IN THE UNITED STATES THIS REPORT IS AVAILABLE ONLY TO PERSONS WHO HAVE RECEIVED THE PROPER OPTION RISK DISCLOSURE DOCUMENTS.

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1 European Equty Dervatves Strategy 4 May 005 N THE UNTED STATES THS REPORT S AVALABLE ONLY TO PERSONS WHO HAVE RECEVED THE PROPER OPTON RS DSCLOSURE DOCUMENTS. Correlaton Vehcles Technques for tradng equty correlaton Correlaton reflects the dfference between ndex and sngle stock volatltes Short correlaton trades perform well, explotng the relatve rchness of ndex volatlty Overvew n ths note we compare alternatve vehcles for tradng equty correlaton, how well they have performed, and what drves ther p/l. n general, short correlaton trades often n the form of dsperson trades have performed well, wth return/rsk ratos above. Such trades are usually constructed by sellng ndex varance swaps and buyng sngle stock varance swaps. They seek to take advantage of the relatve rchness of ndex volatlty wthout beng outrght short volatlty. However, dependng on trade constructon, a short correlaton trade can actually be an effcent way to own volatlty. Fgure 1: Euro Stoxx 50 3-month mpled and realsed correlaton... mpled tends to trade above realsed correlaton mpled correlaton realsed correlaton N ov -00 Nov -01 N ov -0 Nov -03 N ov -04 Equty Dervatves Strategy Stephen Enchcomb Ncolas Granger Smon Mguez nvestor Dervatves Marketng Andrea Morres Flow Dervatves France/Belgum Germany/Austra taly Span/Portugal Swtzerland Netherlands & Scandnava U Mddle East, E. Europe South Afrca North Amerca Lsted Dervatves Corporate Dervatves Marketng Antono Polverno

2 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Table of contents Overvew... 1 ntroducton... 3 Vehcles for tradng correlaton - summary... 4 What s Correlaton?... 5 Dfference between correlaton measures... 5 mpled Correlaton... 6 The Correlaton Proxy... 7 Example: the correlaton proxy... 7 Correlaton versus Volatlty... 9 Tradng Correlaton Tradng Correlaton - Correlaton Swaps Example: correlaton swaps Varance Swaps... 1 Example: varance swaps... 1 Tradng Correlaton - Dsperson trades Tradng correlaton - Vanlla dsperson trades Example: a vanlla dsperson trade s long volatlty Tradng correlaton - Correlaton-weghted dsperson trades Example: a correlaton-weghted dsperson trade s ntally vega neutral Drvers of p/l for a correlaton-weghted dsperson trade Correlaton s the maor drver for correlaton-weghted dsperson trade p/l The level of exposure to correlaton s scaled accordng to prevalng volatlty condtons Volatlty Dsperson: an addtonal drver of correlaton-weghted dsperson trade p/l... 1 Example: effect of convexty on dsperson p/l... 3 Alternatve weghtng schema and dynamc replcaton... 5 Rsks and returns of tradng correlaton... 7 Lookng forward... 8 What are the rsks of tradng correlaton?... 9 Correlaton across sectors and ndces Correlaton across ndces nter-sector and ntra-sector Correlaton Correlaton between ndces... 31

3 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 ntroducton Equty correlaton measures how much stock prces tend to move together. t provdes the lnk between the volatlty of an ndex and the volatltes of component stocks; descrbed approxmately by the formula: ndex Vola tlty correlaton Average Sngle Stock Volatlty Ths formula shows that correlaton s an mportant factor n ndex volatlty: for example, backtestng shows that changes n ndex volatlty are at least 5 drven by changes n correlaton. By examnng the relatve levels of mpled ndex and mpled sngle-stock volatltes, t s possble to back out the level of forward-lookng correlaton beng prced n by the market. As wth mpled ndex volatlty, ths mpled correlaton tends to trade at a premum to that actually delvered (Fgure ). Lke volatlty, correlaton can tself be treated as an asset class and traded n ts own rght. Sellng correlaton has hstorcally been proftable, captalsng on the relatve rchness of ndex volatlty. The spread between ndex mpled and realsed volatlty has been generally postve and has consstently exceeded the spread between average mpled and realsed sngle-stock volatlty. To take advantage of both of these opportuntes we can sell ndex volatlty to proft from ts rchness, whlst buyng sngle-stock volatlty n order to hedge out some or all of the short volatlty exposure. Why trade correlaton? A short correlaton poston can be a good source of alpha, snce mpled correlaton usually trades at a premum to that delvered Tradng correlaton allows dversfcaton of a portfolo returns from correlaton are somewhat antcorrelated wth ndex returns Correlaton allows a vew to be taken on the relatve moves of ndex volatlty n comparson to snglestock volatlty Long correlaton (mpled or realsed) can be a good hedge aganst market rsk Fgure : Euro Stoxx 50 3-month mpled and realsed correlaton... mpled tends to trade above realsed correlaton 1.00 mpled correlaton 0.90 realsed correlaton Nov-00 Nov-01 Nov-0 Nov-03 Nov-04 Table 1: Returns and volatlty of strateges (n vegas) and rsk-return ratos. Long vanlla dsperson Long correlatonweghted dsperson Short ndex varance (for comparson) 3m 6m 1y Return (annualsed) 8.6% 4.8% 3.1% Volatlty (of return) 4.5%. 1.3% Rsk-Return Return (annualsed) 8.8%.9% 1.4% Volatlty (of return) 3.4% 1.3% 0.5% Rsk-Return.6..6 Return (annualsed) 3.8% -0.1% -.% Volatlty (of return) 9.9% 5.4%.7% Rsk-Return Results asuume 3% bd-offer on dsperson trades and 1.5% bd-offer on varance swaps. Data August 000 to date Returns from short correlaton trades have been largely postve over the last few years (Table 1). For example, sellng 6-month correlaton va correlaton-weghted dsperson has generated an average annual return of.9 vegas, wth a rsk-return rato of.. However, over the last two years, absolute levels of returns from dsperson trades have been markedly reduced, sufferng from the reducton n levels of volatlty. Despte ths, rsk-return ratos of correlaton-weghted trades have managed to reman roughly stable due to a correspondng reducton n the volatlty of the strategy. See pages 7-9 for detals. 3

