moreover, regardlessof when the option knocks out during the term of the option, the trader,who is short this knock-out call, would either need to sell shares overthe barrier or not do anything at all on the underlying stock.2 This meansthat his barrier risk is basically nonexistent as he can always put an offerin at 1 cent below the barrier for the amount of shares that he wouldneed to sell. This means that the most the trader can lose is 1 cent on theamount of shares he would need to sell.
1The time value of the knock-out call will be minimal at the barrier as almost any path of stock price willpredict the stock to go through the barrier.2The ladder structure as a whole might force the trader to buy shares over the barrier because the trader isshort gamma from the higher strike calls. However, this short gamma risk is already priced in and therefore doesnot need to be used as an excuse to apply a barrier shift.
13Lookback OptionsThe principle of a lookback option is to give the investor the maximumpayoff based on perfect hindsight. The lookback takes two forms. Onewhere the settlement price of the option is chosen with perfect hindsightof the stock’s path during the term of the option and the strike if fixed.And one where the strike is chosen with perfect hindsight and the settlement price is the stock price at maturity of the option. In other words,there are four different types of lookback options:
r Fixed strike call lookback (Max lookback)– Pays the difference between the strike and the highest stock levelduring the term of the option.r Fixed strike put lookback (Min lookback)– Pays the difference between the strike and the lowest stock levelduring the term of the option.r Floating strike call lookback– Pays the difference between the stock price at maturity and thelowest stock level during the term of the option.r Floating strike put lookback– Pays the difference between the stock price at maturity and thehighest stock level during the term of the option.
13.1 PRICING AND GAMMA PROFILE OF FIXEDSTRIKE LOOKBACK OPTIONSA fixed strike lookback option will typically be priced with a MonteCarlo process. However, the fixed strike lookback can also be priced asa ladder option where the lock in levels are one tick apart. For example,one can price a fixed strike € 45 call lookback on BMW, where the tickvalue is 1 cent, in the following way:
r Investor is long a € 45 call which knocks out at € 45.01 to then pay arebate of € 0.01.
r Investor is long a € 45.01 call which knocks out at € 45.02 to thenpay a rebate of € 0.01.
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Exotic Options Trading
r Investor is long a € 45.02 call which knocks out at € 45.03 to thenpay a rebate of € 0.01.
...
r Investor is long a € 50 call which knocks out at € 50.01 to then pay arebate of € 0.01.
...Obviously a lookback put option can be priced as a ladder option equivalent to a series of down-and-out puts with a cent rebate on each.The gamma profile of a Max lookback option becomes intuitive whenviewing it as a ladder option. Indeed, as long as the stock goes up therewill be gamma on the lookback option and the gamma will decreasequickly when the stock goes down, as the options below have alreadyknocked out and therefore have no gamma on them any more. Thereforethe gamma of a Max lookback is highest when the stock is at its high,taken from the start of the option term, and increases over time as longas the stock is at or close to its high. For the Min lookback it is exactlythe opposite. The gamma is highest when the stock is at its low, takenfrom the start of the option trade.Therefore the gamma and vega exposure of a Max lookback optioncan be hedged by selling a series of plain vanilla upside calls that are,say, 5 % apart. The Min lookback would be hedged by selling a seriesof downside plain vanilla puts.
13.2 PRICING AND RISK OF A FLOATING STRIKELOOKBACK OPTIONLike the fixed strike lookback, the floating strike lookback would always be priced as a Monte Carlo process. However, unlike the fixedstrike lookback, it is virtually impossible to replicate the floating strikelookback as a set of knock-out options. Since the strike of a floatingstrike lookback call resets if the share price goes down, one can see thatthe closest way to replicate the floating strike lookback call option isas a set of forward starting options. However, the situation occurs thatthese forward starting options are only struck if the share price is lowerthan the share price of all the previous days since the inception of thelookback option. Unfortunately, a knock-in feature that is dependent onthe future share price cannot be captured by the forward starting option.
Lookback Options
81
Nonetheless, to understand the risk of a floating strike lookback optionit is very useful to think in terms of forward starting options. It basicallymeans that a floating strike lookback call option is effectively exposed toforward starting risk for as long as the share price goes down. A floatingstrike lookback put is exposed to forward starting risk for as long as theshare price goes up. This means that the floating strike lookback optionhas a degree of risk associated with forward starting options. To makethis more clear consider the following example.Consider a trader who sells a one year floating strike lookback calloption. The trader is obviously short skew, because as long as the shareprice goes down forward starting options are struck and he therefore getsshorter vega. On top of that, the forward starting option that is struck inthis case is on a higher implied volatility than the ‘at the money’ impliedvolatility. If the share price goes up he will in turn get longer vega. Unlikea regular forward starting option (see sub-section 11.2), the trader wouldnot go short delta against the lookback option since it is not certain thatany forward starting option will get struck. For example, if the shareprice never goes below the initial share price, no forward starting optiongets struck. For a floating strike lookback put option the exact oppositeholds. Consider a trader who sells a floating strike lookback put option.In this case the trader is long skew, as he goes shorter vega if the shareprice goes up because new options get struck.
14CliquetsCliquet options are options where the strike price of the option potentiallyresets at predetermined points in time. The most common cliquet optionis the ratchet option. The ratchet option is an option where the strike priceresets at predetermined points in time while at the same time lockingin the performance of the previous period. This chapter discusses thepricing and risk of the ratchet option and will show that it is nothingmore than a set of forward starting options.
14.1 THE RATCHET OPTIONConsider a 3 year 100 % ratchet call option on BMW that locks in theperformance of BMW in each year. In other words, the strike price ofthe call resets every year to the prevailing stock level of BMW while atthe same time locking in the performance of that year. Suppose that theinitial stock level of BMW is S0 = € 40, after 1 year it is S1 = € 45,after the second year it is S2 = € 50 and after the third year