oser option is European means that,once the buyer has chosen whether the chooser should be a put or a call,the chosen option is a European style option and can therefore only beexercised at maturity. The European property has therefore nothing to dowith the timing of the buyer’s choose date. Namely, the buyer can pickany day up to 2 months to decide whether the chooser should be a callor a put. However, the buyer would always pick the last possible date todecide whether the chooser should be a put or a call as there is no benefitin choosing earlier. Suppose that the 2 year forward is € 40 when BMWis worth € 38 after 2 months, i.e. the 22 months forward in 2 months is€ 40 if BMW’s share price is equal to € 38 in two months’ time. Armedwith this assumption the 2 year chooser option on BMW can be priced
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Table 8.1 Terms of the chooser optionUnderlyingOption styleOption typeMaturityChoose dateStrike
BMWEuropeanChooser2 yearsUp to two months€ 40
as follows by the trader. In other words, if the trader buys the followingposition he is fully hedged against the short chooser option.
r Buy a 2 year € 40 strike European call plusr Buy a 2 months € 38 strike European put.To see that the sum of the above two options prices the chooser optionon BMW accurately, consider the following two scenarios1. After two months BMW’s share price is above € 38.Since the 2 months put expires worthless, the trader is just left witha € 40 call option expiring in 22 months. This mirrors exactly theposition as a result of the choice of the buyer of the chooser option.Indeed, the buyer of the chooser option decides to turn the chooseroption into a € 40 call option instead of a put option since the 22months forward is higher than € 40 and therefore the call is worthmore than the put.2. After two months BMW’s share price is below € 38.In this case the put expires ‘in the money’ and therefore the traderexercises the € 38 put within his hedging portfolio, which results ina short share position for the trader. This means that the trader isleft with a short share position combined with a long call position.Put–call parity means that a short share position combined with along call position is equal to a long put position. Again, this mirrorsexactly the short chooser position. Indeed, the buyer of the chooserposition decides to turn the chooser into a 22 months € 40 put as the22 months forward is below € 40 and therefore the 22 months € 40put is worth more than the 22 months € 40 call.The above shows that a simple chooser is easy to price. Consider achooser option with a strike price K c , a maturity T1 and a maturity upto the choose date of T2 . To price this chooser option, one just has to
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determine the stock level, S f , such that the forward on the choose datewith an expiry equal to the maturity date of the chooser is equal to thestrike, K c , of the chooser, i.e. the forward value of a stock price S f anda time to maturity of T1 − T2 is equal to K c . A put and a call optionwith a strike equal to the forward and the same maturity are worth thesame. Hence, on the choose date, S f determines the tipping point for thechooser call option to be worth more than the chooser put option. Thechooser option can now be priced as
r A call option with a time to maturity of T1 and a strike K c plusr A put option with a time to maturity T2 and a strike price S f .Alternatively, the chooser option can also be priced as
r A put option with a time to maturity of T1 and a strike K c plusr A call option with a time to maturity T2 and a strike price S f .This is because being long a put option and a share is the same as beinglong a call option, which mirrors exactly the position of the short chooseroption when, on the choose date, the share price is above S f in which casethe owner of the chooser turns the chooser into a call option. Obviously,when, on the choose date, the share price is below S f the owner of thechooser decides to choose the put option. Again, mirroring the positionof the hedged portfolio as the call with strike S f and maturity T2 is ‘outof the money’ and therefore not exercised. This results in a long putposition in the hedged portfolio.
8.2 RATIONALE BEHIND CHOOSER OPTIONSTRATEGIESThe investors in chooser options apply a similar strategy to investorsin straddles. They both believe that a stock is about to go through avolatile period but are unsure about the direction of the stock price. Thedifference between an investor in a chooser option and an investor instraddles is that the first is confident that the direction is revealed withina certain time frame and is therefore willing to choose between a putand a call after this time frame. As a result the investor pays less for thechooser option than he would have paid for the equivalent straddle.