4 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Vehcles for tradng correlaton - summary Ths secton summarses the prncpal vehcles for tradng correlaton and outlnes ther characterstcs. Ths note contnues by dscussng the fundamentals behnd correlaton (pages 5 9). We nvestgate correlaton tradng vehcles n detal and dssect ther p/l on pages Returns and rsks of tradng correlaton are revewed on pages 7 9. There are two prncpal vehcles for tradng correlaton, namely correlaton swaps and varance dsperson trades. Correlaton swaps gves drect exposure to correlaton, payng out on the dfference between the strke of the swap, and the subsequent realsed average parwse correlaton of a pre-agreed basket of stocks. Although correlaton swaps are the easest and most drect method of tradng correlaton, they have been less lqud than dsperson trades and harder to mark to market. Furthermore, due to the drect nature of ther correlaton exposure and the fact that mpled correlaton tends to trade at a premum to that delvered, the level of correlaton whch can be sold through a correlaton swap has generally been lower than the level of mpled correlaton backed out from ndex and sngle-stock volatltes. The most common vehcle for tradng correlaton has hstorcally been the dsperson trade. Ths trade conssts of takng opposng postons n the volatlty of an ndex and the volatlty of ts consttuents. Typcally a dsperson trade would be effected through varance swaps known as a varance dsperson trade, where a long poston would be constructed by sellng varance on an ndex, and buyng varance on ts consttuents. A long dsperson poston wll be short correlaton but long the dsperson of volatltes of the ndex consttuents. The relatve weghtngs of the consttuent varance swaps are of crucal mportance n a dsperson trade. n ths note we dscuss two dfferent weghtng schemas; a vanlla dsperson trade (the smplest) whch has both correlaton and outrght volatlty exposures, and a correlaton-weghted dsperson trade, whose p/l more closely reflects changes n correlaton. t turns out that the vanlla dsperson trade has n fact been an effcent way to own volatlty, n effect usng the alpha avalable from short correlaton to subsdse the cost of beng long volatlty. The short correlaton exposure to some extent hedges the long volatlty poston but also allows the trade to proft from a decrease n correlaton relatve to volatlty. On the other hand, the correlaton-weghted dsperson trade has profted more drectly from moves n correlaton, although there are other mportant drvers of p/l. n partcular, though the trade s ntally vega neutral, the p/l arsng from correlaton wll be magnfed or attenuated by prevalng levels of volatlty, causng the poston to develop a vega exposure as correlaton moves. Fgure 3: vanlla dsperson payoff s closely related to long varance returns vega payoff 15% 1 5% -5% vega payoff long vanlla dsperson pay off (rhs) 4 long v arance returns (lhs) Aug-00 Aug-01 Aug-0 Aug-03 Aug Fgure 4: whereas the correlaton-weghted dsperson payoff s more related to correlaton vega payoff 1% 8% 4% -4% correlaton ponts long 6m correlaton-w eghted dsperson trade pay off m mpled mnus subsequent 0.40 realsed correlaton 0.30 Aug-00 Aug-01 Aug-0 Aug-03 Aug

5 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 What s Correlaton? The realsed correlaton of a par of stocks measures how much the stock prces tend to move together. The realsed correlaton of an ndex s smply an average across all possble pars of consttuent stocks. There are two ways of computng ths average, the results of whch drve the payoffs of the two prncpal vehcles for tradng correlaton. Average parwse correlaton: Ths s a smple equally weghted average of the correlatons between all pars of dstnct stocks n an ndex. t s the payoff of a correlaton swap. ndex correlaton: Ths s a measure of correlaton derved from the volatlty of an ndex and ts consttuent sngle stocks. Realsed ndex correlaton turns out to be equal to the weghted sum of consttuent parwse stock correlatons, where the correlatons are weghted by both the stock weghts and ther respectve volatltes (see Box 1, page 8 for detals). Ths measure of correlaton s the prncpal drver for the p/l of a varance dsperson trade. n practce these two correlaton measures are very smlar. The dfference between the two represents the spread of volatltes and correlatons across the ndex. f ether all volatltes are the same, or f all parwse correlatons are the same, the dfference wll be zero. Dfference between correlaton measures Whatever the maturty, ndex realsed correlaton wll be greater than average parwse correlaton f the more volatle (and hgher weghted) stocks n the ndex are more hghly correlated. Through backtestng, we have found that for short-dated correlaton, the two measures are almost dentcal, wth the realsed ndex correlaton slghtly greater than average parwse correlaton (Fgure 5). Ths s supported by the fact that the average 3- month parwse correlaton has tended to be somewhat hgher for the hgher volatlty stocks (Fgure 6). Fgure 5: Euro Stoxx 50 3-month realsed correlaton has been slghtly above average parwse correlaton correlaton Fgure 6: because the hgh volatlty stocks n the ndex have tended to be more correlated. correlaton av erage correlaton of hgh-v ol stocks av erage correlaton of low v ol stocks av erage parw se correlaton hstorc ndex correlaton 0.00 Jan 01 Jan 0 Jan 03 Jan 04 Jan Jan 01 Jan 0 Jan 03 Jan 04 Jan 05 For 6-month and especally 1-year correlaton, averaged parwse correlaton has been slghtly greater than ndex realsed correlaton. However, the ndex realsed correlaton of a hypothetcal equally weghted ndex s always very close to average parwse correlaton, suggestng, at least for longer-dated correlaton, that t s the stock weghtngs rather than the volatlty weghtngs that make the dfference. 5

