ium paid for the composite option at inceptionof the trade. For the call option the FX hedge on the paid premiumworks in the opposite direction to the FX hedge on the notional of thedelta hedge. For the put option the FX hedge on the paid premiumof the composite option works in the same direction as the FX hedgeon the notional of the delta hedge. Obviously, the reason that a traderwants to hedge the FX exposure on the premium is that in order tobuy the composite option in the composite currency the trader wouldfirst need to sell the local currency to buy the composite currency. Thiseffectively gives him an FX position which still needs to be hedged.Counterintuitively, no FX hedge needs to be executed on the premium ifthe trader sells a composite option, as the trader receives the premium inthe composite currency and the model used to determine the price and thedelta of the option is equation 20.1, which assumes financing in thecomposite currency, i.e. USD in the above example. The only reason that
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Table 20.1 FX hedge on the composite optionPosition compositeoption
FX hedge ondelta notional
Long composite calloptionShort composite calloptionLong composite putoptionShort composite putoption
Sell local currency to buycomposite currencyBuy local currency to sellcomposite currencyBuy local currency to sellcomposite currencySell local currency to buycomposite currency
FX hedge on premiumpaid for composite optionat inception of tradeBuy local currency to sellcomposite currencyNoneBuy local currency to sellcomposite currencyNone
the trader needs to execute an FX hedge on the premium paid if he buysa composite option is to offset the FX he executes to actually be able tobuy the option and therefore ensure financing in the composite currencyas the model in equation 20.1 prescribes. Therefore the full FX hedgecan be summarised by Table 20.1.
21Outperformance OptionsThe outperformance option measures the outperformance, whether it beon the upside or downside, of one stock against another and pays thisdifference. Obviously, an outperformance option can also be structuredbetween two different assets, like a stock and the oil price, in whichcase it measures the outperformance of the stock against the oil price.The beauty of outperformance options is that their pricing and riskslook complicated but can easily be understood when comparing themto a composite option. The next sub-section gives an example of anoutperformance option.
21.1 EXAMPLE OF AN OUTPERFORMANCE OPTIONConsider an investor who believes that BMW’s share price will outperform the share price of Volkswagen (VW) in the next year. Therefore hebuys a 1 year outperformance option on BMW versus VW. Suppose thatBMW’s share price is € 40 and VW’s share price is € 70. The investordecides to buy this outperformance of BMW versus VW on a notionalof € 7 million. To demonstrate the payoff of this option two differentprice trends are considered. First, assume that in 1 year BMW’s shareprice will be trading at € 50 and VW’s share price at € 63. In this casethe payout on the outperformance option should be 35 % of € 7 million,as BMW went up 25 % and VW went down 10 % and therefore BMWoutperformed VW by 35 %. Secondly, assume that in 1 year BMW’sshare price is at € 50 and the share price of VW is at € 91. In this casethe payout on the outperformance option is zero as VW outperformedBMW by 10 %.Another way to think about the payout of the outperformance option isto say that the holder of the outperformance option gets paid the greaterof zero and the notional growth of BMW shares minus the notionalgrowth/decline of VW shares. Viewing it like this makes it intuitive thatthe outperformance option is nothing more than a composite option,because the payout of a composite call option in the currency of theunderlying stock is equal to the notional growth of the share minus the
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notional growth/decline of the composite currency, quoted in the localcurrency. So, if one views a currency as an asset that is quoted in thecurrency of the underlying stock of a composite option, one sees thatthe definition of the composite option and the outperformance optionare exactly equivalent. In other words, a composite option is a type ofoutperformance option where the performance of the stock is measuredagainst a specific currency.Another way to look at the outperformance option on two stocks isto say that one is allowed to swap the stock the investor thinks willunderperform for the other stock in the ratio of the two stock prices atinception of the trade.