9Digital OptionsA digital option is one of the most straightforward options. It is nothingmore than an option that gives a fixed payout if it is below or above acertain point and does not give a payout at all in all other cases. Despitebeing so simple, the digital option could still classify as an exotic optionas its payout cannot be replicated by a set of standard options. However,rather than viewing this as an exotic option and using a Monte Carloprocess to calculate the price of a digital option, one can make a goodestimate of the price of a digital option by using option spreads. On top ofthat one will find that the only real way to risk manage the digital optionis with option spreads. The vast majority of digital options are European.Therefore this chapter only considers European digital options.Consider the following example. An investor buys a 3 month Europeandigital option on BMW which pays € 10 if after 3 months the stock isabove € 50 and pays 0 if BMW is below € 50 at maturity. Figure 9.1shows the payoff profile at maturity for this digital option on BMW.The trader who sells this digital option on BMW can easily replicatethe price of this digital by using a geared call spread. The gearing ofthis call spread depends on the width of the call spread. The wider thecall spread the lesser the gearing and the more conservative the price,i.e. the trader would charge the investor more to buy the digital. Thetrader believes that a € 2.5 wide call spread is enough to be able torisk manage this position. In other words, the trader will price this byselling a € 47.5/50 call spread that is 4 times geared. To see that thisreplicates the behaviour of the digital, consider three different BMWshare prices at maturity. First, suppose BMW’s share price at maturityis € 55. Under the terms of the digital the investor is supposed to get€ 10. This is exactly what the trader’s position prescribes. Indeed, thetrader has a 4 times geared € 47.5/50 call spread and each call spreadgives a payout of € 2.5 if the stock price ends up at € 55, resultingin a total payout to the investor of € 10. Secondly, If the share priceof BMW is € 49 at maturity, the investor is not supposed to receiveanything under the terms of the digital. However, the 4 times geared€ 47.5/50 call spread prescribes a payout of € 6, resulting in a windfall
Exotic Options Trading
Payoff
54
10
BMW stock price at maturity
0
50
Figure 9.1 Payoff at maturity from buying a digital call option on BMW. Strike price =€ 50 and digital payout € 10
of € 6 to the trader. This obviously shows that the trader has priced thedigital conservatively by choosing this € 47.5/50 call spread and wouldhave been more aggressive by choosing a tighter call spread. However,the width of the call spread is necessary to make up for the large gammaand pin risk around € 50. Indeed, just before expiry with the stock priceexactly at € 50 the digital is not worth anything, but if the stock pricegoes up to € 50.01 the digital is all of a sudden worth € 10. This isobviously extremely hard to risk manage because the trader would notknow what delta to put against this option. Therefore the best way toprice this digital is by means of a geared call spread. Even with a callspread the trader would still need to manage a large pin and gamma risk,but the fact that he prices the digital as a call spread gives him a provisionagainst this risk. However, since the call spread gives the trader a shortposition in the € 47.5 call and a long position in the € 50 call, the traderhas shifted his pin risk to the 47.5 strike and enjoys the benefit of pin onthe 50 strike. In a way, the trader has given himself a cushion to managethe pin risk of the digital by forcing himself to be hedged from € 47.5onwards because of the call spread. Thirdly, when the share price ofBMW is € 46 neither the digital nor the call spread prescribes a payoffto the investor.
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9.1 CHOOSING THE STRIKESA natural question would be ‘why is the digital in the previous section not replicated as a € 50/52.5 call spread’? The answer is that the€ 50/52.5 call spread achieves exactly the opposite of what the traderwants to achieve, namely giving himself a cushion. The € 50/52.5 callspread starts to act once the digital event has already occurred. In a waythe € 50/52.5 call spread makes the trader feel ‘richer’ than he actuallyis. For example, if the stock price ends up at € 51 the 4 times gearedcall spread would specify a payoff of € 4 whereas the digital woulddemand a payoff of € 10. In other words, the hedging scheme of the€ 50/52.5 call spread does not build up to a € 10 payout at a stock priceof € 50 whereas a € 47.5/50 call spread does exactly that. In a way,the € 50/52.5 call spread is chasing the tail of the digital whereas the€ 47.5/50 is forcing the trader to act before the digital actually kicks inand is therefore well positioned to deal with the ‘explosive’ digital event.