6 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Fgure 7: 6-month Euro-Stoxx 50 delvered correlaton correlaton Aug-00 Aug-01 Aug-0 Aug-03 Aug-04 6m av erage parw se correlaton 6m ndex realsed correlaton 6m equally -w eghted ndex realsed correlaton Fgure 8: 1-year Euro Stoxx 50 delvered correlaton correlaton Aug-00 Aug-01 Aug-0 Aug-03 1y av erage parw se correlaton 1y ndex realsed correlaton 1y equally -w eghted ndex reasled correlaton mpled Correlaton The advantage of usng the ndex realsed correlaton measure s that t s possble to use the same method to back out an mpled correlaton measure from the mpled ndex and sngle stock volatltes. Ths mpled correlaton measure represents the market s expectatons of future realsed ndex correlaton nherent n the mpled volatlty. Note that we can only compute ths mpled correlaton f the ndex upon whch t s based has actvely traded volatlty/optons. By usng the correlaton proxy examned n the followng secton, we can approxmate mpled correlaton by squarng the rato of mpled ndex to average sngle-stock volatlty. Ths approxmaton makes t easy to estmate the mpled correlaton of an ndex from mpled volatltes avalable n the market. mpled correlaton tends to trade rch compared to realsed correlaton. Ths s a manfestaton of the relatve rchness of ndex to sngle-stock volatlty: the spread of ndex mpled over realsed volatlty has generally exceeded the spread of sngle-stock mpled over realsed volatlty (Fgure 16). Ths rchness s n part drven by structural forces of supply and demand wthn the equty dervatves market. For example, nvestors purchasng ndex puts to protect ther downsde creates demand for ndex volatlty, whereas the sellng of covered calls to earn alpha ncreases the supply of sngle-stock volatlty. There s another knd of mpled correlaton whch s the strke of a correlaton swap. Snce the payoff of such a swap s equal to the average parwse correlaton and ths s usually very close to the realsed ndex correlaton, the strke of a correlaton swap should n theory trade close to the mpled correlaton descrbed above. However, unlke ndex mpled correlaton, correlaton swap mpled correlaton cannot be nferred from other market data and s avalable only n the form of a quoted strke for a correlaton swap. Such quotes can dffer substantally from ndex mpled correlaton levels as a result of market appette and dffcultes arsng from hedgng and replcaton. From now on, unless stated otherwse, by mpled correlaton we wll mean the correlaton backed out from mpled ndex and sngle-stock volatlty and not the strke of a correlaton swap. 6

7 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 The Correlaton Proxy Correlaton s correctly calculated by the formula ω = (see Box 1 for detals). H ω ( ω ) However, f the correlaton s not too close to zero (n practce > 0.0) and the number of members of the ndex s large enough (more than about 0, but really what s mportant s that all the weghts are relatvely small), then the formula = wll be a good proxy for the correlaton (Fgure 10). (See 'Fundamental H ( ω ) relatonshp between an ndex's volatlty and the correlaton and average volatlty of ts components', Sebasten Bossu and Y Gu, 005.) That s, the correlaton s approxmately equal to square of the rato of ndex volatlty to average sngle-stock volatlty, allowng ndex volatlty to be expressed n terms of correlaton and average stock volatltes as follows: ndex Vola tlty correlaton Average Sngle Stock Volatlty. Example: the correlaton proxy n the Euro Stoxx 50, on 1 May 005, average 3-month realsed sngle-stock volatlty was 16.3%, wth 3- month realsed correlaton at Multplyng 16.3% by the square-root of correlaton gves an estmate for the realsed ndex volatlty of 10.7%, close to the true value of 10.8%. n practce the approxmaton gven by the correlaton proxy s wdely used. t s accurate except n tmes of very low correlaton or for small ndces. For example, f stock correlatons were zero, current ndex volatlty would be.5% not the ndcated by the proxy. Note, the correlaton proxy always overestmates correlaton. Ths correlaton proxy can be calculated n exactly the same way for mpled correlaton as for hstorc correlaton, demonstratng that mpled correlaton s approxmately equal to the square of the rato of mpled ndex volatlty to average mpled sngle-stock volatlty. The formulas for correlaton and the correlaton proxy can be used n reverse to predct a value for ndex volatlty gven values for correlaton and average sngle-stock volatlty (as n the above example). The relatonshp between correlaton and volatlty descrbed by ether the formula or the proxy s non-lnear: when correlaton s low, an ncrease n correlaton wll cause a greater ncrease n volatlty than when correlaton s hgh (Fgure 9). Fgure 9: Volatlty as a functon of correlaton: Euro Stoxx 50 weghts and volatltes as of 01/04/005 volatlty 16% 1% 8% 4% v olatlty as a functon of correlaton v olatlty predcted from correlaton prox y correlaton Fgure 10: 6-month Euro Stoxx 50 realsed correlaton and proxy correlaton Aug-00 Aug-01 Aug-0 Aug-03 Aug-04 realsed correlaton realsed correlaton prox y 7

8 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Box 1: Calculatng correlaton ndex varance s descrbed by the equaton: = ω + < respectvely the weght and volatlty of the th stock n the ndex and th stocks. Assumng an ndex of N stocks, average parwse correlaton s gven by ω ω, where ω and are s the parwse correlaton of the th and A = N( N 1) <. ndex realsed correlaton s computed by solvng the ndex varance equaton for a sngle value of : gvng ω H = < ωω. < Usng the dentty ( ω ) = ωω = ω + ω ω we can wrte: = ( ω ) ω ω H, or alternatvely H < < ω ω ω ω =. The mpled correlaton s calculated n exactly the same way as H, replacng realsed volatltes and by the correspondng mpled volatltes throughout the equaton. Note the nterpretatons of ( ) ω and ω. The frst s the square of the average (weghted) snglestock volatlty. The second s what the varance of the ndex would be wth zero correlaton. So we have two nterpretatons of the ndex realsed correlaton, descrbed by the two forms above. The frst shows that realsed correlaton s the rato of ndex varance mnus uncorrelated varance to average sngle stock volatlty squared mnus uncorrelated varance. The second form shows that realsed correlaton s the weghted sum of parwse correlaton, where the correlatons are weghted by both the stock weghts and ther volatltes. The dfference between H and A can be gven explctly by consderng the second form of ( ω ω ω ω ) H A = where ω ω s the average of the product of all dstnct ω ω pars of weghts and volatltes. H. 8

