coupon, the rebate of the downand-out put and the redemption of the zero coupon bond are linked.The payoff profile of the reverse convertible with conditional couponis shown in Figure 25.2. The dotted payoff is to show that this only
110 %100 %80 %
Payoff
60 %
BMW’s share price at maturity60 %
80 % 85 %
100 %
Figure 25.2 Payoff at maturity of a reverse convertible with a 10 % conditional coupon,where the coupon is conditional on BMW >85 % and the investor is short a 100/85 %knock-in put on BMW
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occurs when the down-and-in put has actually knocked in. Obviously,the market conditions under which the conditional coupon is highestare when interest rates are high and when the implied volatility and thedividend yield of the underlying stock are high.There are many iterations of the reverse convertible note with conditional coupon. For example, instead of being protected on the downsidethrough the down-and-in put, one can think of a note where the investoris fully exposed to the downside of BMW. The 10 % coupon is stillconditional on BMW never breaching 85 % of BMW’s initial level. Inreturn for giving up the protection through the down-and-in put andtherefore participating in the full downside of BMW’s share price, theinvestor participates in the full upside over 110 % up to 135 %. Thisstructure can be priced as follows and is also worth 100 %. Althoughthe the structure looks complicated, it can still be priced as a series ofoptions and barrier options. However, the pricing is quite convolutedbut it shows the potential of creating innovative structures by financiallyengineering combinations of existing options rather than creating newoption payoffs. Therefore, the pricing below can better be seen as anexercise than an addition to theory.
r A zero strike call which the trader sells. This zero strike call is worth100 % minus the dividend yield and ensures that the investor is exposedto both the downside and the upside.r A 110/100 % put spread which the trader also sells, where both putsknock out at 85 %. The 110/100 % put spread, which knocks out at85 %, together with the zero strike call ensure that the investor receives a 10 % coupon provided that the share price never breaches85 %, while at the same time fully exposing the investor to thedownside.r A 135 % call which the trader buys and ensures that the investor’sparticipation to the upside is capped to 135 %.r A 100/110 % call spread which the trader buys. However, both calls inthe call spread are down-and-in calls and only knock in when BMW’sshare price breaches 85 %. This ensures that the investor does notparticipate in the first 10 % upside in BMW’s share price when the85 % level has been breached. When the 85 % has not been breached,the zero strike call in combination with the 110/100 % put spreadensures that the investor receives a 10 % coupon while being fullyexposed to the downside, i.e. up to 100 % the payout to the investoris 10 % higher than one would expect based on the zero strike call.
Engineering Financial Structures
163
135 %
110 %100 %
Payoff
85 %
BMW’s share price at maturity
85 %
100 % 110 %
135 %
Figure 25.3 Payoff at maturity of a reverse convertible with a 10 % conditional coupon,where the coupon is conditional on BMW >85 % and the investor has full exposure tothe downside as well as the upside from 110 % up to 135 %
Therefore this 100/110 % call spread need only be live once the 85 %level has been breached. Hence, they are down-and-in calls.Theoretically, one could apply a downward barrier shift to the100/85 % down-and-in call. However, the value change over the barrier is minimal and the exposure to any delta change is also partiallyoffset by the short 110/85 % down-and-in call.Figure 25.3 shows the payoff profile of the above iteration of the reverseconvertible with conditional coupon. The dotted line shows the payoffto the investor when the 85 % barrier has been breached, in which casethe investor loses his conditional coupon and hence the 10 % parallelshift in payout.
