.
17Callable and PuttableReverse ConvertiblesA callable reverse convertible is a reverse convertible that can be calledby the issuer (the seller) at his choice at any of the pre-defined observationdates during the term of the callable reverse convertible. This means thatthe issuer holds an option as to whether he wants to keep the short reverseconvertible position till maturity or unwind it early at the observation dateof his choosing. Because the issuer has this additional option the buyerof this callable reverse convertible is compensated by means of a highercoupon than the coupon on a regular reverse convertible. The pricingof such a callable reverse convertible can be seen as an optimisationprocess of an unconditional autocallable, where the coupon is optimised(maximised) versus the barrier level of this autocallable.A puttable reverse convertible is the opposite of the callable reverseconvertible in the sense that it is not the issuer but the buyer who hasthe choice of unwinding the reverse convertible early at the observationdate of his choosing. Again, the puttable can be priced as an optimisation process of an autocallable. However, in this case the coupon isminimised against the barrier level. The fact that the buyer can redeemhis investment plus a part of the coupons early, and therefore has accessto his capital earlier, results in a slightly lower coupon.The next two sub-sections discuss the pricing of the callable andputtable reverse convertibles in more detail.
17.1 PRICING THE CALLABLE REVERSECONVERTIBLEConsider a 2 year callable reverse convertible on BMW which can becalled (unwound) by the issuer at any of the three semi-annual observation dates. The put within the reverse convertible that is used to enhancethe coupon is a 100 % put knocking in at 80 %. Chapter 15 showed thata regular reverse convertible on BMW pays a semi-annual coupon of3.5 % with a re-offer of 100 %. Therefore, one expects the callable reverse convertible to pay a semi-annual coupon of at least 3.5 %. In order
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to solve for the semi-annual coupon of this callable reverse convertible,one prices this structure as an autocallable where the barrier levels atthe respective semi-annual observation dates are optimised to give thelargest coupon for a re-offer or fair value of 100 %. To begin with onecan start with a barrier level of 110 % for all the three semi-annual observation dates which, according to Chapter 16, sub-section 16.3, givesa coupon of 3 %. Secondly, one can try to price an autocallable wherethe first two observation dates have a barrier of 1000 %1 and the thirdsemi-annual observation date has a barrier of 104 %. This reverse convertible autocallable appears to give a coupon of 3.65 % for a fair valueof the whole reverse convertible structure of 100 %. Proceeding in thisway one finds that the most optimal barrier levels at which the structureautocalls are the following for the respective semi-annual observationdates, and with this barrier structure the callable reverse convertible paysa semi-annual coupon of 3.8 %:
r The barrier for the first observation date, above which the structureknocks out, is 120 %. This means that, if BMW’s share price is higherthan 120 % of its initial value at the first observation date, it is optimalfor the issuer of the callable reverse convertible to call the structure andrepay the buyer 103.8 %. The reason that it is optimal for the issuer tocall the full structure and repay 103.8 % is that the remaining value ofthe down-and-in put has reduced so much that it is more economical topay the full 100 % early plus the first coupon of 3.8 % and give up thedown-and-in put than to keep the structure with the future liability topay three artificially high2 additional coupons but pay the full 100 %in 1.5 years’ time.r The barrier for the second observation date, above which the structureknocks out, is 112 %. This means that, if BMW’s share price is higherthan 112 % of its initial value at the second observation date, it isoptimal for the issuer of the callable reverse convertible to call thestructure and repay the buyer 107.6 %. The reason that it is optimalfor the issuer to call the full structure and repay 107.6 % is that theremaining value of the down-and-in put has reduced so much thatit is more economical to pay the full 100 % early plus the first twocoupons of 3.8 % and give up the down-and-in put than to keep thestructure with the future liability to pay two artificially high additionalcoupons but pay the full 100 % 1 year later.1Such a high barrier makes sure that it does not autocall on either of the first two observation dates. Thereforeit allows for the optimisation of the barrier on the third semi-annual observation date.2The coupons are artificially high as they are increased because of the initial value of the down-and-in put.