9 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Correlaton versus Volatlty Correlaton s closely related to volatlty (Fgure 11 and Fgure 1). t s not surprsng that t s related to ndex volatlty as correlaton s a component of ndex volatlty wth the relatonshp approxmately descrbed by: ndex Vola tlty correlaton Average Sngle Stock Volatlty. However, t s also true that correlaton and average sngle-stock volatlty are correlated (Fgure 13). Ths s not entrely surprsng ether snce correlaton and volatlty, both mpled and hstorc, are drven by smlar factors. Correlaton measures how much stocks tend to move together and ths s more lkely n tmes of hgh volatlty for example durng a sell-off prompted by some knd of negatve surprse. Note that whlst correlaton has fallen snce 00/003 and especally n recent months, both mpled and realsed correlaton are stll lookng hgh relatve to ther hstory when compared to volatlty. Ths could be because correlaton s based to fall further or perhaps because the underlyng envronment has changed and we should expect correlaton to be hgher, even n a low-volatlty world (see page 30). Fgure 11: Euro Stoxx 50 6-month mpled correlaton and volatlty volatlty/correlaton 10 mpled correlaton mpled v olatlty Aug-00 Aug-01 Aug-0 Aug-03 Aug-04 Fgure 13: Correlaton of 6m Euro Stoxx 50 realsed correlaton and 6m realsed average sngle stock volatlty has been strong snce the begnnng of 00 realsed correlaton Fgure 1: Euro Stoxx 50 6-month realsed correlaton and volatlty volatlty/correlaton realsed correlaton realsed v olatlty Aug-00 Aug-01 Aug-0 Aug-03 Aug-04 Fgure 14: On September , mpled correlaton reached 1.00 as ndex mpled volatlty overtook average sngle-stock volatlty volatlty 55% 5 ndex v olatly av g sngle-stock v olatly correlaton (rh ax s) correlaton Jan 00 - Apr 005 Aug Dec realsed sngle-stock volatlty 45% 4 35% 3 5% 01-Sep 08-Sep 15-Sep -Sep 9-Sep mpled correlaton s especally senstve to changes n volatlty. Ths s probably due to the greater lqudty n tradng ndex volatlty compared to sngle-stock volatlty. n September 001, excess demand for putprotecton at the ndex level combned wth a lagged remarkng of sngle stock volatlty drove up mpled correlaton above ts theoretcal maxmum of mpled correlaton then fell as sngle-stock volatltes caught back up wth ndex volatlty (see Fgure 14). 9

10 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Tradng Correlaton Sellng correlaton has hstorcally been proftable, captalsng on the relatve rchness of ndex volatlty. The spread between ndex mpled and realsed volatlty has been generally postve (Fgure 15) and has consstently exceeded the spread between average mpled and realsed sngle-stock volatlty. To take advantage of both of these opportuntes we can sell ndex volatlty to proft from ts rchness, whlst buyng sngle-stock volatlty n order to hedge out some or all of the short volatlty exposure. Fgure 15: mpled ndex volatlty has generally traded above hstorc volatlty except durng perods of hgh volatlty and volatlty 45% 3 15% -15% Dfference -3 Euro Stox x 50 3m mpled v olatlty Euro Stox x 50 3m realsed v olatlty Aug-00 Aug-01 Aug-0 Aug-03 Aug-04 Fgure 16: the spread between mpled and realsed ndex volatlty has exceeded that between mpled and realsed sngle stock volatlty volatlty ndex v olatlty : mpled/realsed spread av erage sngle-stock v olatlty : mpled/realsed spread Aug-00 Aug-01 Aug-0 Aug-03 Aug-04 There are two prncpal vehcles for tradng correlaton, namely correlaton swaps and varance dsperson trades. Correlaton swaps are the easest and most drect method, but they are less lqud and harder to mark to market. The market for varance dsperson trades has been much larger than that for correlaton swaps, although the correlaton swap market s growng as the result of exotcs desks tradng correlaton n ths more drect form to hedge ther correlaton exposures. Due to hedgng dffcultes and the drect nature of the correlaton exposure, correlaton swap strkes have tended to trade around the levels of correlaton realsed by the relevant basket of underlyng stocks. For ths reason, and because mpled correlaton generally exceeds that delvered, the level of correlaton that can be sold va a correlaton swap has tended to be lower than that sold va a varance dsperson trade Below we dscuss the mechancs of correlaton swaps and varance dsperson. We also dssect the p/l of a varance dsperson trade to dentfy the factors other than correlaton whch drve p/l. Tradng Correlaton - Correlaton Swaps Correlaton can be traded drectly va a correlaton swap. A correlaton swap gves exposure to the average parwse correlaton of a pre-determned basket of stocks. The swap strke s the level of (average parwse) correlaton that s bought or sold, and s typcally scaled by a factor of 100 (.e. a correlaton of 0.55 s quoted as a strke of 55). The payout of a correlaton swap s the notonal amount multpled by the dfference between the swap strke and the subsequent realsed average parwse correlaton on the basket of underlyngs. payout = notonal (realsed average parwse correlaton strke) 10