21.2 OUTPERFORMANCE OPTION DESCRIBEDAS A COMPOSITE OPTIONSub-section 21.1 made clear that an outperformance option is nothingmore than a composite option. Although this is the case, it might notbe easy to see how this translates into a comparison between the actualoutperformance option where the strike is defined and a composite option. This is because with a composite option one is only interested inthe payout of the option in the composite currency, and to compare theoutperformance option with the composite option one needs to convertthis payout to the currency of the underlying stock.Before actually going into the pricing of an outperformance option itis good to first clearly state the payoff at time t for an outperformanceoption on BMW, BMWt , versus VW, VWt . This payoff is the following BMWtVWtoutCt = max, 0 · N,(21.1)−BMW0VW 0where N is the notional of the outperformance option, which was takento be 7 million. By multiplying equation 21.1 byBMW0 VWt·VWtBMW0equation 21.1 can be rewritten as BMW0NBMWtoutCt = max, 0 · VWt ·−.VWtVW0BMW0
(21.2)
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To see how a price for Ctout can be determined, one just has to be able toprice BMWtBMW0out,VWCt,0 .(21.3)= max−VWtVW0However, closer investigation of equation 21.3 gives that Ctout,VW expresses nothing more than a call option on BMW where the payout is ina different currency. The currency is obviously one VW share. In thistcurrency the value of one BMW share at time t is BMW, and as the initialVWtBMW0value of a BMW share in the VW currency is VW0 one sees that equation21.3 is the payoff of an ATM call option on BMW in the VW currency.In turn, equation 21.3 can be viewed as a call option on a newly definedtstock Ft = BMW, which according to Appendix A can be described byVWtthe following processdFt= σout dWt ,Ft
(21.4)
222= σBMW+ σVW− 2ρσBMW σVW .σout
(21.5)
where
The Black–Scholes formula gives a price for a call option on a processFt , which is defined by equations 21.4 and 21.5, and has a strike price F0 .What makes it difficult to compare the outperformance option with thecomposite option is that with the composite option one is only interestedin the value of the composite option in the composite currency, whereaswith the outperformance option one has to convert the price of the optionin the currency of VW back into a sensible euro price. In other words, inthe case of the outperformance option on BMW versus VW, one needsto multiply Ctout,VW by VWt to get the actual value of the outperformanceoption on BMW versus VW. Multiplying this by the number of options,N /BMW0 , one exactly gets the payoff of the outperformance option asdescribed by equation 21.2.
21.3 CORRELATION POSITION OF THEOUTPERFORMANCE OPTIONEquation 21.5 makes clear that the owner of an outperformance option has a short position in the correlation between the two assets.In other words, the more inversely correlated the two stocks in the
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outperformance option are, the more valuable is this outperformance option. This is intuitive as the holder of an outperformance option stands toreceive the highest payout if the two stocks move in opposite directions,i.e. the outperforming stock moves up and the other stock moves down.
21.4 HEDGING OF OUTPERFORMANCE OPTIONSDelta hedging the outperformance option is very similar to the hedging ofthe composite option. Firstly, one can easily find a delta on the compositeBMW call in the VW currency, as described by equation 21.4. So, if atrader is long a BMW outperformance option versus VW, he is basicallylong a composite call option on BMW in the VW currency on a notionalNthat is equal to VWt · BMW. The parameters of equation 21.4 provide a0delta, δout , from which he can calculate the euro notional of BMW hewould need to sellNδout · VWt · BMW0
BMWt
.
(21.6)
According to Table 20.1 the trader would then need to buy the VWcurrency in the same notional as his delta hedge on BMW. Buying theVW currency obviously means that the trader needs to buy VW sharesand is therefore effectively paying (selling) euros in the same notionalas his delta hedge on BMW. However, the hedge on the premium paidfor the outperformance option works differently to the hedge for thecomposite option. Equation 21.4 shows that the financing of the outperformance option should be flat, which means that the dollar amountTable 21.1 Hedges on the outperformance option and net euro cash flowSecurity
Amount bought or sold
Euro cash flow
Outperformance option
1
−C Tout,VW · VWt ·
BMW share
−
VW share
+
VW share
−
Nδout · VWt · BMW0
BMWtNδout · VWt · BMW0VWtNC Tout,VW · VWt · BMW0
VWtNet cash flow
NBMW0
+δout · VWt ·
NBMW0
−δout · VWt ·
NBMW0
C Tout,VW · VWt ·0
NBMW0
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spent and received should be the same. This means that to hedge thisoutperformance option fully the trader would also need to buy eurosto sell the VW currency on the premium paid for the outperformanceoption. In other words, sell VW shares and therefore buy euros on thepremium paid for the outperformance option. Table 21.1 summarisesthe full hedge of the ourperformance option on BMW over VW andalso shows that the actual cash euro value adds up to zero. Obviously,when hedging an outperformance option, a trader would always net outthe two hedges on VW in Table 21.1. In Table 21.1 the two hedges onVW are shown separately for the sake of clarity, as each line serves adifferent aspect of the hedge of the outperformance option.
