cash position of 500 thousandeuros. In other words, when BMW is already down 0.5 % the holder ofthe variance swap benefits more if the stock goes down even more thanwhen it goes back up again. This is logical since the holder of the varianceswap wants to have as large a move as possible. The profit on the varianceswap when it goes down another 0.5 % to finish the first business daydown 1 % can now be calculated as either:1.12· γc = 5000200or2. gamma profit on the first 0.5 % move + profit on 500 thousand deltacash for the second 0.5 % move (delta profit) + gamma profit on thesecond 0.5 % move (equals the first 0.5 % move), mathematically0.520.52· γc + 0.5 × 5 · 105 +· γc = 5000200200Obviously, the profit on the move for the second business day is also5000 euros as the gamma cash is constant through time and stock price.In other words, the investor has made 10 000 euros on the two respectivemoves, but lost 40 000 euros in theta, i.e. an overall loss of 30 000 euros.
23.5 REALISED VARIANCE VOLATILITY VERSUSSTANDARD DEVIATIONThe realised volatility, σ R , as defined in the payoff of the variance swap,is not defined as a regular standard deviation. The volatility in the payoffof the variance swap is really defined as the square root of the lognormalreturns squared times the annualisation factor. Mathematically, 2Nd256Si2σR =·,(23.15)lnSi−1Ndi=1where,
r Si is the stock price at the close of business on day i;r 256/Nd is the annualisation factor;r Nd is defined as the number of days during the term of the varianceswap.
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The reason that the volatility in the variance payoff is defined according to equation 23.15 is to make sure that every move contributesto the variance payoff regardless of any trend the stock price mighthave. Indeed, the standard deviation reduces when a stock has a specifictrend. For example, a stock that goes up by 1 % for 10 days in a rowhas a standard deviation of 0. In other words, the standard deviation ofthe lognormal returns compensates for any trend the stock might haveby subtracting the average return from each return. Mathematically thestandard deviation, sd , of the lognormal returns is defined as  ⎤2⎡  Nd ln SiNdi=1Si−1⎣ln Si⎦ · 256 .sd =−(23.16)Si−1NdNdi=1
23.6 EVENT RISK OF A VARIANCE SWAP VERSUSA SINGLE OPTIONIt has already briefly been discussed that the event risk on a varianceswap is far greater than the event risk on a single option, except for singleoptions close to expiry and close to the strike price. The reason being thata variance swap has a constant gamma cash whereas the gamma cash ofa single option is concentrated around the strike price. This means thatif a large move occurs the profit impact is far greater on a variance swapthan for a single option. The single option first needs to be relativelyclose to the strike for the move to have any impact at all. Even then thesingle option loses its gamma quickly for very large moves as it movesaway from the strike price. Another very distinct feature of a varianceswap is that the profit impact for a specific move does not depend onwhen the move occurs during the term of the option. A 10 % move thatoccurs a year away from maturity has the same profit impact as a 10 %move that occurs 1 day before maturity. This can easily be seen fromthe profit formula in combination with the fact that the gamma cash ona variance swap is constant. The profit formula is stated below. In thisformula z is the percentage movement of the underlying stock price.γc · z 2200The fact that the profit on a move is independent of time for a varianceswap is in stark contrast to a single option, where the timing of the moveheavily affects the profitability of the move. This is for the simple reason
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that the gamma on a single option changes quickly over time. Assumingthat the option is at the money, a 10 % move that occurs a year awayfrom maturity is far less profitable than a move that occurs a day beforeexpiry. Indeed, the gamma of a 1 day option is a multiple of a 1 yearoption.