25.4 LEVERAGED PRODUCTS FOR THE RISKSEEKING INVESTORThe real risk seeking investor is prepared to give up any coupon forthe sake of leveraging his view on a stock or the market. The perfectnote such an investor would invest in is a structure where the investorbuys a bond and, when he is bullish, he sells a put to gain additional
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130 %
100 %
Payoff
80 %
BMW’s share price at maturity80 %
100 %
120 %
Figure 25.4 Payoff at maturity of a geared bullish note where the investor is fullyexposed to the downside and participates in 150 % of the upside
participation in a call. When the investor is bearish, he sells a call optionto gain additional participation in a put option.As an example, consider an investor who is bullish on BMW andtherefore sells a 1 year ‘at the money’ put in order to gear his exposureto the upside. In other words, the investor buys a 1 year zero couponbond, sells a 1 year ‘at the money’ put and buys X % of a 1 year ‘at themoney’ call. X is solved for, such that the full structure is worth 100 %.It appears that X % is 150 %, which means that the investor participatesin 1.5 times the upside of BMW’s share price. The payoff profile ofthis note, which is sometimes referred to as a supertracker, is shown inFigure 25.4.Now consider an investor who is bearish on BMW and therefore buysa bearish geared note. In this case the investor buys a 1 year zero couponbond, sells a 1 year ‘at the money’ call and buys a 1 year X % geared ‘atthe money’ put. When X % is equal to 233 %, the full bearish geared noteappears to be worth 100 %.2 The payoff profile of the bearish geared noteis shown in Figure 25.5. The above shows that the bearish note works2For the sake of completeness the numbers that are used in both the bullish and the bearish geared notes arethat the 1 year zero coupon bond is worth 97 %, the 1 year ‘at the money’ call is worth 4 % and the 1 year ‘at themoney’ put is worth 3 %.
Engineering Financial Structures
165
144.66 %
100 %
Payoff
80 %
BMW’s share price80 %
100 %
120 %
Figure 25.5 Payoff at maturity of a geared bearish note where the investor is fullyexposed to the upside and participates in 233 % of the downside
along the same lines as the bullish note with the caveat that the bearishnote has some credit exposure to the investor that the bullish note doesnot have. Since the investor is owed 100 % at maturity and the put cannever be worth more than 100 %, the bullish note gives no credit exposureto the investor in the note. However, with the bearish note the investor isshort an ‘at the money’ call which is worth more than 100 % if BMW’sshare price is higher than 200 % and, since the investor is only owed100 %, this bearish note does give some credit exposure to the investor.For this reason, the bearish note typically has a 200 % call embedded init which the investor buys. This will slightly decrease the gearing on theput in the bearish note.
Appendix AVariance of a Composite Optionand Outperformance OptionIt is relatively easy to prove that the variance of a composite option isequal to22σcompo= σ S2 + 2ρσ S σFX + σFX.
(A.1)
One just has to compare the movement of the underlying in the compositecurrency with the movement of a geometric basket option composed ofthe underlying in the local currency and the FX rate. The formula forthis basket option is as follows:Ft = St,local St,FX ,
(A.2)
dSt,local= rlocal dt + σ S dW1,t ,St,local
(A.3)
dSt,FX = rcompo − rlocal dt + σFX dW2,t .St,FX
(A.4)
where
and1
Since
and
 Var [ln (Ft )] = Var ln St,local St,FX = Var ln St,local + ln St,FX2t,= σ S2 + 2ρσ S σFX + σFX E [ln (Ft )] = E ln St,local St,FX= rlocal t + rcompo − rlocal t= rcompo t,
(A.5)
(A.6)(A.7)(A.8)
1To get an intuitive feel for equation A.4, imagine an exchange rate with zero volatility and therefore the valueof the exchange rate only changes because of differences in risk free interest rates between the currencies. In this1+rcompo dtcase St+dt,FX = St,FX. Taylor expansion gives that this is equal to St,FX (1 + (rcompo − rlocal) dt).1+rlocal dt
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the movement of the stock in the composite currency can be modeledaccording to equation 20.1.Assuming that another stock Yt behaves according to the followingprocessdYt,local= rlocal dt + σY d W3,t ,Yt,local
(A.9)
one can easily prove that the variance of the process Ft = YStt is equal toσ F2 = σ S2 + σY2 − 2ρσ S σY .
(A.10)
If the dividend yield is zero, the mean for Ft is zero. Namely, St,localVar [ln (Ft )] = Var lnYt,local = Var ln St,local − ln Yt,local
and
= σ S2 + σY2 − 2ρσ S σY ,
(A.11)
 E [ln (Ft )] = E ln St,local ) − ln Yt,local=0
(A.12)
Appendix BReplicating the Variance SwapThe variance swap can be replicated by K12 options on all different strikesK . Since out of the money options are generally more liquid, a varianceswap is replicated by both calls and puts. Let S c be the cutoff above whichthe replicating option strikes are calls and below S c the replicating optionstrikes are puts. Now it is easy to prove that the payoff of this replicatingportfolio at maturity is:ST − S c (ST ) =− lnSc

STSc
.
