MAY 5 JU WHA YOU NEED O KNOW AOU VARIANCE WAP ebasen ossu Eva rasser Regs Guchard EQUIY DERIVAIVE Inal publcaon February 5 Equy Dervaves Invesor Markeng JPMorgan London Quanave Research & Developmen IN HE UNIED AE HI REPOR I AVAILALE ONLY O PERON WHO HAVE RECEIVED HE PROPER OPION RIK DICLOURE DOCUMEN
Overvew In hs noe we nroduce he properes of varance swaps, and gve deals on he hedgng and valuaon of hese nsrumens. econ gves quck facs abou varance swaps and her applcaons. econ s wren for raders and marke professonals who have some degree of famlary wh he heory of vanlla opon prcng and hedgng, and explans n nuve mahemacal erms how varance swaps are hedged and prced. econ 3 s wren for quanave raders, researchers and fnancal engneers, and gves heorecal nsghs no hedgng sraeges, mpac of dvdends and jumps. Appendx A s a revew of he conceps of hsorcal and mpled volaly. Appendces and C cover echncal resuls used n he noe. A YOU NEED O KNOW AOU VARIANCE WAP JU WH We hank Cyrl Levy-Marchal, Jeremy Weller, Manos Venardos, Peer Allen, mone Russo for her help or commens n he preparaon of hs noe. hese analyses are provded for nformaon purposes only and are nended solely for your use. he analyses have been derved from publshed models, reasonable mahemacal approxmaons, and reasonable esmaes abou hypohecal marke condons. Analyses based on oher models or dfferen assumpons may yeld dfferen resuls. JPMorgan expressly dsclams any responsbly for ( he accuracy of he models, approxmaons or esmaes used n dervng he analyses, ( any errors or omssons n compung or dssemnang he analyses and ( any uses o whch he analyses are pu. hs commenary s wren by he specfc radng area referenced above and s no he produc of JPMorgan's research deparmens. Research repors and noes produced by he Frm's Research Deparmens are avalable from your salesperson or a he Frm's webse, hp://www.morganmarkes.com. Opnons expressed heren may dffer from he opnons expressed by oher areas of JPMorgan, ncludng research. hs commenary s provded for nformaon only and s no nended as a recommendaon or an offer or solcaon for he purchase or sale of any secury or fnancal nsrumen. JPMorgan and s afflaes may have posons (long or shor, effec ransacons or make markes n secures or fnancal nsrumens menoned heren (or opons wh respec hereo, or provde advce or loans o, or parcpae n he underwrng or resrucurng of he oblgaons of, ssuers menoned heren. he nformaon conaned heren s as of he dae and me referenced above and JPMorgan does no underake any oblgaon o updae such nformaon. All marke prces, daa and oher nformaon are no warraned as o compleeness or accuracy and are subjec o change whou noce. ransacons nvolvng secures and fnancal nsrumens menoned heren (ncludng fuures and opons may no be suable for all nvesors. Clens should conac her salespersons a, and execue ransacons hrough, a JPMorgan eny qualfed n her home jursdcon unless governng law perms oherwse. Enerng no opons ransacons enals ceran rsks wh whch you should be famlar. In connecon wh he nformaon provded below, you acknowledge ha you have receved he Opons Clearng Corporaon's Characerscs and Rsks of andardzed Opon. If you have no receved he OCC documens and pror o revewng he nformaon provded below, conac your JPMorgan represenave or refer o he OCC webse a hp://www.oponsclearng.com/publcaons/rsksoc.pdf Copyrgh 5 J.P. Morgan Chase & Co. All rghs reserved. JPMorgan s he markeng name for J.P. Morgan Chase & Co. and s subsdares and afflaes worldwde. J.P. Morgan ecures Inc. s a member of NYE and IPC. JPMorgan Chase ank s a member of FDIC. J.P. Morgan Fuures Inc. s a member of he NFA. J.P. Morgan ecures Ld. and J.P. Morgan plc are auhorsed by he FA and members of he LE. J.P. Morgan Europe Lmed s auhorsed by he FA. J.P. Morgan Eques Lmed s a member of he Johannesburg ecures Exchange and s regulaed by he F. J.P. Morgan ecures (Asa Pacfc Lmed and Jardne Flemng ecures Lmed are regsered as nvesmen advsers wh he ecures & Fuures Commsson n Hong Kong and her CE numbers are AAJ3 and AA6 respecvely. Jardne Flemng ngapore ecures Pe Ld s a member of ngapore Exchange ecures radng Lmed and s regulaed by he Moneary Auhory of ngapore ("MA". J.P. Morgan ecures Asa Prvae Lmed s regulaed by he MA and he Fnancal upervsory Agency n Japan. J.P.Morgan Ausrala Lmed (AN 5 888 s a lcensed secures dealer. In he UK and oher EEA counres, hs commenary s no avalable for dsrbuon o persons regarded as prvae cusomers (or equvalen n her home jursdcon.