11 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Correlaton swap cashflows Buyer pays correlaton swap strke Buyer of Correlaton Swap Seller of Correlaton Swap Seller pays average parwse realsed correlaton Example: correlaton swaps Suppose that on October 1st 004 we sell a 6-month correlaton swap on an equally weghted basket of stocks consstng of the members of the Euro Stoxx 50 ndex. The strke of the correlaton swap s 55, and the notonal amount s 10,000 After sx months, on Aprl 1st 005, we calculate the realsed 6-month average parwse correlaton of the stocks n our basket as 0.4. Then the p/l s calculated as: P/L = notonal x ( strke realsed parwse correlaton ) = 10,000 x (55 4) = 130,000 The advantages of a correlaton swap over other forms of correlaton trade are: The correlaton swap gves drect exposure to the level of delvered (average parwse) correlaton wth no dynamc hedgng/replcaton requred. A correlaton swap can be set up on any basket of underlyngs, not necessarly a traded ndex. The buyer and seller smply agree on the basket, the notonal and the level of correlaton traded (the strke) at the ncepton of the trade. Bd-offer spreads are much lower on correlaton swaps than on dsperson trades. A spread of around 3 correlaton ponts would be typcal for a correlaton swap, equatng to between about 0.5 and 1 vega. n contrast the bd-offer spread on a dsperson trade s about to 3 vegas. Due to the drect nature of the correlaton exposure the correlaton swap strke has tended to trade at around the levels of correlaton realsed by the relevant basket of underlyngs, as opposed to tradng around the mpled levels of correlaton backed out from the ndex. As a result of ths correlaton swap strkes have tended to trade below ndex mpled correlaton. For example n late Aprl 005, 1-year ndex mpled correlaton was around 60 whereas correlaton swap strkes were tradng nearer to

12 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Varance Swaps Varance swaps are the buldng blocks of varance dsperson trades, the most common form of dsperson trade. Ths secton provdes a summary of the most mportant propertes of varance swaps, before gong on to consder the varous forms of varance dsperson trade n the followng sectons. A varance swap s an OTC nstrument offerng nvestors drect exposure to the volatlty of an underlyng asset. The strke of the swap s the level of volatlty bought or sold and s agreed at trade ncepton. The notonal s typcally expressed as a vega amount whch s the average p/l for a 1% move n volatlty. The true notonal s the varance amount whch s the vega amount dvded by twce the strke. The p/l of the swap at expry s equal to the varance amount multpled by the dfference between the strke squared and the realsed varance. Buyer pays varance swap strke Buyer of Varance Swap Seller of Varance Swap Seller pays realsed varance at expry The p/l for a (short) varance swap wth strke and subsequent realsed varance s gven by p / l = NVega = ( ) NVarance where NVega s the vega notonal and N Varance s the varance notonal. By conventon, volatlty s scaled by a factor of 100 (.e. a strke of 0 represents a volatlty of ) Payoffs from varance swaps are convex n volatlty. That s, a long poston wll gan more from an ncrease n realsed volatlty than t wll lose for a correspondng decrease. For ths reason varance swap strkes trade at a premum to a pure volatlty swap. n addton a varance swap can be statcally hedged by a (contnuous) portfolo of optons across a range of strkes, but more strongly weghted towards the downsde strkes. Thus (due to postve skew) the strke of a varance swap tends to trade above at-the-money mpled volatlty typcally around the 90-95% strke level dependng on maturty. (See Just What You Need To now About Varance Swaps, Sebasten Bossu, Eva Strasser and Regs Guchard 005.) Example: varance swaps Suppose a 1-year varance swap s stuck at 0 wth a vega amount of 100,000 (whch mples a varance amount of,500). f the ndex then realses 5% volatlty over the next year, the long wll receve 56,500 =,500 x (5 0 ). However f the ndex only realses 15%, the long wll pay 437,500 =,500 x (15 0 ). So the average exposure for a realsed volatlty beng 5% away from the strke s 500,000 or 5 tmes the vega amount, as requred. 1

13 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Tradng Correlaton - Dsperson trades Ths secton ntroduces dsperson trades and outlnes two types of varance dsperson trade: vanlla dsperson trades and correlaton-weghted dsperson trades. Subsequent sectons examne these trades n more detal. Although correlaton s traded most drectly through a correlaton swap, a correlaton exposure wll naturally arse when a portfolo of (long/short) optons on an ndex and (short/long) optons on consttuents of that ndex s held. A (long) dsperson trade s a trade whch s short ndex volatlty and long volatlty on ts consttuents. Typcally such a trade would be done through varance swaps usng a varance dsperson trade: sell varance on the ndex, buy varance on ts consttuents. But t could also be acheved through delta-hedged optons, straddles (delta-hedged or not) or any other sutable volatlty vehcle. We wll concentrate on varance dsperson trades. A dsperson trade wll have exposure to correlaton, but also to other factors for example volatlty dependng on the weghtngs chosen for the consttuent legs of the trade. Although a pure exposure to correlaton s not achevable through such a statc strategy, other exposures can be hedged to some extent by alterng the weghtngs and/or employng a dynamc replcaton strategy. The weghtngs of the sngle-stock varance swaps are of crucal mportance n a dsperson trade. Here we dscuss two dfferent weghtng schemas: a vanlla dsperson trade (the smplest) whch has both correlaton and outrght volatlty exposures, and a correlaton-weghted dsperson trade, whose p/l more closely reflects changes n correlaton. A vanlla dsperson trade s one where the varance swaps on the ndex members are bought n exact proporton to the weghts of the members n the ndex. A long vanlla dsperson trade s short correlaton, but t s also long volatlty and the p/l from changes n volatlty s generally a more mportant drver of overall p/l. A correlaton-weghted dsperson trade uses mpled correlaton to weght the sngle stock varance swaps. Such a dsperson trade wll maxmse exposure to correlaton whle ensurng that ntal vega exposure s zero. Correlaton-weghted dsperson trades have tracked correlaton much better than vanlla dsperson trades. Both types of dsperson trade have hstorcally been proftable. Whch type of dsperson trade s preferable depends on desred exposures. The vanlla dsperson trade has n fact been an effcent way to own volatlty, n effect usng the alpha avalable from short correlaton to subsdse the cost of beng long volatlty. On the other hand, the correlaton-weghted dsperson trade has profted drectly from correlaton returns and s less drectly affected by moves n volatlty. Below we dscuss n detal the drvers of p/l and hstorcal performance of vanlla and correlaton-weghted dsperson trades. 13