(B.1)
Indeed, there are two possibilities; either ST > S c , in which case the putsare worthless and the payoff is merely the integration of all the calls,or ST < S c and all the calls are worthless and the payoff is merely anintegration of all the puts. First consider the scenario that ST > S c
 (ST ) ==
 Sc0
1max [K − ST , 0] dk +K2
Sc
1(ST − K ) dkK2
 ST ST
1ST+ 2 dkKKScST ST= − ln(K ) −K Sc ST − S cST=− lncSSc
=
−
 ∞Sc
1max [ST − K , 0] dkK2
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If ST < S c one gets (ST ) ==
 Sc0
1max [K − ST , 0] dk +K2
0
1(K − ST ) dkK2
 Sc Sc
 ∞Sc
1max [ST − K , 0] dkK2
ST1− 2 dkKK0 cST S= ln(K ) +K 0 ST − S cST=− lncSSc
=
To prove that a variance swap can at any time be replicated by 1/K 2options on each strike, one not only needs an expression for (ST ), thevalue of the replicating portfolio at maturity, but also for any t. Equation23.4 shows that (St ) can be expressed as (St ) =
 Sc0
1  −r (T −t)KeN(−d)−SN(−d)dk2t1K2
 ∞
1 St N (d1 ) − Ke−r (T −t) N (d2 ) dk2Sc K St − S cStσ 2 (T − t)=−ln.+ScSc2+
(B.2)
The proof for expression B.2 is quite prolonged. Nonetheless this appendix gives it to first of all allow the mathematically interested readerto check the validity of the replicating method and secondly it can serveas a point of reference.The proof is split up into two parts. First it is shown that (St ) with acutoff equal to S c , (St , S c ), minus (St ) with a cutoff of St , (St , St ),is equal toSt − S c− lnSc

StSc
.
(B.3)
Replicating the Variance Swap
171
The second part of the proof shows that (St , St ) is equal toσ 2 (T − t).(B.4)2Adding the first and second parts is obviously equal to the replicatingportfolio of 1/K 2 options per strike where a cutoff of S c , (St , S c ), isused and is in turn equal to the expression in equation B.2.For simplicity, both parts are proved assuming that the interest rateis zero. The first part makes use of the following feature of a standardnormal distribution1 − N (d1 ) = N (−d1 )1 − (N (d2 ) = N (−d2 ).Using this knowledge it is easy to prove the first part Sc1(St , S c ) − (St , St ) =[K − KN(d2 ) − St + St N (d1 )] dk2K0 St1−[K − KN(d2 ) − St + St N (d1 )] dkK20 ∞1+[St N (d1 ) − KN(d2 )] dk2Sc K ∞1+[St N (d1 ) − KN(d2 )] dk2St K Sc1=[K − KN(d2 ) − St + St N (d1 )] dk2St K Sc1−[St N (d1 ) − KN(d2 )] dk2St K Sc1=[K − St ] dk2St K St − S cSt=.− lncSScThe proof for the second part is quite daunting and is therefore only forthe readers that are interested in the rigorous mathematics behind thereplicating portfolio of the variance swap. The proof starts by substituting d1 and d2 into their respective positions and subsequently applying
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several shifts to the integral.1 St√1α=σ T −t(St , St )=[KN(−d2 ) − St N (−d1 )] dkK20 ∞1+[St N (d1 ) − KN(d2 )] dk2St K Stln( SKt ) α1KN−+=K2α20− St N −+
 ∞St
ln( SKt ) α−α2
1St NK2
dk
ln( SKt ) α+α2
ln( SKt ) α−dkα2 0 ααdlllK =St ell+− St N−St e N=α2α2St el−∞ ∞ααdllll−St e N − −St N − ++α 2α 2St el0 0  llαα−lN=+−e N−dlα2α2−∞ ∞lαlα−l+e N − +−N − −dlα2α20 ∞ lααllN − +−e N − −dl=α2α20 ∞αlαl−l+−N − −dle N − +α2α20 ∞lα=1 + e−l N − +α20αldl− 1 + el N − −α2−KN
1
This proof is thanks to Alex Boer.