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r The barrier for the third observation date, above which the structureknocks out, is 104 %. This means that, if BMW’s share price is higherthan 104 % of its initial value at the third observation date, it is optimalfor the issuer of the callable reverse convertible to call the structure andrepay the buyer 111.4 %. The reason that it is optimal for the issuer tocall the full structure and repay 111.4 % is that the remaining value ofthe down-and-in put has reduced so much that it is more economicalto pay the full 100 % early plus the first three coupons of 3.8 % andgive up the down-and-in put than to keep the structure with the futureliability to pay one additional coupon but pay the full 100 % 6 monthslater.This means that the full callable reverse convertible can be priced as:
r Early redemption plus coupons– The trader sells a zero strike knock-out put with a barrier of 120 %,112 % and 104 % for the first three semi-annual dates respectivelyand a 0 % barrier for the fourth semi-annual observation date. Theknock-out put can only knock out 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 3.8 %, equals a rebate of 103.8 %2. The rebate at the second semi-annual observation date is 100 %plus two times 3.8 %, equals a rebate of 107.6 %3. The rebate at the third semi-annual observation date is 100 % plusthree times 3.8 %, equals a rebate of 111.4 %4. The rebate at the fourth and last semi-annual observation date is100 % plus four times 3.8 %, equals a rebate of 115.2 %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 out atrespectively 120 %, 112 % and 104 % for the first three semi-annualobservation dates. Obviously the trader would price the downsidebarrier with a barrier shift, i.e. pricing it as a 100 % put with a 77 %knock in. It is important to recognize that the downside barrier at80 % can knock in continuously, i.e. throughout the full life of thecallable reverse convertible, whereas the upside barriers can onlyknock out at one of the three semi-annual observation dates.In pricing the callable reverse convertible it is important to recognisetwo things. First of all the structure only knocks out if the share price
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is above a certain level. In other words, it is only optimal for the issuer to call the callable reverse convertible if the share price is abovea certain level. This is obvious, since the issuer of the callable reverseconvertible owns a down-and-in put that becomes more valuable whenthe underlying share price is lower and it is therefore less economicalfor the issuer to give up this down-and-in put. Secondly, one has toobserve that the process of finding the most optimal barriers at whichthe issuer should call the callable reverse convertible is not a randomprocess based on trial and error but can be done in a systematic waywith the Newton–Raphson3 method, provided one starts to solve for theoptimal barrier with the penultimate observation date and works backwards to the first observation date. In the above example one woulduse the Newton–Raphson to first find the optimal barrier for the thirdsemi-annual observation date, while keeping the barriers of the first twosemi-annual observation dates so high that the structure would never getcalled on either of these first two observation dates. Knowing the optimal barrier for the third semi-annual observation date, one then goeson to find the optimal barrier for the second semi-annual observationdate, again using a Newton–Raphson process. To be able to optimisethe barrier for the second semi-annual observation date one has to usean unrealistically high barrier for the first semi-annual observation date.Lastly, knowing the optimal barriers for both the third and second observation dates, one can finish by solving the optimal barrier for the firstsemi-annual observation date using the same Newton–Raphson process.
17.2 PRICING THE PUTTABLE REVERSECONVERTIBLEWith the puttable reverse convertible it is the buyer who has the optionto unwind the reverse convertible early, in which case the buyer sees hisfull investment plus a part of his coupons returned early.Consider a 2 year puttable reverse convertible on BMW which canbe unwound by the buyer at any of the three semi-annual observationdates. The put within the reverse convertible that is used to enhancethe coupon is a 100 % put knocking in at 80 %. Chapter 15 showedthat a regular reverse convertible on BMW pays a semi-annual couponof 3.5 % with a re-offer of 100 %. Therefore, one expects the puttable3The Newton–Raphson process solves for the barrier by decreasing the barrier until it gives a worse couponthan the previous barrier and then subsequently halving the spread between these last two barriers. With thisprocess one can solve for the optimal barrier to any required accuracy.