9.2 THE CALL SPREAD AS PROXY FOR THE DIGITALA call spread is not only used to price a digital but, from the perspective of the trader, it is also the product that he actually trades. In otherwords, when a trader sells a digital he books a call spread in his riskmanagement system instead of the exact terms of a digital. In this way,he can accurately risk manage the digital as a call spread. The previoussub-section showed that a call spread is a conservative proxy for thedigital and therefore at expiry of the European digital the trader checksthe payout of the call spread against the payout according to the termsof the digital. A payout of the call spread versus a non payout of thedigital subsequently results in a windfall to the trader.
9.3 WIDTH OF THE CALL SPREAD VERSUSGEARINGThe narrower the width of the call spread the higher the gearing necessaryto replicate the digital. In turn, the higher the gearing of the call spreadthe larger the risk that needs to be managed on the short strike, andsince the strikes of the call spread are so close together the trader giveshimself less of a cushion to prepare for the digital event. In other words,the narrower the width of the replicating call spread the higher the riskand the more the call spread behaves like the digital.
10Barrier OptionsBarrier options are very popular amongst retail investors as the barrierfeature provides the investor with additional protection or leverage. Froma risk management perspective, barrier options are interesting becausethe risks are discontinuous around the barrier and therefore the Greeksbecome less predictable and very often even change sign around thebarrier. Barrier options are priced with Monte Carlo processes, but beforeone can price barrier options one has to be fully aware of the risks aroundthe barrier. The risks associated with the barrier are typically of such anature that the terms of the barrier option have to be adjusted slightlyto be able to capture all these risks in the price and to also be able tomanage the barrier risk properly. This section discusses in depth the risksembedded in barrier options and shows how these risks should be takeninto account in the price. This section also emphasises the skew riskassociated with a barrier option. Although there are basically 8 types ofsingle barrier options, the down-and-in put is used as a leading examplein order to get across all the risks within a barrier option.The reason behind this approach is that the underlying causes forbarrier option risk are generic and once the drivers of barrier optionrisk are understood for the down-and-in put, one will be able to derivethe risks for other types of barrier options. However, in this regard it isimportant to distinguish between the drivers behind risk and the actualrisk. The fact that the drivers behind risk are the same for the differenttypes of barrier options does not mean that the risks are the same. Infact, between different types of barrier options the symptoms of risk canbe completely opposite in nature. There are in total 8 different types ofsingle barrier options and the down-and-in put is chosen as a leadingexample because of its popularity amongst retail investors. The 8 typesof single barrier options, the typical retail demand1 and the barrier risk tothe trader for each type of barrier option are summarised in Table 10.1.
1Whether it is a barrier option or not, retail investors will typically sell puts and buy calls. However, it doesoccur that a retail investor effectively buys a down-and-out put as part of a structured product.
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Table 10.1 Types of single barrier optionsBarrier type
Typical retail demand
Trader’s barrier risk
Down-and-in putDown-and-out putUp-and-in putUp-and-out putDown-and-in callDown-and-out callUp-and-in callUp-and-out call
SellSell/BuySellSellBuyBuyBuyBuy
SignificantSignificantLowLowLowLowSignificantSignificant
Aside from single barrier options another class of barrier optionsexists, namely the double barrier options. Double barriers are discussedin sub-section 10.13.
10.1 DOWN-AND-IN PUT OPTIONA down-and-in put option is an option that only becomes a European putoption when the share price breaches a downside barrier. For example,consider a trader who buys a 1 year 100 % European put option onBMW that only becomes a put option if BMW’s share price ever dropsbelow 70 % of BMW’s initial share price during the one year term ofthe option. In other words, the trader is long a 100/70 % down-and-input and its payout at maturity is shown graphically in Figure 10.1. Thedotted diagonal line in Figure 10.1 indicates that the payoff in this areaonly occurs at maturity if the barrier of 70 % has been hit during the lifeof the option. If this has not been the case, the payoff would still be 0.Although the pricing of the down-and-in put is nothing more thana Monte Carlo process, there are two phenomena that one needs tounderstand before one can price, let alone, trade a down-and-in put. Themain risk of a down-and-in put lies in the delta change over the barrier.Another risk that needs to be understood is the skew exposure of a downand-in put. The following sub-sections will discuss these risks in moredetail.