able of Conens Overvew... able of Conens.... Varance waps... 3.. Payoff 3 Convexy 4 Rules of humb 5.. Applcaons 5 Volaly radng 5 Forward volaly radng 5 preads on ndces 6 Correlaon radng: Dsperson rades 7.3. Mark-o-marke and ensves 8 Mark-o-marke 8 Vega sensvy 9 kew sensvy 9 Dvdend sensvy 9 A YOU NEED O KNOW AOU VARIANCE WAP JU WH. Valuaon and Hedgng n Pracce..... Vanlla Opons: Dela-Hedgng and P&L Pah-Dependency Dela-Hedgng P&L pah-dependency.. ac Replcaon of Varance waps 4 Inerpreaon 6.3. Valuaon 6 3. heorecal Insghs...8 3.. Idealzed Defnon of Varance 8 3.. Hedgng raeges & Prcng 8 elf-fnancng sraegy 9 Prcng 9 Represenaon as a sum of pus and calls 3.3. Impac of Dvdends Connuous Monorng Dscree Monorng 3.4. Impac of Jumps 3 Appendx A A Revew of Hsorcal and Impled Volaly...4 Appendx Relaonshp beween hea and Gamma...7 Appendx C Peak Dollar Gamma...8 References & blography...9
. Varance waps.. Payoff A varance swap s an nsrumen whch allows nvesors o rade fuure realzed (or hsorcal volaly agans curren mpled volaly. As explaned laer n hs documen, only varance he squared volaly can be replcaed wh a sac hedge. [ee econs. and 3. for more deals.] ample erms are gven n Exhb.. below. Exhb.. Varance wap on &P 5 : sample erms and condons VARIANCE WAP ON &P5 PX INDICAIVE ERM AND CONDIION A YOU NEED O KNOW AOU VARIANCE WAP JU WH Insrumen: rade Dae: Observaon ar Dae: Observaon End Dae: Varance uyer: Varance eller: Denomnaed Currency: wap D D D D (e.g. JPMorganChase D (e.g. Invesor UD ( UD Vega Amoun:, Varance Amoun: 3,5 ( deermned as Vega Amoun/(rke* Underlyng: rke Prce: 6 Currency: Equy Amoun: &P5 (loomberg cker: PX Index UD 3 afer he Observaon End Dae, he Equy Amoun wll be calculaed and pad n accordance wh he followng formula: Fnal Equy paymen Varance Amoun * (Fnal Realzed Volaly rke Prce If he Equy Amoun s posve he Varance eller wll pay he Varance uyer he Equy Amoun. If he Equy Amoun s negave he Varance uyer wll pay he Varance eller an amoun equal o he absolue value of he Equy Amoun. where Calculaon Agen: JP Morgan ecures Ld. Documenaon: IDA 5 N ln P Fnal Realsed Volaly P Expeced _ N Expeced_N [number of days], beng he number of days whch, as of he rade Dae, are expeced o be cheduled radng Days n he Observaon Perod P he Offcal Closng of he underlyng a he Observaon ar Dae P Eher he Offcal Closng of he underlyng n any observaon dae or, a Observaon End Dae, he Offcal elemen Prce of he Exchange-raded Conrac 3
Noe: Reurns are compued on a logarhmc bass: P ln. P he mean reurn, whch normally appears n sascs exbooks, s dropped. hs s because s mpac on he prce s neglgble (he expeced average daly reurn s /5 nd of he money-marke rae, whle s omsson has he benef of makng he payoff perfecly addve (3-monh varance 9-monh varance n 3 monhs -year varance. I s a marke pracce o defne he varance noonal n volaly erms: Varance Noonal Vega Noonal rke Wh hs adjusmen, f he realzed volaly s vega (volaly pon above he srke a maury, he payoff s approxmaely equal o he Vega Noonal. A YOU NEED O KNOW AOU VARIANCE WAP Convexy he payoff of a varance swap s convex n volaly, as llusraed n Exhb... hs means ha an nvesor who s long a varance swap (.e. recevng realzed varance and payng srke a maury wll benef from boosed gans and dscouned losses. hs bas has a cos refleced n a slghly hgher srke han he far volaly, a phenomenon whch s amplfed when volaly skew s seep. hus, he far srke of a varance swap s ofen n lne wh he mpled volaly of he 9% pu. Exhb.. Varance swaps are convex n volaly $5,, $4,, $3,, $,, $,, $ -$,, -$,, -$3,, Payoff rke 4 Varance 3 4 5 Volaly Realzed Volaly JU WH Readers wh a mahemacal background wll recall Jensen s nequaly: E( Varance E( Varance. 4
Rules of humb Demeerf Derman Kamal Zou (999 derved a rule of humb for he far srke of a varance swap when he skew s lnear n srke: K AMF skew var 3 where AMF s he a-he-money-forward volaly, s he maury, and skew s he slope of he skew curve. For example, wh AMF %, years, and a 9- skew of vegas, we have K var.3%, whch s n lne wh he 9% pu mpled volaly normally observed n pracce. For log-lnear skew, smlar echnques gve he rule of humb: K β 4 4 ( 3 var AMF β AMF AMF 5 where AMF s he a-he-money-forward volaly, s he maury, and β s he slope of he log skew curve 3. For example, wh AMF %, years, and a 9- skew of % ln(.9 vegas, we have β. 9 and K var.8%. Noe ha hese wo rules of humb produce good resuls only for non-seep skew. AMF A YOU NEED O KNOW AOU VARIANCE WAP.. Applcaons Volaly radng Varance swaps are naural nsrumens for nvesors akng dreconal bes on volaly: Realzed volaly: unlke he radng P&L of a dela-hedged opon poson, a long varance poson wll always benef when realzed volaly s hgher han mpled a ncepon, and conversely for a shor poson [see econ. on P&L pah-dependency.] Impled volaly: smlar o opons, varance swaps are fully sensve a ncepon o changes n mpled volaly Varance swaps are especally aracve o volaly sellers for he followng wo reasons: Impled volaly ends o be hgher han fnal realzed volaly: he dervave house has he sascal edge. Convexy causes he srke o be around he 9% pu mpled volaly, whch s slghly hgher han far volaly. Forward volaly radng ecause varance s addve, one can oban a perfec exposure o forward mpled volaly wh a calendar spread. For example, a shor -year vega exposure of, on he Eurooxx 5 sarng n year can be hedged as follows [levels as of Aprl, 5]: JU WH 3 he skew curve s hus assumed o be of he form: ( K β ln( K / F where F s he forward prce. AMF 5