14 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Tradng correlaton - Vanlla dsperson trades A vanlla dsperson trade s short an ndex varance swap and long sngle stock varance swaps, where the vega amount of each sngle stock swap s proportonal to the weght of that stock n the ndex. The ndex varance swap vega notonal s the sum of sngle stock vega notonals. A vanlla dsperson trade s short correlaton but long volatlty. t s long volatlty because, wth correlaton less than one, a gven ncrease n sngle stock volatltes wll lead to a smaller ncrease n ndex volatlty. Example: a vanlla dsperson trade s long volatlty We sell 10,000 vega notonal of an ndex varance swap at a strke of 0. We buy a total of 10,000 vega notonal of sngle-stock varance swaps at an average strke of 30. n order to set up a vanlla dsperson trade the vega notonal of each stock wll be proportonal to ts weght n the ndex. Usng the correlaton proxy we estmate correlaton at 0.44 = ndex volatly sngle stock volatlty 0 =. 30 Suppose that average sngle-stock volatlty ncreases by 1% whlst correlaton stays constant. The correlaton proxy tells us the level of ndex volatlty: ndex volatlty = sqrt(correlaton) x average sngle stock volatlty = sqrt(0.44) x 31% = 0.67% That s, a 1% ncrease n sngle-stock volatltes wll lead to only a 0.67% ncrease n ndex volatlty. Therefore (wth correlaton less than one) a vanlla dsperson trade (short ndex volatlty, long sngle-stock volatlty) has a postve vega senstvty. n fact, wth correlaton constant, the ntal vega of the dsperson trade s equal to one mnus the square root of the correlaton. Ths long vega exposure means that the p/l of a vanlla varance dsperson trade has been closely lnked wth the p/l of a long volatlty poston (Fgure 17). n fact the exposure to correlaton has been less mportant than the exposure to volatlty. Fgure 17: The p/l from a vanlla dsperson trade s closely related to long varance p/l p/l (vegas) 15% 1 5% -5% p/l (vegas) long vanlla dsperson p/l (rhs) 4 long v arance p/l (lhs) Aug-00 Aug-01 Aug-0 Aug-03 Aug Fgure 18: Vanlla dsperson p/l s postvely correlated wth long varance p/l vanlla dsperson p/l (vegas) 15% 1 5% % long varance p/l (vegas) 14

15 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Fgure 18 shows the strong postve correlaton between the p/l from a long vanlla dsperson trade and the p/l from a long varance swap on the ndex. The lne of best-ft demonstrates that whlst the varance swap s more leveraged, by a factor of almost :1, the dsperson trade p/l has been much more consstently postve as a result of the alpha earned from beng short correlaton. Although the vanlla dsperson trade s techncally short correlaton, ths exposure has been domnated n practce by the long volatlty poston. n fact there has been a slghtly negatve correlaton between p/l from a long vanlla dsperson trade and moves n correlaton (mpled mnus subsequent realsed). Ths s probably due to the fact that short correlaton p/l (mpled mnus realsed) s generally postvely correlated wth short varance p/l, snce smlar factors drve both of them. Therefore, the prncpal exposure of the vanlla dsperson trade to long volatlty cancels out the smultaneous exposure to short (mpled mnus realsed) correlaton. Despte the fact that a vanlla dsperson trade fals to provde a drect exposure to correlaton, t has a number of propertes whch can make t an attractve asset, especally n comparson to a varance swap. The advantages of the vanlla dsperson trade n comparson to an outrght varance swap are: 1. The short correlaton exposure to some extent hedges the long volatlty exposure helpng to protect aganst adverse (downwards) moves n volatlty and reduce the overall volatlty of the trade.. The alpha earned from sellng correlaton (snce mpled correlaton trades at a premum to realsed) s used to fund the long volatlty poston (snce volatlty also trades at a premum) thus makng t a more effcent way of beng long volatlty. 3. Although the drect correlaton exposure s lmted, the trade can proft from a decrease n correlaton relatve to volatlty. At best a vanlla dsperson trade can combne the advantages both from beng long volatlty and from beng short correlaton. The short correlaton poston has hstorcally earned a premum whch can offset the cost of enterng nto a long volatlty poston. Furthermore the short correlaton exposure offers some protecton aganst adverse downward moves n volatlty: such decreases n volatlty are often accompaned by correspondng reductons n correlaton from whch the dsperson trade wll proft. Although the converse also apples and profts from ncreases n volatlty wll be somewhat offset by losses due to the short correlaton exposure, the gans from volatlty wll tend to domnate due to the convexty of the consttuent varance swaps. t should be stressed that whlst a vanlla dsperson trade has hstorcally been an effcent way of ownng volatlty, ths s n large part due to the premum of mpled over delvered correlaton. f ths premum was not present the vanlla dsperson trade would look less attractve. n ths scenaro such a trade would smply be long volatlty wth a lesser exposure to short correlaton, whch would act to reduce both large profts and large losses. Some market partcpants vew a vanlla dsperson trade as deal because the long volatlty component s seen as a partal hedge aganst adverse moves n correlaton. n fact backtestng shows that the volatlty exposure tends to domnate that of correlaton, makng the poston more smlar to a subsdsed long volatlty poston. 15