Replicating the Variance Swap
=
=
=
1 + e−l − x 2e 2 dx√2π0−∞ − l −αα21 + e−l − x 2e 2 d x dl−√2π−∞ ∞  021 + e−l − (x− αl + α2 )2edx√2π0−∞ 021 + e−l − (x− αl − α2 )2−ed x dl√2π−∞ ∞  − l +αα
−
l2
e +e
−∞
 ∞  0 −∞
0
=
2
 ∞  0 0
x=−x
173
l2
− 2l
e +e

e
− 2l
− 2l

e−
22xx 2 + l 2 + α4 − 2lxα + α −lα2
l2
e e
−

22xx 2 + l 2 + α4 − 2lxα − α +lα2
 ∞  ∞ 
d xdl√2π
d xdl√2π
 x 2 l 2 α2lle 2 + e− 2 e− 2 − 2α2 − 8
0
0
 d xdlxlxα× e− e− α − 2√2π ∞  ∞22 α2l=2αl, x=2x=eαl + e−αl e−2x −2l − 8004αd xdl× eαx − e−αx e−4xl √2π ∞  ∞2−2(x+l)2 − α8 16αd xdlcosh(αl) sinh(αx)ee=√2π00 ∞  px−lp= x+l22 , q= 2=e−8 p cosh(α( p − q)) sinh(α( p + q))0
=

− xlα + xα2
−p
α 2 32α× e− 8 √dqdp2π ∞ p2−8 p 2 − α8 32αeecosh(α( p − q))√2π − p0× sinh(α( p + q))dq] d p
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Exotic Options Trading
=
 ∞e0
===p= 4z
=
=
=
===
2
−8 p 2 − α8
e
32α√ sinh(2αp) · p d p2π
 ∞  16αα2α222e−8 p +2αp− 8 − e−8 p −2αp− 8 p d p√2π 0 ∞22 √√16α− 8 p− √α8− 8 p+ √α8pedp−pe√2π 0 ∞(4 p− α2 )2(4 p+ α2 )216α−−22− pepedp√2π 0 ∞22z − (z− α2 )16αz (z+ α2 ) dz2e− e− 2√4442π 0 ∞ αα  − z2 z+e 2 dz√22π− α2 ∞ α  − z2 −e 2 dzz−α22 ∞ ∞2αz2− z2zedz −ze− 2 dz√α2π− α22 ∞ ∞αα − z2α − z2 +√e 2 dz +e 2 dzα222π− α22 ∞ ∞αα − z2 α − z2 e 2 dz +e 2 dz√α222π− α22√αα 2π·√22π22σ (T − t)α=.22
BibliographyAarts, P. (1999). Pricing the basket option and exploring its volatility structure, Maththesis Utrecht University.Demterfi, K., Derman, E., Kamal, M. and Zou, J. (1999). More Than You Ever WantedTo Know About Volatility Swaps, Quantitative Strategies Research Notes, GoldmanSachs.Derman, E. (1992). Outperformance Options. Goldman Sachs & Co.Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury Press, BelmontCalifornia.Grimmett, G.R. and Stirzaker, D.R. (1992). Probability and Random Processes, 2nd ed.Clarendon Press, Oxford.Hull, J.C. (1993). Options, Futures, and other Derivative Securities, 2nd ed. PrenticeHall Inc.Hull, J.C. (1997). Options, Futures and other Derivatives, Prentice-Hall Inc.Lamberton, D. and Lapeyre, B. (1996). Introduction to Stochastic Calculus Applied toFinance, Chapman & Hall, London.Roelfsema, M.R. (2000). Exploiting the Dependencies between Index and Stock Options,Delft University Press.de Weert, F.J. (2002). Basket en skew arbitrage, Math thesis Utrecht University.de Weert, F.J. (2006). An Introduction to Options Trading, John Wiley & Sons Ltd,Chichester.Wilmott, P. (1998). Derivatives: The Theory and Practice of Financial Engineering,University Edition, John Wiley & Sons Ltd, Chichester.