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103
reverse convertible to pay a semi-annual coupon of less than 3.5 %.In order to solve for the semi-annual coupon of this puttable reverseconvertible, one prices this structure as an autocallable where the barrierlevels at the respective semi-annual observation dates are optimised togive the smallest coupon for a re-offer or fair value of 100 %. However,in contrast to the callable reverse convertible, the barriers are notbreached if the stock price is above the barrier but when the stockprice is below the barrier. In other words, the barriers are downand-out rather than up-and-out. To begin with one can start with abarrier level of 100 % for all the three semi-annual observation dates,which appears to result in a coupon of 2.4 %. Secondly, one can try toprice an autocallable where the first two observation dates have a barrierof 0 %4 and the third semi-annual observation date has a barrier of 90 %.This reverse convertible autocallable appears to give a coupon of 2.2 %for a fair value of the whole reverse convertible structure of 100 %.Proceeding in this way one finds that the most optimal (lowest coupon)barrier levels at which the structure autocalls are the following for therespective semi-annual observation dates, and with this barrier structurethe puttable reverse convertible pays a semi-annual coupon of 2 %:
r The barrier for the first observation date, below which the structure
knocks out, is 85 %. This means that, if BMW’s share price is lowerthan 85 % of its initial value at the first observation date, it is optimalfor the buyer of the puttable reverse convertible to unwind the structureand receive 102 % from the issuer. The reason that it is optimal for thebuyer to unwind the full structure and receive 102 % is that the valueof the down-and-in put is sufficiently large that it is more economicalto get paid the full 100 % early plus the first coupon of 2 % and losethe short down-and-in put position than to keep the structure withthe prospect of receiving three additional coupons but receiving thefull 100 % 1.5 years later and having a potential liability on the shortdown-and-in put.r The barrier for the second observation date, below which the structureknocks out, is 87 %. This means that, if BMW’s share price is lowerthan 87 % of its initial value at the second observation date, it is optimalfor the buyer of the puttable reverse convertible to unwind the structureand receive 104 % from the issuer.r The barrier for the third observation date, below which the structureknocks out, is 90 %. This means that, if BMW’s share price is lower4
Such a low barrier allows for the optimisation of the third semi-annual observation date.
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than 90 % of its initial value at the third observation date, it is optimalfor the buyer of the puttable reverse convertible to unwind the structureand receive 106 % from the issuer.In other words, the full puttable reverse convertible can be priced as:
r Early redemption plus coupons– The trader sells a zero strike knock-out put with barriers of 85 %,87 % and 90 % for the first three semi-annual dates respectivelyand a 0 % barrier for the fourth semi-annual observation date. Theknock-out put can only knock out if BMW’s share price is belowthe barrier level at one of the first three semi-annual observationdates and if BMW’s share price is above the barrier at the fourth(maturity) semi-annual observation date and pays a rebate of1. The rebate at the first semi-annual observation date is 100 % plusone times 2 %, equals a rebate of 102 %2. The rebate at the second semi-annual observation date is 100 %plus two times 2 %, equals a rebate of 104 %3. The rebate at the third semi-annual observation date is 100 % plusthree times 2 %, equals a rebate of 106 %4. The rebate at the fourth and last semi-annual observation date is100 % plus four times 2 %, equals a rebate of 108 %A barrier of 0 % for the fourth semi-annual observation date ensuresthe 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 out(down-and-out) at respectively 85 %, 87 % and 90 % if BMW’s shareprice is below the barrier level at one of the first three semi-annualobservation dates. Obviously the trader would price the 80 % knockin barrier with a barrier shift, i.e. pricing it as a 100 % put with a77 % knock in. It is important to recognize that the downside barrierat 80 % can knock in continuously, i.e. throughout the full life ofthe puttable reverse convertible, whereas the other three solved foroptimal barriers can only knock out at one of the three semi-annualobservation dates.
18Asian OptionsAsian options are options whose strike price or settlement price1 does notdepend on one observation but on an average of observations. Thereforethere are two types of Asian options. The Asian out option is a Europeanoption where the settlement price is determined by an average of the underlying on a set of predetermined observation dates. This means that, atexpiry, the Asian out option pays the difference between the average ofthe underlying (floating settlement price) on these predetermined observation dates and the fixed strike. The Asian in option is a European optionwhere the strike price is determined by an average of the underlying ona set of predetermined observation dates. This means that, at expiry,the Asian in option pays the difference between the underlying price atexpiry and the average of the underlying at these predetermined observation dates (floating strike). The next sub-sections discuss the pricingand risk management of the Asian out and Asian in options respectively.
