BMW stockis trading at S3 = € 45. The contribution of each year to the payoff atmaturity of this ratchet option can be summarised by
r the contribution of year 1 is max [S1 − S0 , 0] = € 5;r the contribution of year 2 is max [S2 − S1 , 0] = € 5;r the contribution of year 3 is max [S3 − S2 , 0] = € 0.The payoff formula for each year clearly shows that the strike price ofthe ratchet option resets each year to the prevailing stock price, i.e. theinitial strike is S0 , after one year the strike resets to S1 and after twoyears the strike resets to S2 . The full payoff of an X % ratchet call optionat maturity with n + 1 observations and therefore n resets isXSi , 0 ,max Si+1 −100i=0
n

(14.1)
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where Si is the stock level after the i-th period. Obviously, the payoff atmaturity for an X % ratchet put option looks likenXmax(14.2)Si − Si+1 , 0 .100i=0Since the strike price resets every year and the performance over theprevious period gets locked in, the ratchet option can be priced as a set offorward starting options. Obviously, since the payoff occurs at maturitythe value of each forward starting option needs to be discounted by theperiod from expiry of each forward starting option till the maturity ofthe ratchet option. In other words, the 3 year ratchet call option on BMWcan be priced by adding the following forward starting options:1. A one year at the money call option with a delayed payout date oftwo years.2. A forward starting at the money call option with a forward start datein one year’s time and maturing in two years’ time with a delayedpayout date of one year.3. A forward starting at the money call option with a forward start datein two years’ time and maturing in three years’ time.In a perfectly normally distributed world where there is no skew andBMW is a non-dividend paying stock, the price of the 100 % ratchet calloption would be2
e−r ·i S0 N (d1 ) − S0 e−r N (d2 ) e−(3−(i+1))r =
i=02
e−2r S0 N (d1 ) − S0 e−r N (d2 ) ,
(14.3)
i=0
where r is expressed on an annualised basis, i.e. the unit of time is oneyear. Equation 14.3 is derived from the Black–Scholes formula for a calloptionct = St N (d1 ) − K e−r (T −t) N (d2 ),where
 St 
+ r + 12 σ 2 (T − t),d1 =√σ T −t√d2 = d1 − σ T − t.ln
(14.4)
K
(14.5)(14.6)
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In equation 14.3, e−r ·i represents the discounting in order to get theprice of the forward starting option and e−(3−(i+1))r takes care of thediscounting as a result of the delayed payoff date. Also, in equation14.3, d1 and d2 are calculated with T − t being equal to 1 and K = S0 .In practice skew does exist and therefore a ratchet option would alwaysbe priced as a Monte Carlo process.
14.2 RISKS OF A RATCHET OPTIONSince a ratchet option can be priced as a set of forward starting options,the risks involved are very similar to those of a forward starting option. Therefore, the main risk of a ratchet option manifests itself in theskew, see Chapter 11. For example, numerous investment banks havelost money by selling ratchet call spreads. They were, for example, selling 100/130 % ratchet call options. An n + 1 year 100/130 % ratchet callspread with yearly resets has the following payoff:n
max [min {Si+1 − Si , 0.3 · Si } , 0] ,
(14.7)
i=0
where Si is the stock level after the i-th period. However, selling callspreads means selling skew, as the bank sells the lower strike andbuys the higher strike. This obviously means that the bank has to takeinto account all the issues with forward starting options discussed inChapter 11. For a long period several banks have been pricing theseforward starting options using the skew of the specific maturity ratherthan the skew associated with the maturity equal to the term of the forward starting option, i.e. the maturity minus the forward start date, seeChapter 11, sub-section 11.3. This way of pricing has resulted in largelosses for some banks.