23.7 RELATION BETWEEN VEGA EXPOSURE ANDVARIANCE NOTIONALThe standard for trading variance swaps is to agree on the vega exposure of the variance swap rather than the variance notional. In order tocalculate the variance notional associated with the vega exposure of avariance swap, one has to derive the variance notional needed to havethe change in volatility times vega equal the change in variance timesthe variance notional, N . In other words, if the variance notional isN , the variance strike is σ K2 and the change in volatility is y, one wantsto solve the following equation(σ K + y)2 − σ K2 × N = vega × y × 100.(23.17)The factor 100 in the right hand side of equation 23.17 stems from thefact that volatility is expressed as a percentage. Since y is a small number,Taylor3 expansion on (σ K + y)2 reduces equation 23.17 to (square termsin y or higher are negligibly small)σ K2 · N + 2σ · y · N − σ K2 · N = vega · y · 100⇓vegaN=· 1002σ
(23.18)
23.8 SKEW DELTAAll the derivations and conclusions about variance swaps in the previoussub-sections are based on the assumption that the implied volatility isconstant in strike price. Chapter 5 shows that this is not a valid assumption as the effect of skew results in higher implied volatilities for lowerstrike prices. This skew effect means that σ K is higher than the ‘at the3
If f (y) is a function, Taylor expansion can rewrite the function f (y) asf (y) = f (0) + f  (0)y +
f  (0)y 2f  (0)y 3++·····2!3!
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money’ volatility. More importantly, it means that the variance swap hasa delta after all, because the buyer of a variance swap benefits from themarket going down as the market will likely become more volatile andthe ‘at the money’ implied volatility will go up. The delta on a varianceswap induced by the effect of skew is better known as the skew delta. Theskew delta of a variance swap can easily be quantified and is clarifiedby the following example.Consider a trader who sells a 1 year variance swap on BMW at σ K =20 and the vega he sells is 25 thousand euros. The skew is such thatevery 1 % down move in BMW’s share price translates to a 0.20 %increase in the ‘at the money’ implied volatility. Since the trader is short25 thousand euros of vega, each 1 % down move in the BMW pricetranslates to a 5 thousand euros loss on the variance swap (σ K goesup to 20.20 and hence a loss of 0.2 × 25 = 5 thousand euros). To hedgethis exposure to the dynamics of the variance swap as a result of skew,the trader has to sell 500 thousand euros delta cash of BMW, whichserves as his skew delta. This skew delta will exactly offset any losseson the BMW variance swap as a result of BMW going down by 1 %, ashe will make 5 thousand euros on the skew delta for a 1 % down move(500 ∗ 0.01 = 5).
23.9 VEGA CONVEXITYThe vega of a variance swap is not constant in implied volatility. In otherwords, whenever the implied volatility changes the vega exposure of thevariance swap changes. This can be seen by taking the derivative of ν inequation 23.9 with respect to σ . This results in:∂ 2 T2 (St )∂ν(T − t)==2·.(23.19)2∂σ∂ σTEquation 23.19 shows that, if the implied volatility goes up, the vegaexposure of a variance swap goes up as well.4 By the same token, if theimplied volatility goes down, the vega of a variance swap goes downas well. Because of skew the ‘at the money’ implied volatility goesup when the market goes down, which in turn translates into a highervega exposure. This means that the vega exposure goes up when the4
Vomma is positive.
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149
underlying goes down and vice versa.5 This principle is referred to asvega convexity. In other words, if the market goes down, the vega of avariance swap goes up and if the market goes up, the vega of a varianceswap goes down. This can also be seen from formula 23.18. On anygiven day, when the implied volatility goes up, the vega needs to go upfor the variance notional to remain constant.
5
Vanna is negative.
24DispersionDispersion literally means scattering or spreading widely. In trading, dispersion means that one trades the basket volatility against the volatilitiesof the constituents in that basket. The reason that being short volatilityon the basket and long the volatilities of the basket constituents is calleddispersion is because this trade makes money when the constituents areheavily inversely correlated, i.e. −1. This can be seen by considering abasket with two constituents A and B with equal weights. If both A andB are very volatile, in which case being long volatility on these namesmakes money, and A and B are perfectly inversely correlated, i.e. Amoves opposite to B, which means that, de facto, the basket does notmove and therefore being short volatility on the basket makes moneyas well. This section discusses trading dispersion in more detail andshows how the volatility of the basket is derived from its constituents.More importantly, this chapter shows that being long dispersion meansbeing short correlation plus long volatility rather than just being shortcorrelation.