18.1 PRICING THE GEOMETRIC ASIAN OUT OPTIONAsian out options can be subdivided into a class where the average partof the Asian options is geometric and one where the average part isarithmetic. The price of an Asian out option where the average part isgeometric is easy to price, as the geometric average of a lognormallydistributed underlying has a lognormal distribution. Although Asian outoptions with an arithmetic average part are more common, this subsection will first discuss geometric Asian out options. Understanding thegeometric Asian out option makes it easier to understand the workingsof an arithmetic Asian out option.Consider a 1 year geometric average out Asian call option withmonthly observations and a strike price K on a stock St . This means
1The settlement price is the share price against which an option settles, which, for a regular option, is theshare price at the close of business on the expiry day.
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g
that the payout at maturity of this Asian out call option, C T , is:121gSi12 − K , 0 ,(18.1)C T = maxi=1
where Si is the stock price of St at the end of the i-th month. Supposeσi are the implied volatilities of Si with corresponding time units ofi/12-th of a year. And ri are the interest rates for the i-th month. Sincethe correlation between σi and σi+1 is zero, one derives that 121211= VarVar lnln (Si )Si1212i=1i=1
and
 E ln
12
=
121 Var [ln (Si )]12 i=1
=
121 σ 2.12 i=1 i
112
Si
=E
i=1
121
12i=1
(18.2)
ln (Si )
=
121 E [ln (Si )]12 i=1
=
121 ri .12 i=1
(18.3)
This means that the geometric Asian out call option can be priced as acall option on the process12
1
Si12 ,
(18.4)
121 σ 2.12 i=1 i
(18.5)
i=1
which has a a volatility of σ gσg =
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107
The interest rate r g , which is effectively the financing of the hedge, forpricing the geometric Asian out option isrg =
121 ri .12 i=1
(18.6)
This means that the geometric Asian out option can be priced with theBlack–Scholes model using a volatility as in equation 18.5 and an interestrate as in equation 18.6.It is important to recognise that σi is always smaller than σi+1 and ri isalways smaller than ri+1 . The longer the period over which the volatilityis measured the larger it gets, since any deviation in period i is automatically in period i + 1. However, the annualised volatility of σi canstill be larger than the annualised volatility of σi+1 . In other words, thevolatility of an average of observations is always smaller than thevolatility of the share price itself. Obviously, the more observationsthe Asian option has, the smaller the implied volatility.2 This basically means that the price of a geometric average out option is alwayssmaller than the price of a regular option with the same maturity.3
18.2 PRICING THE ARITHMETIC ASIANOUT OPTIONUnlike the geometric Asian out option, the arithmetic mean of a lognormally distributed underlying does not have a lognormal distribution.However, one can use the geometric average as an estimate for the arithmetic mean and hence get a good understanding of the Greeks and otherfeatures of the arithmetic Asian out option. The actual pricing of thearithmetic Asian out option would always be done with a Monte Carloprocess. For the sake of completeness, the payout at maturity for boththe arithmetic Asian out call and put are given in formulae 18.7 and 18.8
2It is very important to distinguish between annualised implied volatility and the implied volatility associatedwith the actual term of the option in this regard. The volatility or implied volatility as referred to in this section isalways the volatility associated with the actual term of the option and is therefore not expressed as an annualisedvolatility.3There is a very special situation where the price of a geometric Asian out put is larger than the regular putwith the same maturity. This is when the interest rate is so large that the regular put is worth very little. Sinceequation 18.6 shows that the interest rate for the geometric Asian out option is substantially smaller than theinterest rate of the regular put, this geometric Asian out put can be worth more, as the decrease in interest ratehas a bigger impact on the price than the decrease in volatility. A geometric average out call option is alwaysworth less than the regular call option with the same maturity.
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respectively.