16 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Tradng correlaton - Correlaton-weghted dsperson trades t s possble to weght the consttuent legs of a varance dsperson trade to make the trade vega-neutral at ncepton and much more closely related to changes n correlaton thereafter. Although the maor contrbutor to the p/l of ths trade s the change n correlaton, between mpled and subsequent realsed, over the lfetme of the swap, there are stll other sgnfcant factors affectng the p/l. Vega Exposure: deally, we would lke to be able to choose the weghts n a varance dsperson trade so that the trade pad out on the bass of the change n correlaton whlst remanng ndependent of volatlty. t turns out that ths s mpossble (at least wthout a dynamc hedgng strategy). We can however choose the weghts so that the trade starts off as vega-neutral, although a volatlty exposure wll develop as correlaton subsequently moves. Back-testng shows that the p/l from ths trade s closely related to the dfference between mpled and subsequent realsed correlaton (Fgure 19). Weghtng: n order to construct ths trade n terms of vega notonals we weght each sngle-stock varance swap by the weght of the stock n the ndex multpled by the mpled correlaton. We then further multply by the rato of the stock s varance strke to the ndex varance strke. A varance dsperson trade weghted n ths way wll be vega-neutral at trade ncepton and a long poston wll proft from ncreases n sngle stock varances n proporton to the stock s weght n the ndex. See dscusson below for detals. A trade weghted n ths way wll be referred to as a correlaton-weghted (varance) dsperson trade. Note that n practce a smpler weghtng scheme for correlaton-weghted dsperson trades s often employed. Ths scheme works by weghtng each sngle-stock varance swap by ts ndex weght multpled by the square root of the mpled correlaton. t turns out that ths alternatve scheme s also vega-neutral at trade ncepton (Box 5) and results n an almost dentcal p/l to that descrbed here (see page 5). Fgure 19: P/L from a long correlaton-weghted dsperson trade s related to correlaton p/l vega p/l 1% 8% correlaton ponts long 6m correlaton-w eghted dsperson trade p/l short 6m correlaton p/l Fgure 0: but the relatonshp between them s non-lnear correlaton-weghted dsperson p/l (vegas) 1 5% 4% -4% Aug-00 Aug-01 Aug-0 Aug-03 Aug % mpled subsequent realsed correlaton (correl ponts) 16

17 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Box : Dsperson trade weghtngs The p/l of a dsperson trade wth arbtrary weghts α (for the sngle-stock vega notonals) s gven by: p / l = α where and represent the ndex and sngle-stock varance swap strkes; and ndex and sngle-stock realsed varances. and For a vanlla dsperson trade the weghts of the vega notonals would be the stock weghts: represent the α = ω. For a correlaton-weghted dsperson trade the weghts are gven by multplyng the stock weghts by the mpled correlaton and the rato of stock to ndex mpled volatlty: α = ω. Note that usng the alternatve weghtng α = ω acheves a vrtually dentcal effect (see pages 5 and 6 for detals). Example: a correlaton-weghted dsperson trade s ntally vega neutral Consder our prevous example wth ndex volatlty at, average snge-stock volatlty at 3 and correlaton at approxmately We frst demonstrate how to set up a correlaton-weghted dsperson trade under these crcumstances. For a vanlla dsperson trade the weghtng of any sngle-stock varance swap n the trade (n vega-notonal terms) would be the weght of the stock n the ndex. To compute the weghtng for the correlaton-weghted dsperson trade we multply ths stock weght by the correlaton (0.44) and then by the rato of the stock s volatlty to ndex volatlty. Assumng the stock has the average sngle-stock volatlty ths wll gve a weghtng factor of 0.44 x (3/) = 0.66 to be multpled by the stock s ndex weght. Therefore (assumng all stocks have volatltes equal to the average sngle-stock volatlty) we wll be short 3 parts ndex varance and long parts sngle-stock varance. Suppose as before that sngle-stock volatlty ncreases by 1% whlst correlaton stays constant. Then accordng to the prevous example, ndex volatlty wll ncrease by 0.66% whlst sngle-stock volatlty wll ncrease by 1%. Snce we are short ndex to sngle-stock volatlty n a 3: rato our net p/l wll be zero. Suppose nstead that correlaton decreases by 4 correlaton ponts whlst stock volatltes stay constant. Our trade wll stll be weghted n a 3: rato. Sngle-stock volatlty wll reman unchanged, but, usng the usual correlaton proxy formula, ndex volatlty wll ncrease to 3 multpled by the square root of 0.40, equal to 19%. Therefore, snce we are short ndex volatlty, our trade wll make a proft of 1 vega, as a result of the favourable move n correlaton. n the followng secton we dscuss the drvers of p/l for correlaton-weghted dsperson trades. 17

18 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Drvers of p/l for a correlaton-weghted dsperson trade t turns out that f you are long a correlaton-weghted dsperson trade you wll want the followng to occur: 1. Correlaton to decrease.. Average sngle-stock volatlty to ncrease - f correlaton has decreased. (f correlaton has ncreased you would want average sngle-stock volatlty to decrease n order to mnmse your losses.) 3. Dsperson of stock volatltes wthn the ndex to ncrease. n the followng paragraphs we explan why the p/l s drven by these factors and how ther mportance vares under dfferng volatlty condtons. Box 3 and Box 4 below summarse the maths. 1. Correlaton s the maor drver for correlaton-weghted dsperson trade p/l The maor drver for p/l of a correlaton-weghted varance dsperson trade s correlaton (Fgure 1). That s, the returns from a long correlaton-weghted dsperson trade are closely related to the dfference between the mpled correlaton at trade ncepton and the subsequent realsed correlaton (Fgure 19). Note that a long correlaton-weghted dsperson trade s short correlaton: f you have sold ndex varance aganst sngle-stock varance you want correlaton to decrease. Termnology: n the followng dscusson we use the term correlaton p/l to refer to the dfference between mpled correlaton and subsequent realsed correlaton over the approprate term. Whlst ths s really a hypothetcal p/l, as t s not possble to trade ths drectly, t represents the dfference between the correlaton prced n by the market and that actually delvered. n ths sense t s the target p/l of a correlaton-weghted dsperson trade (although the latter s expressed n vegas rather than correlaton ponts). The correlaton between correlaton-weghted dsperson trade returns and long volatlty returns s almost zero: ndeed slghtly negatve (Fgure ). That s, unlke the vanlla dsperson trade, the correlaton-weghted dsperson trade s strongly exposed to changes n correlaton but roughly neutral to changes n volatlty. Fgure 1: There s a strong relatonshp between p/l and correlaton p/l correlaton-weghted dsperson p/l (vegas) 1 Fgure : but no clear relatonshp between p/l and volatlty p/l correlaton-weghted dsperson p/l (vegas) 1 5% 5% % mpled subsequent realsed correlaton (correl ponts) -5% long volatlty p/l (vegas) 18