22Best of and Worst of OptionsThe best of option is an option where the investor receives the gainon the best performing stock among a predefined number of stocks.In other words, the investor is long a call on the best performing stockamong the predefined number of stocks. The worst of option is an optionwhere the investor is long a put on the worst performing stock amongst apredefined number of stocks. The pricing of either the best of or the worstof option is typically done by using a Monte Carlo process. There is evena closed form solution for the best of and worst of options. However,since both the best of and worst of options are, in practice, always pricedas a Monte Carlo process, this section focuses on explaining the risksassociated with such options. The main risk of either a best of or a worstof option is the correlation between the underlying assets on which thebest of or worst of option is based. Luckily, the correlation risk can beeasily derived from an outperformance option.
22.1 CORRELATION RISK FOR THE BEST OF OPTIONConsider an investor who buys a best of option between VW and BMWexpiring in 1 year. This means that after 1 year the investor gets paidthe greater of the percentage gain in VW and BMW. In other wordsthe investor is long a call option on either VW or BMW, whichever hasperformed better after one year. Mathematically, the payoff at maturitylooks like this:VWT − VW0 BMWT − BMW0Best of VW/BMWCT= max,,0 .(22.1)VW0BMW0The best way to understand the correlation risk embedded in this bestof option is to view this option as an ‘at the money’ call on VW plusan outperformance option of BMW over VW. Unfortunately this representation is only equal to the best of option under the condition thatif VW is down after 1 year it has at least outperformed BMW. In otherwords, if VW is lower after one year and BMW has outperformed VW,
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the representation of an ‘at the money’ call on VW plus an outperformance option of BMW over VW gives a payout, whereas the best ofoption does not give a payout at all. Nonetheless, this representationproves very useful in determining whether the seller of a best of optionis long or short correlation. Mathematically, the representation of an ‘atthe money’ call on VW plus an outperformance option on BMW overVW looks like thisVWT − VW0 BMWT − BMW0Best of VW/BMWCT= max,,0VW0BMW0VWT − VW0≤ max,0VW0 BMWTBMW0+ max,0 .(22.2)−VWTVW0Whereas the payoff of an ‘at the money’ call on VW plus an outperformance option of BMW over VW is greater than the best of optionof BMW and VW, the payoff of a long ‘at the money’ forward on VWplus an outperformance option on BMW over VW is smaller than thepayoff of the best of option between BMW and VW. Mathematically,the payoff of the best of option is therefore encapsulated by these tworepresentations and looks like thisVWT − VW0max,0VW0 VW0 − VWTBMWTBMW0− max,0, 0 + max−VW0VWTVW0≤ C TBest of VW/BMWVWT − VW0≤ max,0VW0 BMWTBMW0+ max−,0 .VWTVW0(22.3)Since any outperformance option is short correlation, each side of inequality 22.3 is short correlation. The first term in formula 22.3 at acorrelation of −1 is greater than the third term at a correlation of 1. Alsoat a correlation of −1, C TBest of VW/BMW is equal to the first term in formula22.3. At a correlation between BMW and VW of 1, C TBest of VW/BMW is
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equal to the third term of formula 22.3. This means that the value ofC TBest of VW/BMW increases as the correlation goes down. Hence, the holderof a best of option is short correlation. In other words, the lower thecorrelation between the constituents of a best of option, the higher thevalue of this best of option.