24.1 PRICING BASKET OPTIONSBasket options are typically priced as a Monte Carlo process. However, to better understand the theory of dispersion it is imperative tobe able to use the Black–Scholes formula to price a basket option.Unfortunately a basket option is not lognormally distributed eventhough its constituents are. Therefore it is impossible to use the Black–Scholes formulaprice a basket option. Nonetheless, onen to waccuratelyican use lnS , which is lognormally distributed, to estimatei=1n i,tln.Thisapproximation enables one to use the Black–wSi=1 i i,tScholes formula to price a basket option. Although this method willnever provide the most accurate price for the basket option, it does givethe opportunity to derive the specific features associated with a basketoption and hence the features that are so pivotal in understanding dispersion trading.
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24.2 BASKET VOLATILITY DERIVED FROMITS CONSTITUENTSConsider a basket, Bt , with n constituents, Si,t , and each constituent hasa weight wi . Mathematically this basket looks likeBt =
n
wi Si,t ,
i=1
n
wi = 1.
(24.1)
i=1
If the ‘at the money’ implied volatility of each constituent Si,t is σi ,1statistics show that the ‘at the money’ implied volatility, σb , of the basketis determined by2 n2σb = Var lnwi Si,t(24.2) 
i=1n
wiSi,t
≈ Var ln
(24.3)
i=1
===
nn i=1 j=1nn 
wi w j Cov (ln (Si,t ), ln (S j,t ))wi w j σi j
i=1 j=1nn 
wi w j σi σ j ρi, ji=1 j=1nnn=wi2 σi2 +wi w j σi σ j ρi, j ,i=1i=1 j=1, j=i
(24.4)
where
r σi j is the covariance between ln(Si ) and ln(S j );r σi is the implied volatility of Si ;r ρi j is the correlation between ln(Si ) and ln(S j ).1This notation is purely for the purposes of simplicity. Unfortunately, it can be a bit deceiving as it seemsas if the implied volatility is independent of time, which it obviously is not. It would be better to assume that theimplied volatility equals σi (T − t), but for simplicity the (T − t) part is left out.2Equation 24.2 makes implicit use of Jensen’s inequality, which states that
n
ni=1

Si,t
wi
≤
n1wi Si,tn i=1
n
wi = 1.
i=1
 n nwi to estimate Var lnand therefore uses Var lni=1 Si,ti=1 wi Si,t .
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24.3 TRADING DISPERSIONIn practice it is very easy to trade dispersion. Consider a basket with 10equally weighted stocks. Suppose an investor wants to be long the volatilities of the constituents in respect to their weighting and short volatilityon the basket. The way the investor achieves this is by buying vega onthe stocks according to their weights in the basket and selling vega on thebasket itself. The investor can either buy vega through buying options orthrough buying variance swaps (see Chapter 23) on the stocks. In orderto be short vega on the basket he has to sell options on the basket or sellvariance swaps on the basket. This investor wants to be short one millioneuros of vega on the basket and therefore long € 100 thousand vega oneach of the individual stocks. Suppose that the weighted average impliedvolatility of the stocks is 25 and the implied volatility of the basket is 17.In this case the investor makes money if the spread between the weightedaverage implied volatility of the stocks and the implied volatility of thebasket widens.3
24.4 QUOTING DISPERSION IN TERMSOF CORRELATIONDispersion is typically quoted in terms of correlation. Equation 24.2shows that one can express the volatility of the index or basket in terms ofits constituent volatilities and the correlations between the stocks withinthe basket. In order to measure dispersion in terms of one correlationnumber, one assumes that ρi, j is constant in all i and j and is equal toρb . From equation 24.2 one can derive this correlation ρb asnσb2 − i=1wi2 σi2ρb = n n.(24.5)i=1j=1, j=i wi w j σi σ j
24.5 DISPERSION MEANS TRADING ACOMBINATION OF VOLATILITY AND CORRELATIONConsider an investor who is long dispersion. In other words, he is longvega on the stocks in the basket according to their weights and short3Sub-section 24.5 explains that a widening of the spread between the weighted average volatility and theimplied volatility of the index is not always a recipe for making money on a long dispersion position. Namely,when the overall volatility goes down, one can even lose money on a widening of the spread. To make surethat a long dispersion position benefits from a widening spread, dispersion is often traded as a ratio wherethe vega position on the index is larger than the cumulative vega position on the stocks. This is discussed insub-section 24.6.