Long -year varance sruck a 9.5 on a Vega Noonal of, (.e. a Varance Noonal of 5,8 hor -year varance sruck a 8.5 on a Varance Noonal of 5,8 /,564 (.e. a Vega Noonal of 94,868 Impled forward volaly on hs rade s approxmaely 4 : 9.5 3 { - 8.5 3 {.5. year vol enor year vol enor herefore, f he -year mpled volaly s above.5 n one year s me, say a, he hedge wll be approxmaely up ½ a vega, or 5,, whle he exposure wll be down by he same amoun. However, keep n mnd ha he far value of a varance swap s also sensve o skew. A YOU NEED O KNOW AOU VARIANCE WAP Forward volaly rades are neresng because he forward volaly erm srucure ends o flaen for longer forward-sar daes, as llusraed n Exhb.. below. In hs example, we can see ha he -year forward volales exhb a downard slopng erm srucure. hus, an nvesor who beleves ha he erm srucure wll rever o an upward slopng shape mgh wan o sell he x and buy he x mpled volales, or equvalenly sell 3m and buy 4m, wh approprae noonals: uy x uy 4m and ell m ell x ell 3m and uy m uy spread uy 4m and ell 3m Exhb.. po and forward volaly curves derved from far varance swap srkes ource: JPMorgan. 4 3 9 8 7 6 5 4 3 preads on ndces po 3m fwd 6m fwd m fwd m m 3m 4m 5m 6m 7m 8m 9m m m m Varance swaps can also be used o capure he volaly spread beween wo correlaed ndces, for nsance by beng long 3-monh DAX varance and shor 3-monh Eurooxx 5 varance. Exhb.. below shows ha n he perod -4 he hsorcal spread was JU WH 4 An accurae calculaon would be: PV (y y vol y vol, where PV( s he presen value of pad a me PV ( y 6
almos always n favor of he DAX and somemes as hgh as vegas, whle he mpled spread 5 ranged beween -4 and 4 vegas. Exhb.. Volaly spread beween DAX and Eurooxx 5: hsorcal (a and mpled (b a A YOU NEED O KNOW AOU VARIANCE WAP b ource: JPMorgan DaaQuery. Correlaon radng: Dsperson rades A popular rade n he varance swap unverse s o sell correlaon by akng a shor poson on ndex varance and a long poson on he varance of he componens. Exhb..3 below shows he evoluon of one-year mpled and realzed correlaon. JU WH 5 Measured as he dfference beween he 9% srke mpled volales. Acual numbers may dffer dependng on skew, ransacon coss and oher marke condons. 7
Exhb..3 Impled and realzed correlaon of Eurooxx 5 ource: JPMorgan DaaQuery. A YOU NEED O KNOW AOU VARIANCE WAP JU WH More formally he payoff of a varance dsperson rade s: n w Noonal Noonal Index Index Resdual rke where w s are he weghs of he ndex componens, s are realzed volales, and noonals are expressed n varance erms. ypcally, only he mos lqud socks are seleced among he ndex componens, and each varance noonal s adjused o mach he same vega noonal as he ndex n order o make he rade vega-neural a ncepon..3. Mark-o-marke and ensves Mark-o-marke ecause varance s addve n me dmenson he mark-o-marke of a varance swap can be decomposed a any pon n me beween realzed and mpled varance: Varwap Noonal PV ( ( Realzed Vol(, ( Impled Vol(, rke where Noonal s n varance erms, PV ( s he presen value a me of $ receved a maury, Realzed Vol(, s he realzed volaly beween ncepon and me, Impled Vol(, s he far srke of a varance swap of maury ssued a me. For example, consder a one-year varance swap ssued 3 monhs ago on a vega noonal of $,, sruck a. he 9-monh zero-rae s %, realzed volaly over he pas 3 monhs 8
was 5, and a 9-monh varance swap would srke oday a 9. he mark-o-marke of he one-year varance swap would be:, 3 Varwap 5 9.75 ( % 4 4 $359,69 Noe ha hs s no oo far from he vega loss whch one obans by compung he weghed average of realzed and mpled volaly:.5 x 5.75 x 9 8, mnus srke. Vega sensvy he sensvy of a varance swap o mpled volaly decreases lnearly wh me as a drec consequence of mark-o-marke addvy: Varwap Vega Noonal ( mpled mpled Noe ha Vega s equal o a ncepon f he srke s far and he noonal s vega-adjused: Noonal Vega Noonal rke A YOU NEED O KNOW AOU VARIANCE WAP kew sensvy As menoned earler he far value of a varance swap s sensve o skew: he seeper he skew he hgher he far value. Unforunaely here s no sraghforward formula o measure skew sensvy bu we can have a rough dea usng he rule of humb for lnear skew n econ.: var 3 ( skew K AMF kew ensvy 6 Noonal AMF skew For example, consder a one-year varance swap on a vega noonal of $,, sruck a 5. A-he-money-forward volaly s 4, and he 9- skew s.5 vegas. Accordng o he rule of humb, he far srke s approxmaely 4 x ( 3 x (.5/ 6.6. If he 9- skew seepens o 3 vegas he change n mark-o-marke would be: Dvdend sensvy MM,.5 3.5 6 4 $, 444 54444 3 443 ensvy kew Dvdend paymens affec he prce of a sock, resulng n a hgher varance. When dvdends are pad a regular nervals, can be shown ha ex-dvdend annualzed varance should be JU WH 9