22.2 CORRELATION RISK FOR THE WORSTOF OPTIONConsider an investor who buys a worst of option between VW and BMWexpiring in 1 year. This means that after 1 year the investor gets paidthe greater of the percentage decline in VW and BMW. In other wordsthe investor is long a put option on either VW or BMW, whichever hasperformed worse after one year. Mathematically, the payoff at maturitylooks like this:VW0 − VWT BMW0 − BMWTWorst of VW/BMWCT= max,,0 .VW0BMW0(22.4)To determine the correlation risk associated with a worst of option, thesame analysis as in sub-section 22.1 is needed. The payoff of the worstof option has a lower bound equal to a short forward position in VW plusan outperformance option of VW over BMW and an upper bound equalto a long ‘at the money’ put position plus an outperformance option ofVW over BMW. Mathematically this can be presented asVW0 − VWTmax,0VW0 VWT − VW0VWTVW0− max,0, 0 + max−VW0BMWTBMW0≤ C Tworst of VW/BMWVW0 − VWT≤ max,0VW0 VWTVW0+ max−,0 .BMWTBMW0(22.5)Again, the first term in formula 22.5 at a correlation of −1 is greaterthan the third term at a correlation of 1. Also at a correlation of −1,
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C TWorst of VW/BMW is equal to the first term in formula 22.5. At a correlationbetween BMW and VW of 1, C TWorst of VW/BMW is equal to the third termof formula 22.5. This means that the value of C TWorst of VW/BMW increasesas the correlation goes down. Hence, the holder of a worst of option isalso short correlation. In other words, the lower the correlation betweenthe constituents of a worst of option, the higher the value of this worstof option.
22.3 HYBRIDSAlthough hybrids have not explicitly been discussed so far, the mainproducts used to create a hybrid have already been discussed. It is important to understand that hybrids in themselves are not exotic but it is theproduct that can comprise underlyings from different asset classes thatmakes a structure exotic. For example, the best of and worst of options donot necessarily have to compare the performances of underlyings fromthe same asset class, but can also compare performances from differentasset classes. The same holds true for the outperformance option andeven the basket option, which will be discussed in Chapter 24.
23Variance SwapsIn recent years variance swaps have become increasingly popular as theyare a better way to express a view on the volatility of an underlying stockor index than regular options. When betting on the volatility of a certainunderlying, options have the disadvantage of having vega and gammaexposures concentrated around the strike price of the option. This meansthat, if the stock price moves away from the strike price during the termof the option, any movement in the underlying is irrelevant with respectto the profit of that option and, because the vega has disappeared aswell, changes in implied volatility do not influence the profitability ofthat option either. Therefore, executing a pure bet on the volatility of anunderlying through options is far from perfect because one can be righton the volatility of the underlying during the whole term of the option andstill lose money. For example, a trader wants to take advantage of cheapvolatility by buying an option but the stock drifts away from the strike ina very unvolatile manner after which it becomes very volatile, bringingthe realised volatility of the option during the term of the option far abovethe implied volatility for which the trader initially bought his option. Inthis case the trader obviously loses money despite being right on thevolatility. To get around this local characteristic of volatility exposurefor options, variance swaps were introduced.A variance swap is a contract that pays the difference between theannualised variance of an underlying and the annualised variance strikeagreed upon at inception of the trade. Its payoff at maturity is equal to 2σ R − σ K2 × N ,(23.1)where
r σ R is the annualised realised volatility during the term of the varianceswap;
r σ 2 is the annualised realised variance during the term of the varianceR
swap;r σ K is the annualised volatility strike;
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r σ 2 is the annualised variance strike;r NK is the notional of the variance swap, also called variance notional,in euros per annualised volatility point squared.As with any swap the cash flow at inception is zero. In other words,when pricing a variance swap one solves for variance strike, σ K2 , thatmakes the expected payoff of the swap at maturity zero.The following sub-sections will show how to replicate a varianceswap with options and how it can be shown that, at any moment, thevega is constant in stock price and decreases linearly over time andthe gamma of a variance swap is constant over time and although thegamma is not constant in stock price the gamma cash is. This is instark contrast to the concentrated gamma around the strike of a singleoption.
23.1 VARIANCE SWAP PAYOFF EXAMPLESuppose a trader has sold a 1 year variance swap on BMW at an impliedvolatility level of 20 % and on a notional of € 5 million. After 1 year therealised volatility appears to have been 18 %. In this case the trader willreceive an amount equal to−(0.182 − 0.202 ) ∗ 5 000 000 = 38 000.