15Reverse ConvertiblesA reverse convertible is a structure where an investor buys a bond andsells an ‘at the money’ put option to enhance the coupon. Therefore thecoupon of a reverse convertible is higher than the coupon on a regularbond. Technically, the reverse convertible in itself is not an exotic optionas the structure is nothing more than a bond plus a plain vanilla putoption. However, it is imperative to have a good understanding of thereverse convertible in order to understand more exotic structures likethe autocallable. Moreover, there are many different iterations to thereverse convertible which can turn a plain vanilla reverse convertibleinto an exotic reverse convertible. For example, instead of a regularreverse convertible where the investor buys a bond and sells an ‘at themoney’ put, the investor can decide to sell a down-and-in put whichgives the investor additional protection. Lastly, the reverse convertibleis an extremely popular bullish retail structure and whoever wants tounderstand exotic options needs to be familiar with such an importantstructure as the reverse convertible.
15.1 EXAMPLE: KNOCK-IN REVERSE CONVERTIBLEA very popular iteration to the regular reverse convertible is a structurewhere the investor buys a bond and sells a down-and-in put to enhancethe coupon on the whole structure – a knock-in reverse convertible. Subsection 10.1 shows that the seller of a down-and-in put enjoys additionalprotection over the seller of an ‘at the money’ put. Because the investorreceives additional protection, the knock-in reverse convertible pays aslightly lower coupon than the regular reverse convertible. Nonetheless,the knock-in reverse convertible still pays a higher coupon than a regularbond as a compensation for taking the risk of selling a down-and-in put.Consider a 2 year knock-in reverse convertible on BMW where theinvestor sells a 100/80 % down-and-in put to enhance the coupon on thestructure. In other words, the investor buys a 2 year bond and sells a 2 year100/80 % down-and-in put and in return the investor receives a semiannual coupon. The investor buys the full knock-in reverse convertible
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structure at 100 %. That is to say, a knock-in reverse convertible worksalong the same principles as a regular bond. Whoever buys the reverseconvertible pays the full notional up-front and throughout the life ofthe reverse convertible the investor receives semi-annual coupons. Inorder for the reverse convertible to work it is important that the notionalon the down-and-in put is the same as the notional on the bond. Thiscan be established by having the notional divided by the strike numberof down-and-in puts in the reverse convertible. This also ensures thatif the down-and-in put has knocked in and it is in the money at expiry,the issuer of the reverse convertible can simply exercise the down-and-input for which the issuer should be getting paid an amount equal to thenotional that the issuer owes the investor already through the bond partof the reverse convertible. Thus, if the issuer exercises the down-and-input there is no exchange of cash at expiry. This also shows that a reverseconvertible is not a riskless structure to the investor. If, for example,BMW is trading at 70 % of its initial value, the buyer of the reverseconvertible loses 30 % of his invested notional as he receives shares,through the assignment of the down-and-in put, that are worth 30 %less. To summarise, at expiry there are three different payout scenarios:⎧100 % in cash⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪100 % in cash⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪notional number of shares⎪⎪⎪strike⎪⎪⎪⎪⎪⎪⎪⎪⎩
If BMW never < 80 % during the 2year term of the knock-in reverseconvertibleIf BMW at expiry > 100 % regardlessof whether BMW was ever < 80 %during the 2 year term of the knock-inreverse convertibleIf BMW ever < 80 % during the 2year term of the knock-in reverseconvertible and BMW < 100 % atexpiry
In the third case, the buyer of the reverse convertible obviously losesmoney. Instead of seeing his invested notional returned on the bondhe receives notionalnumber of BMW shares, which he effectively buys atstrike100 % and are worth less than 100 % at expiry. De facto the investor losesthe percentage decline of the BMW share price on the notional invested.
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15.2 PRICING THE KNOCK-IN REVERSECONVERTIBLEPricing a reverse convertible typically means that one solves for thecoupon that makes the value or re-offer of the reverse convertible 100 %.Consider the two year 100/80 % knock-in reverse convertible on BMWfrom sub-section 15.1. Suppose that the semi-annual coupon on a regularbond is equal to 1.75 % and also that the interest rate is constant in everymaturity. Lastly, suppose that a 2 year 100/77 % down-and-in put onBMW is worth 6.7 %. Note that the issuer would use a 3 % barrier shiftto price a 100/80 % down-and-in put, as discussed in Chapter 10. Nowit is easy to determine the semi-annual coupon that makes the knock-inreverse convertible worth 100 %. Indeed, since the semi-annual couponon a regular bond is 1.75 %, the semi-annual discount factors to fourdecimal places are:
r the 6 month discount factor is 1 = 0.98281.0175r the 1 year discount factor is 1 2 = 0.9659(1.0175)r the 18 month discount factor is 1 3 = 0.9493(1.0175)r the 2 year discount factor is 1 4 = 0.9330(1.0175)
the semi-annual coupon, X , for the two year 100/80 % knock-in reverseconvertible on BMW can now be calculated as:0.9828X + 0.9659X + 0.9493X + 0.9330X = 1 − (0.9330 − 0.067)⇓3.831X = 0.134⇓X = 0.035In other words, a cash semi-annual coupon payment of 3.5 % makes the2 year 100/80 % knock-in reverse convertible on BMW worth 100 %.