adjused by approxmaely addng he square of he annualzed dvdend yeld dvded by he number of dvdend paymens per year 6. he far srke s hus: K var exdv ( K var ( Dv Yeld Nb Dvs Per Year From hs adjusmen we can derve a rule of humb for dvdend sensvy: Varwap Dv Yeld µ Noonal Dv Yeld Nb Dvs Per Year K For example, consder a one-year varance swap on a vega noonal of $, sruck a. he far srke ex-dvdend s and he annual dvdend yeld s 5%, pad sem-annually. he adjused srke s hus ( 5 /.5.3. Were he dvdend yeld o ncrease o 5.5% he change n mark-o-marke would be: MM 5/,.3 443 4444 3 Dv Yeld skew sensvy var ( 5.5 5 $, 3 However, n he presence of skew, changes n dvdend expecaons wll also mpac he forward prce of he underlyng whch n urns affecs he far value of varanc. hs phenomenon wll normally augmen he overall dvdend sensvy of a varance swap. A YOU NEED O KNOW AOU VARIANCE WAP JU WH 6 More specfcally he adjusmen s M j j D Τ d M he annualzed average dvdend yeld. ee econ 3.3 for more deals. M where d, d,, d M are gross dvdend yelds and D s
. Valuaon and Hedgng n Pracce.. Vanlla Opons: Dela-Hedgng and P&L Pah-Dependency Dela-Hedgng Opon markes are essenally drven by expecaons of fuure volaly. hs resuls from he way an opon payoff can be dynamcally replcaed by only radng he underlyng sock and cash, as descrbed n 973 by lack choles and Meron. More specfcally, he sensvy of an opon prce o changes n he sock prce, or dela, can be enrely offse by connuously holdng a reverse poson n he underlyng n quany equal o he dela. For example, a long call poson on he &P 5 ndex wh an nal dela of $5, per ndex pon (worh $6,, for an ndex level of, s dela-neuralzed by sellng 5, uns of he &P 5 (n pracce fuures conracs: 6,,/(5 x, Were he dela o ncrease o $5,5 per ndex pon, he hedge should be adjused by sellng an addonal 5 uns ( conrac, and so forh. he eraon of hs sraegy unl maury s known as dela-hedgng. A YOU NEED O KNOW AOU VARIANCE WAP Once he dela s hedged, an opon rader s mosly lef wh hree sensves: Gamma: sensvy of he opon dela o changes n he underlyng sock prce ; hea or me decay: sensvy of he opon prce o he passage of me ; Vega: sensvy of he opon prce o changes n he marke s expecaon of fuure volaly (.e. mpled volaly. 7 he daly P&L on a dela-neural opon poson can be decomposed along hese hree facors: Daly P&L Gamma P&L hea P&L Vega P&L Oher (Eq. Here Oher ncludes he P&L from fnancng he reverse dela poson on he underlyng, as well as he P&L due o changes n neres raes, dvdend expecaons, and hgh-order sensves (e.g. sensvy of Vega o changes n sock prce, ec. Equaon can be rewren: Daly P&L Γ ( Θ ( V (... where s he change n he underlyng sock prce, s he fracon of me elapsed (ypcally /365, and s he change n mpled volaly. We now consder a world where mpled volaly s consan, he rskless neres rae s zero, and oher P&L facors are neglgble. In hs world resemblng lack-choles, we have he reduced P&L equaon: Daly P&L Γ ( Θ ( (Eq. We proceed o nerpre Equaon n erms of volaly, and we wll see ha n hs world he daly P&L of a dela-hedged opon poson s essenally drven by realzed and mpled volaly. JU WH 7 Noe ha n lack-choles volaly s assumed o reman consan hrough me. he concep of Vega s hus nconssen wh he heory, ye crcal n pracce.
We sar wh he well-known relaonshp beween hea and gamma: Γ Θ (Eq. 3 where s he curren spo prce of he underlyng sock and he curren mpled volaly of he opon. In our world wh zero neres rae, hs relaonshp s acually exac, no approxmae. Appendx presens wo dervaons of Equaon 3, one based on nuon and one whch s more rgorous. Equaon 3 s he core of lack-choles: dcaes how opon prces dffuse n me n relaon o convexy. Pluggng Equaon 3 no Equaon and facorng, we oban a characerzaon of he daly P&L n erms of squared reurn and squared mpled volaly: Daly P&L Γ (Eq. 4 A YOU NEED O KNOW AOU VARIANCE WAP he frs erm n he bracke,, s he percen change n he sock prce n oher words, he one-day sock reurn. quared, can be nerpreed as he realzed one-day varance. he second erm n he bracke,, s he squared daly mpled volaly, whch one could name he daly mpled varance. hus, Equaon 4 ells us ha he daly P&L of a dela-hedged opon poson s drven by he spread beween realzed and mpled varance, and breaks even when he sock prce movemen exacly maches he marke s expecaon of volaly. In he followng paragraph we exend hs analyss o he enre lfeme of he opon. P&L pah-dependency One can already see he connecon beween Equaon 4 and varance swaps: f we sum all daly P&L s unl he opon s maury, we oban an expresson for he fnal P&L: n Fnal P&L [ r ] γ (Eq. 5 where he subscrp denoes me dependence, r he sock daly reurn a me, and g he opon s gamma mulpled by he square of he sock prce a me, also known as dollar gamma. Equaon 5 s very close o he payoff of a varance swap: s a weghed sum of squared realzed reurns mnus a consan ha has he role of he srke. he man dfference s ha n a varance swap weghs are consan, whereas here he weghs depend on he opon gamma hrough me, a phenomenon whch s known o opon raders as he pah-dependency of an opon s radng P&L, llusraed n Exhb... I s neresng o noe ha even when he sock reurns are assumed o follow a random walk wh a volaly equal o, Equaon 5 does no become nl. hs s because each squared reurn remans dsrbued around raher han equal o. However hs parcular JU WH