10.2 DELTA CHANGE OVER THE BARRIER FOR ADOWN-AND-IN PUT OPTIONConsider the trader who buys the 100/70 % down-and-in put on BMWfrom an investor. If the delta at inception is 0.4, the trader hedges himself
Payoff at maturity Down-and-in put
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30 %
0
70 %
100 %Share price BMW at maturity
Figure 10.1 Payoff at maturity of a down-and-in put with strike price 100 % and barrierlevel of 70 %
by buying BMW shares in a ratio of 0.4 shares per option. When BMW’sshare price breaches the 70 % barrier level, the down-and-in put optiongoes from not being an option to a put option that is 30 % in the money.This obviously means that the value of the down-and-in put increasessignificantly over the barrier, even if the share price moves down everso slightly, i.e. from 70.1 % to 69.9 %. Since delta measures the sensitivity of the option price to share price movements, the absolute deltaof the down-and-in put becomes extremely large when the share priceapproaches the barrier. This absolute delta can even, and typically does,become greater than 1 when the stock price gets close to the barrier.Since the absolute delta of a regular option can never be greater than1, the trader would inevitably accumulate too many BMW shares whenBMW’s share price approached the barrier. This means that the traderwould need to sell any excess shares when the share price breached the70 % barrier. The initial price of the down-and-in put, which any MonteCarlo price comes to, assumes that these excess shares can be sold exactly at the barrier level of 70 %. However, in practice this proves to beextremely difficult as the share price is already going down for the barrierto be breached in the first place, and the fact that the trader needs to sella large quantity of shares will push the share price down even further.
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Exotic Options Trading
Therefore, the trader almost certainly sells these excess shares belowthe barrier level and as a result incurs a loss on the sale of the excessshares. To avoid this loss, the trader should give himself a cushion to sellany excess shares over the barrier. In order to do that, the trader pricesand risk manages a slightly different option. Namely, an option wherethe barrier is shifted downward in such a way that the trader has enoughroom to sell the excess shares without incurring a loss. This means that,if the trader believes that he needs a 3 % cushion to sell the shares overthe barrier, the trader prices and risk manages a 100/67 % down-and-input rather than a 100/70 % one. The following sub-section discusses thefactors that influence the magnitude of the barrier shift. However, beforethat is discussed it is important to recognise that a long down-and-in putposition goes from being long gamma to short gamma as the share priceapproaches the barrier. In other words, the gamma of a down-and-input changes sign around the barrier.
10.3 FACTORS INFLUENCING THE MAGNITUDEOF THE BARRIER SHIFTThere are 5 main influencing factors that impact the magnitude of thebarrier shift, which are all discussed separately in this sub-section. Thissub-section only discusses the influencing factors for the down-and-input, but the same analysis can easily be extended to other types of barrieroptions. Therefore, this sub-section serves as a blueprint for all barrieroptions.
r The size of the down-and-in put transaction. The larger the size themore shares have to be sold over the barrier and therefore the traderis more likely to move the share price against him, i.e. downward. Inother words, the larger the size the larger the barrier shift.r The difference between strike price and barrier level. If this difference is large, the down-and-in put goes from not being a put to a putthat is far in the money when the share price breaches the barrier.This means that the greater the difference between strike price andbarrier level the greater the barrier shift that a trader would apply. Forexample, a 130/70 % down-and-in put should have a larger barriershift than a 100/70 % down-and-in put.r The volatility of the underlying stock. The larger the volatility of astock the larger is the risk to the trader of the stock price approaching
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61