(23.2)
23.2 REPLICATING THE VARIANCE SWAPWITH OPTIONSThe variance swap can be understood more easily by replicating it withoptions. In this sub-section it will be shown that a portfolio of optionsconsisting of 1/K 2 options of each strike replicates the payoff profile of avariance swap to a factor 2/T . In other words, multiplying this portfolioby 2/T exactly gives the payoff of the variance swap.1Consider a portfolio of options (St ) where each strike has a weighting of 1/K 2 . Since out of the money options are most liquid, all theoptions in the portfolio with a strike lower than the cutoff point S c areput options and the options in (St ) with strikes larger than S c are calloptions. S c will be around the ‘at the money’ level. At expiration, T ,1The following proof is based on the publication More Than You Ever Wanted To Know About VolatilitySwaps (Demterfi et al., 1999).
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Appendix B proves that the value of this portfolio (ST ) is: Sc ∞11 (ST ) =max [K − ST , 0] dk +max [ST − K , 0] dk22KSc K0 STST − S c.(23.3)− ln=cSScAppendix B also shows that the value of the portfolio  for any time tis equal to2 Sc1  −r (T −t) (St ) =N(−d)−SN(−d)dkKe2t1K20 ∞1 St N (d1 ) − K e−r (T −t) N (d2 ) dk+2Sc K St − S cσ 2 (T − t)St=+−ln,(23.4)ScSc2where σ is the implied volatility expressed annually and is assumed tobe the same for all different strikes. Because of the existence of skewthis is not the case in practice, which will be elaborated on in sub-section23.8. The variance exposure of (St ) can be obtained by differentiatingequation 23.4 with respect to σ 2 . This gives the variance exposure ϕ asT −t∂ (St ).(23.5)=2∂σ2Equation 23.5 already shows that the variance exposure of the portfolio(St ) is constant and independent of the stock price. However, to makesure that the variance exposure at inception, i.e. t = 0, is equal to € 1per volatility point squared, one needs to have a quantity of 2/T of theportfolio (St ). Equation 23.4 makes it easy to prove that σ 2 (T − t)22 St − S cSt+(St ) =(23.6)−lnTTScScTϕ=
does indeed replicate the payoff of a variance swap. Indeed, if a traderbuys 2/T of (St ) at t = 0 the fair value is 2 S0 − S cS02(S0 ) =+ σ 2.− lncTTSSc2This can be shown by integrating over K the value of the 1/K 2 put option from 0 to S c plus integratingover K the value of the 1/K 2 call option from S c to infinity.
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If the trader delta hedges the above portfolio perfectly and, over the termof the options in the portfolio, the annualised realised volatility appearsto be σ R , the real value of 2(S0 )/T is 22 S0 − S cS0+ σ R2 .(S0 ) =−ln(23.7)TTScScThis means that the trader’s payoff at maturity is equal toσ R2 − σ 2 .
(23.8)
Expression 23.8 is exactly the payoff formula for a variance swap whenσ K is equal to the implied volatility, σ , of the individual options in theportfolio, which was assumed to be constant for all different strikes.Since the implied volatility is the trader’s best estimate of the futurerealised volatility, the implied volatility should be taken as the fair variance strike. Another way of putting it is that, if σ K is equal to the impliedvolatility, σ , the variance swap can be replicated by 2/(T · K 2 ) numberof options per strike.