121 Si − K , 0 ,max12 i=1
(18.7)

 121 max K −Si , 0 .12 i=1
(18.8)
The reason one can use the geometric average as an estimate for thearithmetic average is the Jensen inequality. Jensen’s inequality statesnnn1n(Si )ai ≤ai Siai = 1.(18.9)n i=1i=1i=1nn(Si )ai is a lower bound for i=1ai Si . AlthoughThis means that i=1using the geometric average as an estimate for the arithmetic averagewill not be helpful in accurately pricing the arithmetic Asian out option,one can use this estimate to derive the dynamics of the Greeks and otherdistinguishing features of the arithmetic Asian out option.The following sub-section will discuss the delta hedging of the arithmetic Asian out option from an economical perspective rather than amathematical perspective. The sub-section thereafter will discuss theGreeks of the arithmetic Asian out option using the geometric averageas an estimate for the arithmetic average.Before moving on to the Greeks it is important to observe that, justlike the geometric Asian option, the implied volatility of the arithmeticAsian out option is smaller than the implied volatility of the regularoption with the same maturity. This can be seen by using the geometricaverage as an estimate for the arithmetic average and hence derivingformula 18.5. However, a more economic way to understand this is byrecognising that the duration of any arithmetic Asian option is smallerthan the duration of the regular option with the same maturity. Thefollowing sub-section gives a more elaborate discussion on the durationof an arithmetic Asian out option. Obviously, since the effective impliedvolatility of an arithmetic Asian out option is smaller than the impliedvolatility of the regular option with the same maturity, the Asian outoption is always worth less4 than the equivalent regular option.4
Apart from the Asian out put option in a very high interest rate environment.
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109
18.3 DELTA HEDGING THE ARITHMETIC ASIANOUT OPTIONThe best way to show the delta hedging of an arithmetic Asian out optionis by means of an example.Consider a 3 month arithmetic Asian out call option on BMW with astrike of € 40 and monthly observations. Suppose a trader sells this Asiancall on BMW to an investor. At inception the delta of the arithmetic Asianout call option is very similar to the equivalent regular three month calloption. Assume that BMW’s stock price at inception is € 40 and hencethe delta for both the arithmetic Asian and the regular call option can beassumed to be 1/2. This means that if the trader sells 90 thousand of theseAsian out call options, he will delta hedge himself by buying 45 thousandBMW shares. Now suppose that, on the day before the observation dateof the first Asian setting, BMW stock is trading at € 48. In this case thedelta of the arithmetic Asian out call will be larger than the equivalentregular call. This is for the simple reason that the first Asian setting willalmost certainly be in the money, i.e. higher than € 40. Suppose thatthe Asian out call has a delta of 5/6, i.e. 75 thousand shares long, andthe equivalent regular call option has a delta of 3/4, i.e. 67.5 thousandshares long. However, since the first Asian setting is assured to be in themoney, which accounts for 1/3 of the weight in the arithmetic average,30 thousand shares of the 75 thousand shares serve as a delta hedge forthe first Asian setting. Once this Asian setting is taken the trader doesnot need to hold these 30 thousand shares long any more and shouldtherefore sell them at the close of the first Asian setting. This meansthat his delta hedge against the arithmetic Asian out call option after thefirst Asian setting has reduced to 45 thousand shares. However, sincethe first Asian setting has been taken, which accounts for 30 thousandshares, the remaining arithmetic average is only on 60 thousand sharesand therefore the remaining Asian out call is effectively on 60 thousandshares only. After the first Asian setting, BMW’s share price goes downand, on the day before the second Asian setting, BMW stock is tradingat € 32. Therefore the delta the trader holds against the Asian call optionhas decreased significantly to 10 thousand shares long. Since the secondAsian setting is pretty much assured to be out of the money, the delta thetrader holds is purely against the third Asian setting and nothing againstthe second one. For that reason the trader does not need to do anythingon the close of the second Asian setting. On the day of the third Asiansetting, BMW stock is trading at € 42 and therefore in the money. This
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means that the trader holds a long position of 30 thousand shares as adelta hedge. Obviously, the trader has to sell these shares at the close onthe day of the third Asian setting.The above example clearly shows the procedure for hedging an arithmetic Asian out option. On the day of an Asian setting, the traderneeds to unwind part of his delta hedge if the Asian setting is in themoney and does not need to do anything if the Asian setting is out ofthe money. If the Asian setting is in the money, the trader needs to unwind as a share position the number of Asian options multiplied bythe weight of the setting. This also shows that the duration of an Asianoption is shorter than the equivalent regular option. Namely, on each ofthe Asian settings the weight multiplied by the full size on the Asian outposition roles off. Effectively, an Asian option is spread out over a set ofregular options with maturities equal to the Asian setting dates. Therefore the term of an Asian option cannot be compared to a regular optionwith the same term but is in fact equivalent to a regular option with ashorter term.5 For any option it holds that the longer the time to maturitythe higher the price. This once again makes clear that the price of anAsian out option should be smaller than its equivalent regular option.