pah-dependency effec s mosly due o dscree hedgng raher han a dscrepancy beween mpled and realzed volaly and wll vansh n he case of connuous hedgng 8. Exhb.. Pah-dependency of an opon s radng P&L In hs example an opon rader sold a -year call sruck a % of he nal prce on a noonal of $,, for an mpled volaly of 3%, and dela-heged hs poson daly. he realzed volaly was 7.5%, ye hs fnal radng P&L s down $5k. Furhermore, we can see (Fgure a ha he P&L was up $5k unl a monh before expry: how dd he profs change no losses? One ndcaon s ha he sock prce oscllaed around he srke n he fnal monhs (Fgure a, rggerng he dollar gamma o soar (Fgure b. hs would be good news f he volaly of he underlyng remaned below 3% bu unforunaely hs perod concded wh a change n he volaly regme from % o 4% (Fgure b. ecause he daly P&L of an opon poson s weghed by he gamma and he volaly spread beween mpled and realzed was negave, he fnal P&L drowned, even hough he realzed volaly over he year was below 3%! a ock Prce (Inal 4% % % radng P&L ($ 'Hammered a he srke'! 75, rke 5, 8% 6% ock Prce 5, A YOU NEED O KNOW AOU VARIANCE WAP JU WH b 4% % % 5 3 45 6 ock Prce (Inal 4% % % 8% 6% 4% % % ock Prce 5 3 % 45 6 75 75 radng P&L 9 5 rke 35 5-day Realzed 9 Volaly 5 35 5 5 65 65 8 8 95 95 3% 5 5 4 43% Dollar Gamma 4 - -5, radng days Volaly 7% 6% 5% 4% 3% % % % radng days 8 ee Wlmo (998 for a heorecal approach of dscree hedgng and Allen Harrs ( for a sascal analyss of hs phenomenon. Wlmo noes ha he daly Gamma P&L has a ch-square dsrbuon, whle Allen Harrs nclude a bell-shaped char of he dsrbuon of fnal P&Ls of a dscreely dela-hedged opon poson. Neglecng he gamma dependence, he cenral-lm heorem ndeed shows ha he sum of n ndependen chsquare varables converges o a normal dsrbuon. 3
.. ac Replcaon of Varance waps In he prevous paragraph we saw ha a vanlla opon rader followng a dela-hedgng sraegy s essenally replcang he payoff of a weghed varance swap where he daly squared reurns are weghed by he opon s dollar gamma 9. We now proceed o derve a sac hedge for sandard ( non-gamma-weghed varance swaps. he core dea here s o combne several opons ogeher n order o oban a consan aggregae gamma. Exhb.. shows he dollar gamma of opons wh varous srkes n funcon of he underlyng level. We can see ha he conrbuon of low-srke opons o he aggregae gamma s small compared o hgh-srke opons. herefore, a naural dea s o ncrease he weghs of low-srke opons and decrease he weghs of hgh-srke opons. Exhb.. Dollar gamma of opons wh srkes 5 o spaced 5 apar Dollar Gamma Aggregae A YOU NEED O KNOW AOU VARIANCE WAP K K 75 K 5 K 5 K K 75 K 5 K 5 5 5 75 5 5 75 5 5 75 3 Underlyng Level (AM An nal, naïve approach o hs weghng problem s o deermne ndvdual weghs w(k such ha each opon of srke K has a peak dollar gamma of, say,. Usng he lack- choles closed-form formula for gamma, one would fnd ha he weghs should be nversely proporonal o he srke (.e. w(k c / K, where c s a consan. [ee Appendx C for deals.] Exhb.. shows he dollar gamma resulng from hs weghng scheme. We can see ha he aggregae gamma s sll non-consan (whence he adjecve naïve o descrbe hs approach, however we also noce he exsence of a lnear regon when he underlyng level s n he range 75 35. JU WH 9 Recall ha dollar gamma s defned as he second-order sensvy of an opon prce o a percen change n he underlyng. In hs paragraph, we use he erms gamma and dollar gamma nerchangeably. 4
Exhb.. Dollar gamma of opons weghed nversely proporonal o he srke Dollar Gamma Lnear Regon Aggregae K 5 K 5 K K 5 K 5 K 75 K 5 5 75 5 5 75 5 5 75 3 Underlyng Level (AM A YOU NEED O KNOW AOU VARIANCE WAP hs observaon s crucal: f we can regonally oban a lnear aggregae gamma wh a ceran weghng scheme w(k, hen he modfed weghs w (K w(k / K wll produce a consan aggregae gamma. nce he naïve weghs are nversely proporonal o he srke K, he correc weghs should be chosen o be nversely proporonal o he squared srke,.e.: where c s a consan. c w ( K K Exhb..3 shows he resuls of hs approach for he ndvdual and aggregae dollar gammas. As expeced, we oban a consan regon when he underlyng level says n he range 75 35. A perfec hedge wh a consan aggregae gamma for all underlyng levels would ake nfnely many opons sruck along a connuum beween and nfny and weghed nversely proporonal o he squared srke. hs s eablshed rgorously n econ 3.. Noe ha hs s a srong resul, as he sac hedge s boh space (underlyng level and me ndependen. JU WH 5