the barrier level. This can be seen by a simple example. Considerthe 100/70 % down-and-in put on a highly volatile stock with thestock currently trading at 71 % of its initial level. Suppose that thetrader has priced this option as a 100/67 % down-and-in put and isalso risk managing it as such. Suppose further that the absolute deltaof the 100/67 % is 2.5 and that the trader is long 200 thousand ofthese 100/70 % down-and-in puts. Since the trader is risk managingthe 100/70 % down-and-in put as a 100/67 % one and the delta of thelatter is 2.5, the trader is long 500 thousand of the underlying shares.If the stock gaps down 10 %, the down-and-in put obviously knocksin and therefore becomes a regular deep in the money 100 % put.Suppose that the delta of this deep in the money 100 % put is 1 andhence the trader needs to have a long position of 200 thousand sharesagainst this put. However, the trader was long 500 thousand sharesand therefore he had to sell 300 thousand shares over the barrier.Because the stock gapped down, the trader can only sell these sharesat 61 % of the initial stock price. Since the trader risk managed thisas a down-and-in put with a barrier of 67 %, he is basically long 300thousand shares outright from 67 % of the initial level to 61 % of theinitial level. If the initial level was, for example, € 100, this wouldamount to a loss of € 1.8 million. The above example shows that, if astock is or can be highly volatile, the buyer of a down-and-in put needsa larger barrier shift in order to be protected against a larger move. Inother words, the larger the volatility the larger the barrier shift.r The barrier level. Since lower stock prices tend to go hand in handwith higher volatilities and higher volatilities result in larger barriershifts, traders need to apply larger barrier shifts to lower barrier levels.For example, the barrier shift for a 100/60 % down-and-in put shouldbe larger than for a 100/70 % down-and-in put.Some cases are ambiguous as to which down-and-in put shouldhave the larger barrier shift. For example, does a 120/70 % downand-in put need a larger barrier shift than a 100/60 % down-and-input? On the one hand one can argue that it does as the differencebetween strike and barrier is larger for the 120/70 % down-and-input than for the 100/60 % one. On the other hand, the barrier level ofthe 100/60 % down-and-in put is lower than the 120/70 % one andtherefore it can be argued that the 100/60 % down-and-in put shouldhave the larger barrier shift. This example shows that determiningthe magnitude of a barrier shift is not an exact science. It is up to
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Exotic Options Trading
Delta down and-in-put
the trader’s own risk assessment which one of the two down-and-inputs he assigns a larger barrier shift to. However, for sure the traderwill assign a larger barrier shift to both the 120/70 % and 100/60 %down-and-in puts than the barrier shift he finds appropriate for the100/70 % and even the 110/70 % down-and-in put.r The time left to maturity. The closer one gets to maturity the largerthe absolute delta will be just before the barrier and therefore thelarger the change in delta over the barrier. This obviously translatesinto larger barrier shifts for shorter maturities and typically tradersuse a scattered barrier table where the barrier shift increases oncethe down-and-in put gets closer to maturity. The reason that, if thestock price is just above the barrier, the absolute delta becomes largerwhen the time left to maturity gets shorter is because there is less timeleft for the down-and-in put to knock in. This means that the barrierbecomes an all or nothing event and starts to mimic the features of adigital option and therefore the change in value of the down-and-input over the barrier becomes larger when the time left to maturityshortens. Since the change in value increases for a barrier breachwhen the time left to maturity is shorter, the absolute delta is largerfor a shorter time to maturity, see Figure 10.2. This all adds up to thefact that the shorter the time left to maturity the larger the barriershift a trader applies to a down-and-in put.
10
Time left to maturity
Figure 10.2 Delta of a down-and-in put plotted against the time left to maturity whenthe share price is just above the barrier level
Barrier Options
63
10.4 DELTA IMPACT OF A BARRIER SHIFTAlthough a barrier shift is mainly there to manage the Greeks close tothe barrier (gamma and delta are larger without a barrier shift) one hasto realise that a barrier shift also affects the delta further away from thebarrier. For example, the larger the barrier shift on a long down-and-input the smaller the absolute delta and therefore the smaller the amountof long shares to delta hedge the down-and-in put. This means that abarrier shift can also be used to express a delta view. If one is bullish onthe underlying stock one should choose to have a smaller barrier shiftand therefore effectively run a longer delta (more long shares) againstthe down-and-in put. If one is bearish on the underlying stock one shouldchoose to increase the barrier shift and therefore run fewer long sharesagainst the long down-and-in put position.
10.5 SITUATIONS TO BUY SHARES IN CASE OF ABARRIER BREACH OF A LONG DOWN-AND-IN PUTThere are specific situations in which the trader, who is long a down-andin put, does not need to sell shares when the stock price goes throughthe barrier level, but actually needs to buy shares. These situations occurwhen the stock price is still far away from the barrier level and thereforehas not accumulated any excess delta after which it goes through thebarrier in a gap move down. In these cases the absolute delta actuallyincreases because of a barrier breach.