15.3 MARKET CONDITIONS FOR MOSTATTRACTIVE COUPONIt is important to understand under which circumstances the coupon ona reverse convertible looks the most attractive. In other words, duringwhich market conditions one can expect the highest demand for reverseconvertibles. There are two inputs that affect the coupon on a reverse
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convertible. The first one is the premium for the put or down-and-input. The higher the premium on the put or down-and-in put the higherthe coupon. The two main contributors to a higher put premium arethe implied volatility and the dividend yield. The higher the impliedvolatility or dividend yield the higher the put premium. Secondly, thelower the discount factors the higher the coupon. Discount factors havean inverse relationship to the interest rate. Therefore, the higher theinterest rates the higher the coupon.1 To summarise, there are threefactors that impact the magnitude of the coupon:
r The higher the interest rates the higher the coupon on the reverseconvertible.
r The higher the implied volatility the higher the coupon on the reverseconvertible.
r The higher the dividend yield the higher the coupon on the reverseconvertible.
15.4 HEDGING THE REVERSE CONVERTIBLEHedging a reverse convertible consists of two hedges. One is the volatility hedge and the second one is the interest rate hedge. There is no exactvolatility hedge for the down-and-in put. However, a good proxy hedgedoes exist. For example, the two year 100/80 % down-and-in put onBMW would typically be hedged with a two year 90 % plain vanilla put.The reason that one would hedge the two year 100/80 % down-and-input with a downside put, i.e. 90 %, is because a down-and-in put is longskew, which makes the effective implied volatility at which one buysa 100/80 % down-and-in put higher than the ‘at the money’ impliedvolatility. In order to hedge this effectively, the buyer of a down-and-input needs to sell some downside options for which he receives an impliedvolatility that is slightly higher than the ‘at the money’ implied volatility.The number of 2 year 90 % put options the buyer of the down-and-input needs to sell can easily be determined by the vega exposure on thedown-and-in put and solving for the number of 2 year 90 % puts thatflattens this vega exposure.1Higher interest rates have a dampening effect on the put premium. This is because a long put is hedgedby buying shares on which the owner of the put incurs financing. If the interest rates go up, the financing of thehedge becomes more expensive and therefore the buyer is not prepared to pay as much for the put. Since an ‘atthe money’ put has a delta less than 1, higher interest rates have a bigger impact on the bond part than on the putoption part of the reverse convertible. Therefore, the result of higher interest rates is that the coupon looks moreattractive.
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91
Hedging any interest rate exposure on a reverse convertible is rathersimple. Suppose the issuer of a € 10 million 2 year 100/80 % reverseconvertible on BMW wants to hedge his interest rate exposure. Supposealso that the delta on the down-and-in put is 0.4. This means that theissuer will buy € 4 million worth of BMW stock on which he incursfinancing. On the € 10 million short bond he actually receives interestand therefore the short bond and the delta hedge work in opposite directions in terms of interest rate exposures. In other words, the issuer ofthe reverse convertible is long floating interest rates on a notional of € 6million. Since it is a 2 year reverse convertible, the issuer is exposed tothe 2 year part of the interest rate curve and will therefore enter into a2 year swap where he pays floating and receives fixed on a notional of€ 6 million in order to hedge his interest rate exposure.DVO1, the price sensitivity of any structure to 1 basis point,2 bp,change in yield curve (the annualised interest rate goes up by 1 bpfor every maturity), is typically used as an interest rate risk measure.Since the issuer of the reverse convertible has a 2 year long interestrate exposure and assuming an annual interest coupon payment, a 1 bpincrease for the 1 year maturity results in a 0.9659 of a bp profit and a 1bp increase for the 2 year maturity results in a 0.9330 of a bp profit. Thismeans that the DVO1 for the issuer of the reverse convertible is equal to€ 6 million ×
2
11 bp = 100of 1 %.