Exhb..3 Dollar gamma of opons weghed nversely proporonal o he square of srke Dollar Gamma K 5 K 5 Consan Gamma Regon K K 5 K 5 K 75 K Aggregae 5 5 75 5 5 75 5 5 75 3 Underlyng Level (AM A YOU NEED O KNOW AOU VARIANCE WAP JU WH Inerpreaon One mgh wonder wha means o creae a dervave whose dollar gamma s consan. Dollar gamma s he sandard gamma mes : $ f Γ ( where f, are he prces of he dervave and underlyng, respecvely. hus, a consan dollar gamma means ha for some consan a: f a he soluon o hs second-order dfferenal equaon s: f ( a ln( b c where a, b, c are consans, and ln(. he naural logarhm. In oher words, he perfec sac hedge for a varance swap would be a combnaon of he log-asse (a dervave whch pays off he log-prce of he underlyng sock, he underlyng sock and cash..3. Valuaon ecause a varance swap can be sacally replcaed wh a porfolo of vanlla opons, no parcular modelng assumpon s needed o deermne s far marke value. he only model choce resdes n he compuaon of he vanlla opon prces a ask whch merely requres a reasonable model of he mpled volaly surface. Assumng ha one has compued he prces p (k and c (k of N pus ou-of-he money pus and N calls ou-of-he-money calls respecvely, a quck proxy for he far value of a varance swap of maury s gven as: 6
Varwap N pus p ( k ( pu pu k PV ( ( K calls N call pu pu c ( k call call ( k k ( k call k where Varwap s he far presen value of he varance swap for a varance noonal of, K V V pu k s he srke, PV ( s he presen value of $ a me, and are he respecve srkes of he -h pu and -h call n percenage of he underlyng forward prce, wh he convenon k. ( k call k In he ypcal case where he srkes are chosen o be spaced equally apar, say every 5% seps, he expresson beween brackes s he sum of he pu and call prces, weghed by he nverse of he squared srke, mes he 5% sep. Exhb.3. below llusraes hs calculaon; n hs example, he far srke s around 6.6%, when a more accurae algorhm gave 6.54%. We also see ha he far srke s close o he 9% mpled volaly (7.3%, as menoned n econ.. A YOU NEED O KNOW AOU VARIANCE WAP JU WH Exhb.3. Calculaon of he far value of a varance swap hrough a replcang porfolo of pus and calls In hs example, he oal hedge cos of he replcang porfolo s.74% (/ * Σ (w p, or 7.4 varance pons. For a varance noonal of,, hs means ha he floang leg of he varance swap s worh,7,397.53. For a srke of 6.65 volaly pons, and a -year presen value facor of.977368853, he fxed leg s worh,7,355.88. hus, he varance swap has a value close o. Wegh 5% rke% Underlyng Call / Pu Forward rke rke (%Forward Maury Impled Volaly Prce (%Noonal.% X5E P,935.,467.5 5% Y 7.6%.4% 6.53% X5E P,935.,64.6 55% Y 6.4%.8% 3.89% X5E P,935.,76. 6% Y 5.%.5%.83% X5E P,935.,97.76 65% Y 4.%.7%.% X5E P,935.,54.5 7% Y.7%.46% 8.89% X5E P,935.,.6 75% Y.4%.75% 7.8% X5E P,935.,348. 8% Y.%.7% 6.9% X5E P,935.,494.76 85% Y 8.7%.79% 6.7% X5E P,935.,64.5 9% Y 7.3%.67% 5.54% X5E P,935.,788.6 95% Y 6.% 3.94%.5% X5E P,935.,935. % Y 4.8% 5.74%.5% X5E C,935.,935. % Y 4.8% 5.74% 4.54% X5E C,935. 3,8.77 5% Y 3.7% 3.37% 4.3% X5E C,935. 3,8.5 % Y.9%.76% 3.78% X5E C,935. 3,375.7 5% Y.%.8% 3.47% X5E C,935. 3,5. % Y.9%.35% 3.% X5E C,935. 3,668.77 5% Y.8%.5%.96% X5E C,935. 3,85.5 3% Y.9%.6%.74% X5E C,935. 3,96.7 35% Y.%.3%.55% X5E C,935. 4,9. 4% Y.5%.%.38% X5E C,935. 4,55.77 45% Y.9%.%.% X5E C,935. 4,4.5 5% Y 3.4%.% ource: JPMorgan. 7
3. heorecal Insghs A YOU NEED O KNOW AOU VARIANCE WAP JU WH 3.. Idealzed Defnon of Varance An dealzed defnon of annualzed realzed varance W, s gven by: W [ ln, ln ], where denoes he prce process of he underlyng asse and [ln, ln ] denoes he quadrac varaon of ln. hs defnon s dealzed n he sense ha we mplcly assume ha s possble o monor realzed varance on a connuous bass. I can be shown ha he dscree defnon of realzed varance gven n econ. converges o he dealzed defnon above when movng o connuous monorng. hs defnon apples n parcular o he classc Io process for sock prces: d µ (,, K d (,, K dw where he drf µ and he volaly are eher deermnsc or sochasc. In hs case, he dealzed defnon of varance becomes: W, (,, K d. However, n he presence of jumps, he negral above only represens he connuous conrbuon o oal varance, ofen denoed [ ln, ln ] c. More deals on he mpac of jumps can be found n econ 3.4. 3.. Hedgng raeges & Prcng For ease of exposure, we assume n hs secon ha dvdends are zero and ha he underlyng prce process s a dffuson process. Moreover, le us assume ha raes are deermnsc. Le us nroduce some noaon: y, we denoe he non-dscouned spo prce process and by Ŝ we denoe he dscouned spo prce process, where refers o he deermnsc money marke accoun. I s mporan o noe ha [ ln,ln ] [ ln,ln ] when raes are deermnsc. Moreover, he connuy of Ŝ ogeher wh Io's formula yelds: ln Defne for all : u [ ln,ln ] d u for all. [ ln,ln ] ln π d u. We now explan how π, whch s closely relaed o he payoff of a varance swap, can be replcaed by connuous radng of he underlyng and cash accordng o a self-fnancng sraegy (V, φ, ψ, where V s he nal value of he sraegy, φ and ψ he quanes o be held n he underlyng and cash a me. he sraegy s sad o be self-fnancng because s mark-o-marke value V V ϕ ψ verfes: u 8
d d dv ψ ϕ A YOU NEED O KNOW AOU VARIANCE WAP (In oher words he change n value of he sraegy beween mes and d s compued as a mark-o-marke P&L: change n asse prce mulpled by he quany held a me. here s no addon or whdrawal of wealh. elf-fnancng sraegy One can verfy ha he followng choce for (V, φ, ψ s self-fnancng: u u d V ψ ϕ Le us pon ou a few mporan hngs: he self-fnancng sraegy only replcaes he ermnal payoff π bu does no replcae π for <. I s ndeed easy o see ha π V : u u u u d d V V π ψ ϕ However π > V for < : u u u u d d V π π < For he self-fnancng sraegy o be predcable (.e. for φ, ψ o be enrely deermned based solely on he nformaon avalable before me, he assumpon ha raes are deermnsc s crucal. Prcng Havng denfed a self-fnancng sraegy we can proceed o prce a varance swap by akng he rsk-neural expecaon of π / : ln ],ln [ln u u d E E E π snce s assumed o be marngale under he rsk-neural measure. Whence: Ŝ 9 JU WH E W E ln, A hs pon, should be noed ha hs represenaon s vald only as long as we assume ha he underlyng sock prce process s connuous and raes are deermnsc. As soon as