23.3 GREEKS OF THE VARIANCE SWAPThe Greeks of the variance swap can easily be derived from equation23.6. Since equation 23.7 assumes that all the options in portfolio (St )are being delta hedged, the delta of the variance swap cannot be derived from equation 23.6. However, assuming that the implied volatilityis constant in strike, the delta of a variance swap is obviously zeroas the variance payoff is independent of stock price. The fact that avariance swap does have a so-called skew delta will be discussed insub-section 23.8.To get to the expressions for vega, gamma and theta of a variance swapone just has to differentiate equation 23.6 to its respective variables.The sensitivity of a variance swap with respect to the implied volatility,i.e. the vega, ν, of a variance swap is∂ T2 (St )σ (T − t)ν==2·.(23.9)∂σTEquation 23.9 shows that the vega of a long variance swap position isalways positive, but also that the vega changes with the changing impliedvolatility. The speed at which the vega changes with the changing impliedvolatility can be derived by taking the derivative of ν with respect to
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σ . The fact that the vega is not constant in implied volatility is calledconvexity and is elaborated on in sub-section 23.9.The exposure, φ, of a variance swap to variance, σ 2 , is equal to∂ T2 (St )T −t=.(23.10)φ=2∂σTEquation 23.10 shows that the variance exposure of a variance swapdecreases linearly over time and is zero at maturity and 1 at t = 0.The sensitivity of a variance swap with respect to time, i.e. the timedecay or theta, θ , of a variance swap is∂ T2 (St )σ2θ==−(23.11)∂tTEquation 23.11 shows that the time decay of the variance swap staysconstant over time. Also, the theta of a variance swap is linearly dependent on the variance. This means that the cumulative theta at expirationis equal to the full variance. This is what one would expect since it showsthat, if the stock does not move during the term of the variance swap,the seller of this variance swap will collect the full variance strike leveltimes the variance notional, N .The rate of change of the delta of the variance swap with respect tothe stock price, i.e. the gamma of the variance swap, is equal to∂ 2 T2 (St )2γ ==(23.12)∂ 2 StT · St2Equation 23.12 shows that the gamma of a variance swap is constantthrough time. It also shows that gamma is not constant in stock price.However, gamma cash, which is defined as γ · St2 /100 (see equation 4.2),is constant in both stock price and time. The fact that gamma cash isconstant through time and in stock price means that the profit ona given percentage move for a long variance position is always thesame regardless of the stock price or when the move occurs. This isan important difference from a single option, where both the stockprice and the time that a move occurs matter significantly for theprofit on a move of a long option position. Indeed, if the stock priceis close to the strike price and the option is close to expiration, theprofit on a move is far greater than when the option has a longtime to maturity or when the stock price is not close to the strikeprice.
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One would expect that the profit formula, as derived in Chapter 3, alsoholds for variance swaps. This can be seen by substituting equation 23.11for theta and equation 23.12 for gamma into the following formula.1 2 2γ σ St + θ = 0.2
(23.13)
23.4 MYSTERY OF GAMMA WITHOUT DELTAIt is paradoxical that a variance swap does not have a delta where it doeshave gamma on it. The reason is that a variance swap does accumulatedelta cash in between the time intervals over which the volatility ismeasured, but resets to zero after each time interval. The delta cashwithin each time interval is given by gamma cash multiplied by thepercentage move. To understand this one has to go back to the rootof how realised volatility is measured and over which time intervals.These time intervals are agreed upon at inception of the variance swaptransaction and are per default defined as the close to close volatility,i.e. the one day volatility. The following example clarifies how the timeintervals over which the volatility is measured impact the relationshipbetween delta cash and gamma cash.Consider an investor who buys a 2 day variance swap on BMWwhere σ K = 32, i.e. an annualised volatility of 32 %. BMW happens tomove 1 % on both of these days. Assuming that a year has 256 days, theannualised realised volatility is therefore 16 and because the varianceswap is bought at 32, the investor’s break-even volatility is 2 % per day.Suppose that the gamma cash on the variance swap is 1 million euros.A gamma cash of 1 million and a break-even volatility of z = 2 % aday give a theta of 20 thousand euros a day. Indeed, since the absolutebreak-even volatility in equation 3.2 is equal to y = zSt , equations 3.2and 4.2 give thatγc · z 2γ · z 2 St2== 20 000.(23.14)2200Although the delta of the variance swap resets to zero at the end of eachbusiness day, one finds that the variance swap does have a delta duringthis business day. This means that at the start of each business day theprofit of a variance swap is independent of the direction of the move.However, once the stock moves during the business day, the profit ofthe variance swap starts to depend on the direction of the move thatfollows. For example, if the buyer of the BMW variance swap snapshotsθ=
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his position halfway through the first business day when BMW is down0.5 %, he finds himself having a short delta