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vega on the basket itself. Mathematically this means that this investorhas the following position:n
wi σi − σb .
(24.6)
i=1
Equation 24.2 shows that this position is equivalent ton
wi σi − σb =
i=1
n
nn wi σi − wi w j σi σ j ρi, j ≥ 0 (24.7)
i=1
i=1 j=1
Equation 24.7 shows that being long dispersion means being long thevolatilities of the individual stocks. Indeed, this is clear since −1 ≤ρi, j ≤ 1 and thereforen
wi σi = 
i=1
nn 
wi w j σi σ j
i=1 j=1
≥
nn 
wi w j σi σ j ρi, j .
i=1 j=1
In other words, everything being equal (correlations and weights remainunchanged), if the volatilities of the individual stocks go up, a long dispersion position makes money. Knowing that a long dispersion positionis like being long the individual stock volatilities, equation 24.7 can nowbe rewritten in terms of ρb , the constant correlation in which dispersionis typically measured, to see what the sensitivity is to correlation for along dispersion position.ni=1
wi σi − σb =
ni=1
nnnwi σi − wi2 σi2 +wi w j σi σ j ρbi=1
i=1 j=1, j=i
(24.8)Equation 24.8 shows that a long dispersion position is short correlation.In other words, if the correlation goes down, a long dispersion positionmakes money.In summary, a long dispersion position is first and foremost a shortcorrelation position. This means that, the more inversely correlated theindividual constituents are, the more more profitable is a long dispersion position. However, it is important to recognise that a long dispersion
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position also has a long exposure to the individual stock volatilities. Tomake sure that a dispersion position is solely a short correlation position, traders typically do not trade the cumulative vega of the stocks inequal size to the vega of the index or basket. The following sub-sectionshows how to ratio the vega of each leg of the dispersion to only havean exposure to correlation.
24.6 RATIO’D VEGA DISPERSIONAs stated in the previous sub-section, dispersion is not a real short correlation trade. Indeed, a decrease in correlation can be offset by a decreasein volatility of the individual stocks. For this reason traders typicallyratio the size in which they trade the vega on the index compared tothe vega on the individual stocks. Determining the required ratio to establish a sole correlation position and therefore take away the residuallong volatility position is rather simple and intuitive. This ratio is indeednothing more than the ratio of the weighted average stock volatility leveland the index volatility level. Consider the example in sub-section 24.3where the weighted average stock volatility is 25 and the index volatilityis 17. To effectively establish a sole short correlation position one has totrade the dispersion in a ratio of 1: 25, meaning that one has to be short1725/17 times the amount of vega on the index compared to the cumulative single stock position. In other words, if the cumulative single stockposition is € 1 million vega, a short position of € 1.47 million vega inthe index effectively gives a pure short correlation position.When trading a dispersion through variance swaps one has to be careful to put the right ratio in place. Since a dispersion through varianceswaps is equivalent to the positionn
wi σi2 − σb2 ,
(24.9)
i=1
the ratio should first of all be expressed in variance notionals rather thanvega. The ratio in which the variance notionals should be traded is thenderived by the square of the ratio between the weighted average singlestock volatility and the index volatility. This means that if the weightedaverage single stock volatility is 25, the index volatility is 17 and thecumulative single stock variance notional is € 200 million, the variancenotional on the index should be 200 · (25/17)2 = € 432.53 million.
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24.7 SKEW DELTA POSITION EMBEDDEDIN DISPERSIONAny dispersion position has an embedded skew delta position. This stemsfrom the fact that the skew on an index is almost always steeper thanthe skew of any of the individual stocks. In turn, the reason for this isthat an individual stock might be volatile on the upside as well as on thedownside, whereas the market as a whole becomes more volatile whenthe market goes down and less volatile when the market goes up. Thefact that an index has a steeper skew than each of the individual stocksresults in the fact that a long ratio’d dispersion position makes money asthe market goes up and loses money if the market goes down, all othervariables being equal. Indeed, whenever the market goes up, the shortindex implied volatility within the long ratio’d dispersion position goesdown more than the implied volatility of the individual stocks. This isdue to a steeper index skew than the skew of the individual stocks. Whenthe market goes down, the steeper index skew results in the reverse, i.e.the short index position loses more than the long single stock volatilitiesmake. In summary, a long dispersion is effectively long delta as a resultof a steeper skew on the index or basket than on the individual stockconstituents. This means that a long dispersion position has an embeddedlong skew delta position, which can be hedged by selling some delta on,for example, the index.