18.4 VEGA, GAMMA AND THETA OF THEARITHMETIC ASIAN OUT OPTIONSince sub-section 18.3 clearly shows that the duration of an Asian outoption is shorter than the maturity of the equivalent regular option, it iseasy to see that the Greeks of the Asian out option change accordingly.This basically means that, compared to a regular option, an Asian outoption has a higher gamma and theta but a smaller vega. For example,the risks and Greeks of a 1 year ‘at the money’ Asian out option withmonthly observations are very similar to the risks and Greeks of a 6months ‘at the money’ option. Obviously, a 6 months ‘at the money’option has in turn a higher gamma and theta than a 1 year ‘at the money’option, but a smaller vega.
18.5 DELTA HEDGING THE ASIAN IN OPTIONIt is not so much the pricing that makes the Asian in option interestingbut the delta hedging of it. Apart from some financing on the delta hedge,5The term of the equivalent regular option can be estimated to be the weighted average of the differentregular option with maturities equal to the Asian setting dates.
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111
the pricing of an Asian in option is really nothing more than the pricingof a forward starting option as discussed in Chapter 11. Also the Greeksof the Asian in option are the same as the Greeks of a forward startingoption.6 What is not obvious for an Asian in option is the delta hedgingup until the date on which the strike is fixed, according to the averageof the Asian in settings. That is exactly what will be discussed in thissub-section for the arithmetic Asian in option.Consider a trader who buys a 3 day Asian in put on BMW with amaturity of 1 year from the date on which the strike price is fixed. Thestock has the following price trend over these 3 days:1. € 302. € 203. € 10The trader buys this Asian in option on a notional of € 9 million. On theday of the first Asian in setting, the Asian in option is obviously an ‘at themoney’ put as the average over the first 1 day is equal to the share price.Therefore, on the first day the trader assumes that the delta of the Asianin option will be 0.5 (a 1 year ATM put has a delta of 0.5) and hedgesthis by buying a third of € 4.5 million, i.e. 50 thousand shares of BMW.On the second day, the average price over the first two days has fallen to€ 25, but the current share price is at € 20 and is therefore already an inthe money put, namely the share price is 80 % of the strike. A one yearput on BMW that is 20 % in the money appears to have a delta of 0.6.This means that with the current information the trader needs to be long216 thousand shares on the third day, the day of the last Asian in setting.This 216 thousand is derived from the fact that, given an average shareprice of € 25, a notional of € 9 million and a delta of 0.6, the delta position should be 0.6 ∗ 9 ∗ 106 /25 = 216 thousand on the third day. Sincethe trader has already bought 50 thousand shares on the first day he stillhas to buy 166 thousand shares over two days. This means that his deltahedge for the second Asian setting is 88 thousand shares. On the thirdday, the average share price over the full three days has fallen to € 20.Since the share price is at € 10, the put is 50 % in the money. A 1 year50 % in the money put appears to have a delta of 80 %. This means thatthe trader’s full delta hedge should be 0.8 ∗ 9 ∗ 106 /20 = 360 thousandshares. Since the trader has already bought 138 thousand shares in the6Buying an Asian in option also means buying gamma embedded in the Asian in part, which affects thepricing of an Asian in option. This effect will be elaborated on in sub-section 18.6.
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Table 18.1 Hedging the Asian in put
BMW share price foreach Asian setting
Delta of 1 yearput with strike priceequal to the average ofthe preceding days
Day 1 setting of € 30
0.5
Buy delta times a third of9 ∗ 106 divided by theaverage 1 day share price,1/2 ∗ 1/3 ∗ 9 ∗ 106 /30 = 50thousand shares
Day 2 setting of € 20
0.6
Buy 1/2 of delta times 9 ∗ 106divided by the average 2 dayshare price minus the sharesbought on the first day,1/2 ∗ (0.6 ∗ 9