IndexADR see American Depository Receiptairbagsconcepts 160–3definition 160American Depository Receipt (ADR)126American optionsconcepts 4, 46–8definition 4An Introduction to Options Trading(author) 1analytically short skew, forward startingoptions 74–5annualised variance, variance swaps139–49, 169–74appendices 167–74arithmetic Asian out optionssee also Asian . . .concepts 105, 107–10delta hedging 109–10duration 108–10geometric averages 108, 110the Greeks 107–10implied volatility 108, 110Monte Carlo process 107pricing 107–8, 110Asian in forwardsconcepts 112–18definition 112gamma 113–15hedging 113–15Monte Carlo simulations 114–18optional early termination 116–18pricing 114–18volatility 115
Asian in optionsconcepts 105, 110–18definition 105delta hedging 110–12pricing 111Asian optionsconcepts 105–18definition 105duration 108–10implied volatility 107, 108Monte Carlo simulations 107, 114–18pricing 105–18risk 105, 109–12types 105Asian out optionssee also arithmetic . . . ; geometric . . .concepts 105–10definition 105pricing 105–8risk 105, 109–10at the money options (ATM) 16, 29–37,41–4, 73–4, 81, 84–91, 93–6,99–104, 110, 119–23, 131, 136,140–2, 148–9, 152–8, 160–3, 164–5gamma 16quanto options 119–23reverse convertibles 87–91, 93–6,99–104, 160–3skew 29–32, 34, 73–4autocallable notesconcepts 93–8, 99–104conditional coupons 94–7definition 93down-and-in put barrier options 93–6,101–4
178
Index
autocallable notes (Continued)early redemption features 95–6, 97–8,101–2examples 93–4interest/equity correlation factors 98maturity dates 94pricing 95–8, 99–104reverse convertibles 93–6, 99–104unconditional coupons 94, 97–8barrier options 57–69, 77–8, 87–90,93–6, 160–3see also down-and-in . . . ;up-and-out . . .autocallable notes 93–6barrier shifts 60–3, 64, 67–9, 96, 104,161–3concepts 57–69, 78, 87–90, 93–6,160–3delta 58–65, 161–3double barrier options 57–8, 68–9the Greeks 57–65ladder options 78lookback options 79–80maturity dates 62, 65–6Monte Carlo simulations 57, 58,59–60, 64, 66, 68–9Newton–Raphson process 102pricing 57–69, 77–8, 89–90, 95–8,162–3retail demand 57–8reverse convertibles 87–90, 93–6,99–104risk 57–69single barrier options 57–68skew risk 57, 58, 66–8time to maturity 62types 57–65, 68–9volatility 60–2barrier parity, knock-out/knock-in barrieroptions 65basket options 138, 151–6, 167–8see also dispersionBlack–Scholes options pricing formula151concepts 138, 151–6, 167–8definition 151hybrids 138implied volatility 152–3Monte Carlo simulations 151pricing 151
vega 153–5volatility 151–6bearish strategies 37–9, 63, 157–9,164–5best of optionsconcepts 135–8correlation risk 135–7definition 135hybrids 138payoff profiles 135–6pricing 135binomial treesconcepts 46–8Monte Carlo simulations 46–7, 48Black–Scholes options pricingformulabasket options 151Brownian motion 45concepts 6–10, 12–14, 45, 84–5, 107,120–1, 131, 151delta 12–14example 7forwards 10implied volatility 8, 45, 107parameters 6, 8–9, 10, 12–14, 107,120–1BMW shares 29–30, 35–6, 39–40, 46–8,49–51, 53–4, 58–60, 63–4, 71–2, 77,79–80, 83–4, 87–9, 91, 93–8,99–104, 109–16, 129–38, 140–9,157–65Boer, Alex 172bondscoupons 87–91, 93–8, 99–104, 157–8,160–3leveraged products for risk-seekinginvestors 157, 163–5reverse convertibles 87–91, 93–6,99–104, 160–3zero coupon bonds 157–8, 160–3BP shares 119–23Brazilian real see BRLbreak-even levels, simple optionstrategies 35–44British poundUSD composite options 125–8USD quanto options 119–23BRL/EUR exchange rate, skew 33–4Brownian motion 45, 121bullish strategies 35–6, 39–40, 63, 157–9,163–5
Index
call optionssee also Greekschooser options 49–51concepts 1–8, 10–21, 23–4, 35–44,158–65definition 3forward prices 10payoff profiles 4–5pricing 6–8, 10put–call parity 11–12up-and-out call barrier options 16–17,58, 63–5, 67–9, 78–80, 103call spread option strategyconcepts 35–7, 53–5, 85, 95–6digital options 1, 53–5, 62, 95–8examples 35–6, 53–4payoff profiles 36–7ratchet options 85skew 36–7, 85callable reverse convertiblesconcepts 99–102definition 99capital guaranteed products for riskaverse investors 157–9attractive market conditions 158–9payoff profiles 158cautious equity investors, exposure equityproducts 157, 1