25Engineering Financial StructuresThis chapter discusses different retail investors and how to financiallyengineer structures that suit each investor’s needs. The structures discussed in this chapter are typically not complicated or even exotic, but,since financially engineered retail products are such an important partof investor demand, this chapter shows how notes can be structuredto achieve the financial needs of different investor profiles. There arebasically three different types of retail investors: the risk averse retailinvestor who wants to invest his money in capital protected products;the retail investor who wants some exposure to the equity markets, butstill wants to have a balanced and well thought out risk profile; the retailinvestor who uses financial structures to leverage his views.
25.1 CAPITAL GUARANTEED PRODUCTSA capital guaranteed product is a suitable product for the risk averseinvestor who does want to have some exposure to the equity markets. Theprinciple of the capital guaranteed product is that the investor buys a zerocoupon bond and, since the present value of a zero coupon bond is lessthan 100 %, the difference between the present value of the zero couponbond and 100 % is invested in buying an option that gives exposure tothe equity markets. In other words, with a capital guaranteed product,the investor gives up a coupon to subsequently invest it in either a call ora put option. For example, a risk averse investor who is bullish on BMWcould buy a capital guaranteed product where, instead of receiving acoupon on the bond, he invests the coupon to buy participation in a calloption. Obviously, if the investor were bearish on BMW, he would giveup the coupon to buy participation in a put option.Consider a risk averse investor who is bearish on BMW for the coming year and therefore wants to buy a capital guaranteed structure thatincreases in value if BMW goes down. Suppose that a one year zerocoupon bond is worth 95.5 %, and a one year ‘at the money’ put onBMW is worth 5.625 %. Therefore theinvestor buys a zero coupon
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140 %120 %
payoff at maturity
100 %
BMW share price at maturity50 %
70 %
100 %
120 %
140 %
Figure 25.1 Payoff at maturity of a bearish capital guaranteed note, where the investorprofits in 80 % of BMW’s decline
bond and 80 % ((100 − 95.5)/5.625) participation in an ‘at the money’put. The investor has € 10 million to invest and BMW’s share price istrading at € 40. This means that the investor buys € 10 million worthof zero coupon bonds and € 8 million worth of BMW ‘at the money’puts, which means that the capital guaranteed note has 200 thousand‘at the money’ puts embedded in it. Since the investor only has 80 %participation in the ‘at the money’ put, the investor profits in only 80 %of BMW’s decline. This means that, if after one year BMW has gonedown by 50 %, the investor only makes 40 % on the invested notionaland therefore gets paid € 14 million after one year, i.e. € 10 millionfrom the zero coupon bond and € 4 million from the embedded put. Thepayoff profile after 1 year looks like Figure 25.1.
25.2 ATTRACTIVE MARKET CONDITIONS FORCAPITAL GUARANTEED PRODUCTSThe market conditions under which capital guaranteed products lookmost attractive are when the option participation and therefore the exposure to the equity markets are largest. This can be achieved in two ways.Either the present value of the zero coupon bond is low, in which case
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there is more premium available to invest in the option and therefore theparticipation in the equity markets is larger. Or the option premium islow, which also increases the participation in the equity markets. Obviously, the present value of the zero coupon bond is lower when interestrates are higher. Option premiums are lower when first of all the impliedvolatility is lower. For a bearish capital guaranteed product, the optionpremium is also lower when the dividend yield of the underlying is lower,since a bearish capital guaranteed product has a put embedded in it. Abullish capital guaranteed product comprises a zero coupon bond plus aparticipation in a call option and hence the participation will be largerfor a higher dividend yield. In summary, these are the market conditionsthat make a bullish capital guaranteed product look most attractive, i.e.the call option participation is highest:
r The higher the interest rate the lower the present value of the zerocoupon bond and therefore the more premium available to invest inthe call option participation embedded in the bullish capital guaranteedproduct.r The lower the implied underlying volatility the lower any option premium and thus the lower the call option premium, which results in ahigher participation in a call option for the bullish capital guaranteedproduct.r The higher the dividend yield the lower the call option premium andtherefore the higher the participation in the call option embedded inthe bullish capital guaranteed product.For a bearish capital guaranteed product the most attractive market conditions can be summarised as follows:
r The higher the interest rate the lower the present value of the zerocoupon bond and therefore the more premium available to invest inthe put option participation embedded in the bullish capital guaranteedproduct.r The lower the implied underlying volatility the lower any option premium and thus the lower the put option premium, which results in ahigher participation in a put option for the bearish capital guaranteedproduct.r The lower the dividend yield the lower the put option premium andtherefore the higher the participation in the put option embedded inthe bearish capital guaranteed product.