we devae from hs assumpon, addonal adjusmens have o be made. For furher deals n hs regard, see econs 3.3 and 3.4. Represenaon as a sum of pus and calls In he prevous paragraphs we showed ha he annualzed realzed varance can be replcaed wh a sac poson n a log conrac on he dscouned sock prce. However n general s no possble o rade log conracs. hus we need o oban an alernave represenaon for he prce of he varance swap usng sandard pu and call opons. For hs purpose, noe ha a wce dfferenable payoff f( can be re-wren as follows: f ( f ( F f ( F [( F ( F ] F f ( y( y dy F f ( y( y Here denoes he spo prce of he underlyng and F denoes he forward prce. For deals we refer o he Appendx n Carr-Madan (. Choosng f(y ln(y and akng expecaons yelds: dy A YOU NEED O KNOW AOU VARIANCE WAP E ln F Pu( y dy Call( y dy y y where y now denoes forward moneyness, and Pu(y or Call(y he prce of a vanlla pu or call exprng a me. Whence: E W y y, Pu( y dy Call( y he nerpreaon of hs formula s as follows: In case he sock prce process s a dffuson process, he annualzed realzed varance can be replcaed by an nfne sum of sac posons n pus and calls. Clearly, perfec replcaon s no possble snce opons for all srkes are no avalable. A more accurae represenaon would hus be a dscrezed verson of he above (see econ.3 for an example. 3.3. Impac of Dvdends When a sock pays a dvdend, arbrage consderaons show ha s prce should drop by he dvdend amoun. hs phenomenon resuls n a hgher varance when he sock prce s no adjused for dvdends, whch s mos ofen he case. From a modelng sandpon, here are hree sandard ways o approach dvdends: connuous dvdend yeld, dscree dvdend yeld, and dscree dollar dvdend. In he followng paragraphs we only focus on he frs wo cases: For connuous dvdend yeld, we consder he prce process: d ( r q d dw dy where r s a deermnsc neres rae, q s a deermnsc dvdend yeld, s eher deermnsc or sochasc. JU WH nce we assume zero dvdends n hs secon, we have F.
For dscree dvdend yeld, we consder he prce process: d r d dw d d j j j where r s a deermnsc neres rae, s eher deermnsc or sochasc, and d,, d M are M dscree connously compounded dvdend yelds pad a daes,, M. Connuous Monorng A connuous dvdend yeld has no mpac on varance when monorng s connuous. In hs regard, observe ha: where F W [ ln,ln ] [ ln,ln ], s he spo prce normalzed by he forward prce. hs s because he dvdend yeld q s assumed o be deermnsc. Hence, here s clearly no mpac due o connuous dvdends. he hedgng sraegy also remans he same. Nex, le us consder he mpac of dscree dvdends. In hs case he sock prce process follows: A YOU NEED O KNOW AOU VARIANCE WAP We now have: d r d dw d d j j W, j c [ ln,ln ] [ ln,ln ] Le us have a closer look a he hedgng sraegy n he conex of dscree dvdend yelds. G exp d where dvdends are j For hs purpose, defne he oal reurn process ( j d j renvesed n he sock. he dscouned oal reurn process G G / beng a marngale we can use a smlar hedgng sraegy as n econ 3. where he sock prce process s now replaced by G. Dscree Monorng Consder a se of samplng daes assume ha he me nervals < < L < N. For smplcy of presenaon, we are all consan and equal o. Recall he dscree defnon of annualzed varance whou mean: Varance N ln j JU WH Noe ha we consder here gross yelds raher han annualzed yelds n he dscree dvdend case.
Conrary o he connuous monorng case, a connuous dvdend yeld has an mpac on varance when monorng s dscree. Consder he log reurn beween - and : A YOU NEED O KNOW AOU VARIANCE WAP z q r ( ( ln where,, (,, ~ q q r r N z. quarng he above yelds: q r z z q r 3 / ( ( ln ecause he expecaon of z s nl and s varance E(z s one, we oban: q r E ( ( ln he relave mpac of dscree monorng on varance s hus: ( ( ln q r E A hs pon, should be noed ha even n he case where neres raes and dvdends are assumed o be zero, we oban some drf conrbuon n case of dscrezaon. hs s due o he erm n he numeraor. Moreover, for, he above expresson mples ha here s no conrbuon due o neres raes and connuous dvdend yelds as already poned ou n he connuous monorng case. We now specalze our consderaons o he case of dscree dvdends. Assumng ha a dscree dvdend d j s pad beween mes - and and carryng ou smlar calculaons as n he prevous paragraph yelds he followng expresson for he expecaon of he log reurn: d r E j ln As can be seen from hs equaon, he conrbuon of dscree dvdends does no converge o zero for. We also oban ha he relave conrbuon of he neres rae and he connuous dvdend yeld whn a me nerval amouns o: d r j JU WH Noe ha hs saemen s rue whn a deermnsc or sochasc volaly framework. In oher frameworks (such as local volaly volaly may depend on and would hus be mpaced by dvdends.