0.9659 + 0.9330= + € 1139.3410 000
16AutocallablesAn autocallable is a note people can invest in that only pays a couponif the underlying index is above or below a certain level and the noteautomatically redeems early if it breaches a second hurdle level, whichcan be the same or different to the one that determines whether theinvestor gets paid a coupon. The reason that the note redeems early isthat, once the underlying index reaches a certain level, the investor wouldprefer to invest his money elsewhere rather than to hold his money in anote that pays a fixed coupon, conditional on a certain level of underlyingindex. To get an enhanced coupon the investor typically also sells anoption, which is then embedded in the note. The following sub-sectionwill give an example of an autocallable note.
16.1 EXAMPLE: AUTOCALLABLE REVERSECONVERTIBLEConsider an investor who has a range bound view on BMW and believesthat in the next 2 years the stock will not drop below 90 % of its valueand will not get above 110 % of BMW’s current value. He also believesthat, if BMW does get above 110 % of its current value, it is a bullishsignal and would like to invest money in BMW stock. The autocallablewould be the perfect strategy for this investor, where the note autocallswhen BMW reaches 110 % of its current value so that he can investthe money of the note in BMW stock directly. The note will pay asemi-annual coupon for as long as BMW stock is not below 90 % ofits current value at these semi-annual observation points. Lastly, theinvestor wants to enhance his coupon by selling a down-and-in put witha strike of 100 % and a continuous barrier of 80 %, which means that hisinvestment is capital guaranteed provided BMW never drops by morethan 20 % at any time during the 2 year term of the note. By sellingthis down-and-in put the investor manages to enhance his coupon toa 4 % semi-annual coupon. Suppose this investor invests 10 millioneuros in this autocallble reverse convertible. The exact terms of this
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note are as follows:
r Coupons
– If at any semi-annual observation point, i, during the term of the noteBMWi > 90 % · BMW0 ,the investor gets paid a coupon of 4 % on his investment at thisparticular semi-annual observation point, i.e. 400 thousand euros.– If at any semi-annual observation pointBMWi < 90 % · BMW0 ,the investor does not get a coupon at this particular semi-annualobservation point.r Early redemption– If at any of the first three semi-annual observation pointsBMWi > 110 % · BMW0 ,the note redeems early and the investor will get his full investmentback, i.e. 10 million euros. Because of this redemption the investorobviously loses his right to coupon payments.r Redemption at maturity– If BMW’s share price has never dropped below 80 % of its initialvalue during the two year term of the note, the note redeems at100 %, i.e. 10 million euros.– If BMW’s share price has at any point during the 2 year term of thenote dipped below 80 % of its initial value, the note redeems atBMW0 − BMWT100 % − max,0 .BMW0It is not straightforward to see whether this conditional autocallable reverse convertible pays a higher or lower coupon than the regular reverseconvertible discussed in Chapter 15. On the one hand, the coupon ofthe above autocallable is conditional and should therefore have a highercoupon than the coupon of the regular reverse convertible, which is unconditional. On the other hand, the autocallable redeems early if, at one ofthe semi-annual observation dates, BMW’s stock price is above 110 %, inwhich case the trader loses a down-and-in put. This would have a dampening effect on the coupon of the above autocallable. However, takingboth aspects into account one can assume that the contingency of thecoupon has a larger increasing impact on the coupon than the downwardpressure on the coupon from potentially losing the down-and-in put.