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25.3 EXPOSURE PRODUCTS FOR THE CAUTIOUSEQUITY INVESTORThe reverse convertible, as discussed in Chapter 15, is typically the typeof investment that a cautious equity investor who has some appetite forrisk would buy. Since the reverse convertible is already discussed inChapter 15, this sub-section takes the opportunity to show yet anotheriteration of a financial note that the cautious equity investor with somerisk appetite would invest in. This product is sometimes referred to asan airbag and, like the reverse convertible, this note gives exposure tothe equity markets through a down-and-in put or regular put. As anadditional feature, the coupon on this note is conditional on a certainlower bound of the stock price and therefore pays a higher coupon thanthe reverse convertible. In effect, the note is slightly leveraged on thedownside as the conditional coupon means that the investor not onlyloses money through the put or down-and-in put on the downside, butcan also lose his coupon if the underlying devalues too much. Thisconditional coupon is often priced as a down-and-out put. The pricingis shown by the following example.Consider an investor who wants to buy a 1 year reverse convertible onBMW with a conditional coupon. Suppose that the one year 100/80 %knock-in reverse convertible pays a bullet coupon at maturity of 7.5 %.The investor is prepared to lose his coupon if, at any time during the lifeof the note, the stock drops below 85 % of BMW’s initial share price.Because the coupon is conditional, the coupon on this note is higher than7.5 %. Unlike the reverse convertible, where one solves for the couponthat makes the fair value or re-offer 100 %, the reverse convertible withconditional coupon does not solve for the coupon directly but solvesfor it indirectly through the redemption of the zero coupon bond andthe rebate of a zero1 strike down-and-out put with rebate. This rebateis equal to the excess redemption over 100 % on the zero coupon bond.In other words, the excess redemption over 100 % and the rebate of thedown-and-out put are used as variables to make the reverse convertiblewith conditional coupon worth 100 %. The conditional coupon appearsto be 10 %, which means that the following structure is worth 100 %.
r The price of a 1 year zero coupon bond with a redemption of 110 %(trader sells) instead of 100 %. Minus1The fact that it is a zero strike down-and-out put with rebate means that it only gives a payout when itknocks out and therefore pays the rebate.
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r The price of a 1 year 100/80 % down-and-in put (trader buys),which the trader prices as a 1 year 100/77 % down-and-in put becauseof the delta change over the barrier. Like the reverse convertible, thedown-and-in put is used to enhance the coupon. Minusr The price of a 0/85 % down-and-out put with a 10 % rebate (traderbuys), which he prices as a 0/83 % down-and-out put because of thedelta change over the barrier (the trader makes money because ofthe rebate as BMW’s stock price goes down and through the barrier).Pricing the barrier shifted down-and-out put decreases the price of thedown-and-out put with rebate and therefore the trader buys it at lowerpremium. The rebate has a delayed payment and is paid at maturity ofthe full structure. The rebate cancels out the excess redemption over100 %, in this case 10 %, from the zero coupon bond and makes surethat the investor only receives 100 % at maturity if at any point the 85 %barrier is breached. It is important to understand that the rebate and theexcess redemption over 100 % from the zero coupon bond are the sameand therefore in solving the conditional