10.6 UP-AND-OUT CALLAn up-and-out call is a call option that ceases to exist if the underlyingstock hits a certain upside barrier during the life of the option. An upand-out call is obviously cheaper than a regular call and can therefore beadvantageous to investors who are bullish on a stock but do not believeit will go above a certain level during the life of the option. Consider atrader who sells a 1 year 100/120 % up-and-out call option on BMW.This means that if BMW’s share price ever breaches 120 % of the initiallevel, the call knocks out and therefore ceases to exist. In this case theup-and-out call loses a lot of value over the barrier as it goes from acall that is 20 % in the money to something that is worthless. The trader,who holds a short position in this up-and-out call, is therefore shortshares against this option as his delta hedge when BMW’s share pricegets close to the 120 % barrier. Note that a regular short call position is
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Exotic Options Trading
hedged by buying shares. This is initially also the case for an up-and-outcall, but when the share price gets closer to the upper barrier level thereis an inflection point where the trader will need to go short shares to bedelta hedged. However, once the up-and-out call knocks out, the traderneeds to buy back these shares as it is no longer a hedge against anything(the option has knocked out). Unlike the long down-and-in put positionthe trader needs to shift the barrier upward for a short up-and-out callposition. However, apart from the direction of the shift the factors thatinfluence the magnitude of the barrier shift are the same for both thelong down-and-in put position and the short up-and-out call position.These influencing factors are described in sub-section 10.3.
10.7 UP-AND-OUT CALL OPTION WITH REBATEConsider a trader who sells the 1 year 100/120 % up-and-out call optionwith the additional feature that when the up-and-out call option knocksout it pays a rebate to the investor of 10 %, i.e. half its intrinsic value. Thiscan obviously still be priced as a Monte Carlo process. A more interestingquestion is how does this rebate affect the barrier shift? Because of therebate, the up-and-out call goes from an option with an intrinsic valueof 20 % to something that is worth 10 % when the stock breaches thebarrier of 120 %. Hence, the change in value over the barrier is far lessin the case where the up-and-out call pays a rebate than if it does not paya rebate. In other words, the discontinuity of the up-and-out call price isless with a rebate than without a rebate. This means that the trader willbe short fewer shares when the share price gets close to the barrier andtherefore needs to buy back less shares when the barrier is breached.For this reason the barrier shift can be smaller when the up-and-out callpays a rebate than if it does not pay a rebate.
10.8 VEGA EXPOSURE UP-AND-OUT CALL OPTIONA long position in an up-and-out call option is not necessarily long vega.In fact, more often than not a long up-and-out call position results in being short vega. This means that, if the implied volatility of the underlyinggoes up, the up-and-out call option actually becomes less valuable. Thereason being that a higher implied volatility results in a higher probability of the call knocking out and therefore a lower chance of theup-and-out call having a payout at maturity. Naturally, an up-and-out
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65
call is not always short vega. Indeed, if it is a very upside barrier theprobability of the up-and-out call knocking out is very low anyway andtherefore an increase in implied volatility has a much bigger impact onthe option part of the up-and-out call than on the fact that there is ahigher chance of the option knocking out. Hence an up-and-out call islong vega for very upside barriers and short vega for lower barriers.
10.9 UP-AND-OUT PUTThere are certain types of barrier options where there is no need to applya barrier shift. For example, a trader who is long an up-and-out put doesnot need to apply a barrier shift. The reason being that the delta hedgeagainst a long up-and-out put position is always a long share position.This means that the trader needs to sell shares over the barrier. But forthat he does not need to shift the barrier as he can always sell out hisshares 1 cent below the barrier in which case he would, at worst, onlylose a cent on the number of shares he needs to sell.
10.10 BARRIER PARITYNow that both knock-in as well as knock-out barriers have been explained, it is time to introduce the barrier parity. Barrier parity statesthat a knock-out (KO) option plus a knock-in (KI) option with the exact same barrier have the same payoff profile as the regular Europeanoption. Mathematically this reads as follows,K