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95
16.2 PRICING THE AUTOCALLABLEThe pricing of the autocallable discussed in sub-section 16.1 is relativelyeasy with the knowledge of digital and barrier options. After all, theconditional coupons are nothing more than a strip of semi-annual 90 %strike digital options which knock out at 110 %, and pay a coupon of4 %. The early redemption feature can be priced by a worthless optionthat knocks out at 110 % to then pay a rebate of 100 %. This worthlessoption is typically taken to be a zero strike put. The barriers are onlylive at the semi-annual observation dates. In other words, the barrierscan only be breached at these semi-annual observation dates. The entirestructure can therefore be priced as:
r Coupons– The trader sells a 1 time geared 86/90 % European call spread expiring after six months with no knock-out feature, as the investor willstill get the 4 % coupon if the note knocks out at the first semi-annualobservation date.Unlike the reverse convertible, this way of pricing the autocallablereverse convertible does not assume that a coupon actually gets paidout in cash at each semi-annual observation date, but is embeddedin the pricing of the call spread. The call spread is a conservativeway of pricing a digital option that gives a payout of 4 % if BMW’sstock price is above 90 % and pays nothing if BMW’s stock price isbelow 90 % at each semi-annual observation date.– The trader sells a 1 time geared 86/90 % European call spread expiring after one year, with a knock-out barrier of 110 %. The barrier islive only at the first semi-annual date, as the coupon needs to be paidat the second semi-annual observation date, regardless of whetherthe structure knocks out at this date.– The trader sells a 1 time geared 86/90 % European call spreadexpiring after 1.5 years, with a knock-out barrier of 110 %. Thebarrier is live only at the first and second semi-annual dates.– The trader sells a 1 time geared 86/90 % European call spreadexpiring after 2 years, with a knock-out barrier of 110 %. Thebarrier is only live at the first three semi-annual dates. The barrier isnot live at the fourth semi-annual observation date because the notecan only redeem at the first three semi-annual observation dates.r Early redemption– The trader sells a zero strike knock-out put with a barrier of 110 %for the first three semi-annual dates and a 0 % barrier for the fourth
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semi-annual observation date. The knock-out put can only knockout at one of the semi-annual observation dates and pays a rebateof 100 %. The barrier for the fourth semi-annual observation dateensures that the client sees his investment returned after the two yearperiod.r Financing through down-and-in put– The trader buys a 100 % put knocking in at 80 % and knocking outat 110 %. The reason that this option knocks out at 110 % is that thedown-and-in put is part of the reverse convertible autocallable note,which knocks out at 110 %. Obviously the trader would price thedownside barrier with a barrier shift, i.e. pricing it as a 100 % putwith a 77 % knock in. It is important to recognise that the downsidebarrier at 80 % can knock in continuously, i.e. throughout the fulllife of the autocallable, whereas the upside barrier at 110 % can onlyknock out at one of the three semi-annual observation dates.When determining how to shift the barrier at 110 % for the different semiannual observation dates, one first has to realise that the different digitaloptions, the zero strike put and the down-and-in put are communicatingoptions when it comes to knocking out. The best way to see this isto imagine BMW’s stock price at 109 % just before the third semiannual observation date. In other words, the autocallable note is close toknocking out. In this case it could well be advantageous to the trader if thestructure knocks out. Firstly, this is because the down-and-in put is veryunlikely to knock in during the next 6 months, making the down-and-input almost worthless at that point. Secondly, if the structure knocks out atthe third semi-annual observation date, the trader will have to pay 100 %6 months early, which works to his disadvantage. However, it wouldalso mean that he would not have to pay the fourth coupon, which worksto the trader’s advantage. Assuming that the rate of interest has notmoved, these two effects will work to the trader’s advantage, as thecoupon plus the present value of 100 % in 6 months is more than 100 %now. This is because the coupon is artificially high as a result of thefinancing through the down-and-in put within the note. Therefore, theabove shows that, if the the knock-in put is close to worthless and theinterest rate has been stable or has moved down, it is advantageous tothe trader if the note knocks out. In this case the trader obviously shiftsthe barrier upwards. However, in most of the cases it is disadvantageousto the trader if the note redeems early, as the consequence is that he losesa down-and-in put which he has paid for and therefore shifts the barrierdownwards at the 110 % level.