19 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 Although there s a strong lnk between p/l from a correlaton-weghted dsperson trade and correlaton p/l, the relatonshp s non lnear (Fgure 1). Ths s due to the nterplay between the correlaton p/l and scalng factor descrbed n the followng secton.. The level of exposure to correlaton s scaled accordng to prevalng volatlty condtons n a long correlaton-weghted dsperson trade, the level of exposure to the hypothetcal correlaton p/l wll be affected by the realsed average sngle-stock volatlty and the ndex mpled volatlty. To compute an estmate of the p/l (at least of that part of the p/l drven by changes n correlaton) we multply the correlaton p/l by a scalng factor equal to the average realsed sngle-stock volatlty squared dvded by twce the ndex varance-swap strke. From a p/l vewpont the ndex varance-swap strke s a constant, so the scalng factor has the effect of makng the trade long volatlty when correlaton decreases and short volatlty when correlaton ncreases. Put another way: at hgher volatltes ths scalng factor causes the p/l resultng from both ncreases and decreases n correlaton to be magnfed. Ths effect s demonstrated by plottng the returns from the correlaton-weghted dsperson trade aganst the average realsed volatlty of the stocks n the ndex (Fgure 3). As average volatlty ncreases, the spread of returns from the correlaton-weghted dsperson trades wdens. Due to the alpha earned from short correlaton, no correlaton-weghted dsperson trade made a (sgnfcant) outrght loss unless average sngle-stock realsed volatlty was over 4. Fgure 3: p/l from a correlaton-weghted dsperson trade s magnfed at hgher volatltes p/l (vegas) 1 Fgure 4: p/l s relatvely well approxmated by multplyng the scalng factor and the correlaton p/l p/l (vegas) 1 5% 5% % 5% 1-5% average sngle-stock realsed volatlty -5% scalng factor x correlaton p/l The scalng factor by whch the change n correlaton s multpled s such that correlaton returns wll be magnfed most n tmes of hgh realsed volatlty, partcularly when mpled volatlty underestmates the future realsed volatlty. Conversely correlaton returns wll be attenuated when realsed volatlty s low, especally when mpled volatlty has overestmated that subsequent realsed. Ths explans why, n the current envronment wth low and generally declnng volatlty, consstently overestmated by mpled levels, we have had very modest returns from long correlaton-weghted dsperson trades, but hgh hypothetcal correlaton p/l (see page 8 for a dscusson of returns). Fgure 4 demonstrates that multplyng the hypothetcal correlaton p/l by the scalng factor closely approxmates the p/l of a correlaton-dsperson trade. Due to the correlaton between correlaton and volatlty, the scalng factor descrbed above turns out to be negatvely correlated wth correlaton p/l. The same factors whch cause mpled correlaton to be large wth respect to realsed correlaton (leadng to postve correlaton p/l) also tend to cause mpled volatlty to be large wth respect to realsed volatlty, leadng to a small scalng factor and hence smaller than expected p/l from the correlaton dsperson trade. Conversely f correlaton p/l s negatve, due to larger realsed correlaton than 19

20 Ncolas Granger (44-0) (44-0) European Equty Dervatves Strategy 4 May 005 mpled, ths would normally be accompaned by realsed volatlty beng greater than that mpled and hence a large scalng factor, magnfyng the losses from the short correlaton poston. Ths accounts for the negatve convexty observed n Fgure 0: as correlaton p/l ncreases, the scalng factor wll tend to decrease, reducng the correlaton gans. Example A: Hgh scalng factor On July 1st 001, mpled ndex volatlty was 5% and mpled correlaton Over the followng sx months the ndex realsed 31% volatlty, wth average realsed sngle-stock volatlty at 43.5% and a realsed correlaton of Thus correlaton p/l was =0.10, and the scalng factor was (43.5%^)/( x 5%) = 0.38, gvng an estmated 6-month correlaton-weghted dsperson trade p/l of 3.8 vegas (the true p/l was 4.1). Note that n ths case everythng moved n our favour: correlaton went down to gve us a postve p/l and volatlty went up to gve a relatvely large scalng factor. Example B: Low scalng factor On October 0th 003, mpled ndex volatlty was 5.6% wth mpled correlaton at Sx months later, realsed ndex volatlty was 14.7% wth average realsed sngle-stock volatlty at.5% and realsed correlaton of n ths case correlaton p/l was 3 correlaton ponts, but the scalng factor was only 0.1 gvng an estmated 6-month correlaton-weghted dsperson trade p/l of 3. vegas (the true p/l was 3.3). n ths case the correlaton p/l was much hgher than n our frst example, but volatlty moved aganst us to produce a very small scalng factor whch heavly reduced our p/l. Ths s qute characterstc of the recent performance of the correlaton-weghted dsperson trade: whlst correlaton p/l has been hgh, volatlty has been low, wth realsed volatlty less than that mpled, leadng to a small scalng factor and reduced p/l (see Fgure 19). The above examples and Fgure 4 show that whlst correlaton-weghted dsperson p/l s well approxmated by multplyng the hypothetcal correlaton p/l by the scalng factor, there are clearly other drvers nfluencng the p/l. Box 3 and Box 4 gve the full pcture: correlaton-weghted dsperson p/l depends on both hypothetcal correlaton p/l and volatlty dsperson p/l wth assocated scalng factors for each. The followng secton dscusses ths p/l drver. 0