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97
16.3 AUTOCALLABLE PRICING WITHOUTCONDITIONAL COUPONAnother, and probably more common, class of autocallables is an autocallable where the coupon is unconditional but still autocalls at a certainlevel. Typically, this autocallable with an unconditional coupon is structured such that the coupons are not paid running but only when thestructure knocks out, or if it never knocks out the coupons are paid atmaturity. Since the coupons are only paid at the observation dates whenthe autocallable actually knocks out, the coupon that is paid at maturityor when it autocalls is the number of observation dates till knock outmultiplied by the agreed coupon. The following example clarifies theautocallable with unconditional coupon.Consider the autocallable example in sub-section 16.2, only then thecoupon is paid regardless of any stock level. One would expect thiscoupon to be lower than 4 % semi-annually. It would most likely alsobe lower than the coupon of the regular reverse convertible, as there isalways the risk to the trader of losing the down-and-in put if the 110 %barrier is breached. Assuming that the fair semi-annual coupon is 3 %for this unconditional autocallable on BMW, the 2 year unconditionalautocallable on BMW would be structured as follows:
r Early redemption plus coupons
– The trader sells a zero strike knock-out put with a barrier of 110 %for the first three semi-annual dates and a 0 % barrier for the fourthsemi-annual observation date. The knock-out put can only knockout at one of the semi-annual observation dates and pays a rebate of1. The rebate at the first semi-annual observation date is 100 % plusone times 3 %, equals a rebate of 103 %2. The rebate at the second semi-annual observation date is 100 %plus two times 3 %, equals a rebate of 106 %3. The rebate at the third semi-annual observation date is 100 % plusthree times 3 %, equals a rebate of 109 %4. The rebate at the fourth and last semi-annual observation date is100 % plus four times 3 %, equals a rebate of 112 %A barrier of 0 % for the fourth semi-annual observation date ensuresthat the client sees his investment returned after the 2 year period.r Financing through down-and-in put– The trader buys a 100 % put knocking in at 80 % and knocking outat 110 %. The reason this option knocks out at 110 % is that thedown-and-in put is part of the reverse convertible autocallable note,which knocks out at 110 %. Obviously the trader would price the
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downside barrier with a barrier shift, i.e. pricing it as a 100 % putwith a 77 % knock in. It is important to recognize that the downsidebarrier at 80 % can knock in continuously, i.e. throughout the fulllife of the autocallable, whereas the upside barrier at 110 % can onlyknock out at one of the three semi-annual observation dates.
16.4 INTEREST/EQUITY CORRELATION WITHINTHE AUTOCALLABLEWhen pricing the autocallable in sub-section 16.2 there was no mentionof an interest/equity correlation. However, if one wants to price theautocallable accurately this interest/equity correlation should be takeninto account. To see this, consider a purely anti-correlated interest/equitycorrelation. In other words, there is a lognormal correlation of −1. Thisobviously has an impact on the pricing of the autocallable. Indeed, inthis case BMW’s share price goes up whenever interest rates are goingdown. This means that, when BMW’s share price reaches 110 %, theinterest rate is lower than at inception of the trade and it is thereforemore advantageous to the trader when the autocallable knocks out thanwhen interest rates have not moved at all.1 For that reason, the trader cangive a higher coupon when he takes this interest/equity correlation of−1 into account. The opposite obviously holds when the interest/equitycorrelation is perfectly correlated, i.e. 1.The reason that no interest/equity correlation was taken into accountwhen pricing the autocallable in sub-section 16.2 is that this correlationis very hard to measure and for short term maturities this correlationalways hovers around zero.2 For that reason traders typically price theautocallable as in sub-section 16.2 and increase or lower the coupondepending on their view on the interest/equity correlation during theterm of the note.
1The reason that it is advantageous to the trader if the autocallable knocks out if interest rates are loweris that the trader effectively borrows money through the autocallable and pays a fixed coupon for it. When thestructure knocks out and interest rates are lower he can refinance himself cheaper.2Over very long periods of time equity and interest rates have a negative correlation. However, over a 2 yearperiod, equity and interest rates might very well be positively correlated. For example, the start of a rate hikecycle after a recession typically features both equity and interest rates going up