3.4. Impac of Jumps he purpose of hs secon s o analyze he mpac of jumps,.e. we no longer assume ha he sock prce process follows a dffuson process and nsead consder a jump dffuson process. For ease of exposure, we gnore neres raes and dvdends: or: d N µ d dw d ( Y n n d ln N µ d dw d ln( Y n n where W, N and Y are ndependen. W s a sandard rownan moon, N s a Posson process wh nensy λ and (Y n are ndependen, dencally dsrbued log-normal varables: A YOU NEED O KNOW AOU VARIANCE WAP dw ~ N(, d dn ~ P( λ d δgn Yn ( k e δ, G n ~ N(, Parameers k, λ, δ can be nerpreed as follows: k s he average jump sze, λ conrols he frequency of jumps, and δ s he jump sze uncerany (sandard devaon. Furhermore, he drf erm µ s chosen such ha s a marngale,.e.: k. We hen have for he annualzed realzed varance: W µ λ N C, [ln,ln ] [ln,ln ] ln ( Yn n And he expeced varance under he rsk-neural measure becomes: C E[ W, ] E[ W, ] λ ln( k δ δ JU WH 3
Appendx A A Revew of Hsorcal and Impled Volaly Hsorcal Volaly he volaly of a fnancal asse (e.g. a sock s he level of s prce uncerany, and s commonly measured by he sandard devaon of s reurns. For hsorcal daly reurns r, r,, r n, an esmae s gven as: Hsorcal 5 n n ( r r A YOU NEED O KNOW AOU VARIANCE WAP JU WH n where r r s he mean reurn, and 5 s an annualzaon facor correspondng o he n ypcal number of radng days n a year. Hsorcal volaly s also called realzed volaly n he conex of opon radng and varance swaps. Here s assumed ha he reurns were ndependen and drawn accordng o he same random law or dsrbuon n oher words, sock prces are beleved o follow a random walk. In hs case, he esmae s shown o be unbased wh vanshng error as he number of daly observaons n ncreases. he daly reurns are ypcally compued n logarhmc erms n he conex of opons o reman conssen wh lack-choles: P r ln P where P s he prce of he asse observed on day, and ln(. s he naural logarhm. Impled volaly Vanlla opons on a sock are worh more when volaly s hgher. Conrary o a common belef, hs s no because he opon has more chances of beng n-he-money, bu because he sock has more chances of beng hgher n-he-money, as llusraed n Exhb A. In a lack-choles world, volaly s he only parameer whch s lef o he apprecaon of he opon rader. All he oher parameers: srke, maury, neres rae, forward value, are deermned by he conrac specfcaons and he neres rae and fuures markes. hus, here s a one-o-one correspondence beween an opon s prce and he lack-choles volaly parameer. Impled volaly s he value of he parameer for whch he lack- choles heorecal prce maches he marke prce, as llusraed n Exhb A. ecause of pu-call pary, European calls and pus wh dencal characerscs (underlyng, srke, maury mus have he same mpled volaly. hs makes he dsncon beween volales mpled from call or pu prces rrelevan. In he case of Amercan opons, however, pu-call pary does no always hold, and he dsncon mgh be relevan. For each srke and maury here s a dfferen mpled volaly whch can be nerpreed as he marke s expecaon of fuure volaly beween oday and he maury dae n he scenaro mpled by he srke. For nsance, ou-of-he money pus are naural hedges agans a marke dslocaon (such as caused by he 9/ aacks on he World rade Cener whch enal a spke n volaly; he mpled volaly of ou-of-he money pus s hus hgher 4
han n-he-money pus. hs phenomenon s known as volaly skew, as hough he marke expecaons of uncerany were skewed owards he downsde. An example of a volaly surface s gven n Exhb A3. Exhb A mulaed payoffs of an a-he-money call when he fnal sock prce s log-normally dsrbued and he volaly s eher % or 4%. mulaed Payoff % vol 4% vol Dsrbuon of Fnal ock Prce Fnal sock prce (% nal prce A YOU NEED O KNOW AOU VARIANCE WAP % 5% % 5% % 5% 3% Exhb A lack-choles and Volaly: a volaly s an npu, b volaly s mpled po Prce rke Prce Maury Ineres Rae Volaly a b lack choles Opon Prce po Prce rke Prce Maury Ineres Rae Impled Volaly lack choles Opon Prce JU WH 5
Exhb A3 Volaly urface of Eurooxx 5 as of December 4 45% 4% 35% 3% 5% % 5% % 5% % 5 55 9 38 A YOU NEED O KNOW AOU VARIANCE WAP ource: JPMorgan. W 3M Y 7Y 54 JU WH 6
Appendx Relaonshp beween hea and Gamma An nuve approach Consder he reduced P&L equaon (Eq. from econ. Daly P&L Γ ( Θ ( (Eq. In a far game, he expeced daly P&L s nl. hs leaves us wh: [( ] Θ Γ E where E[.] denoes mahemacal expecaon 3. Wrng ( ( yelds: Θ Γ E (Eq. A YOU NEED O KNOW AOU VARIANCE WAP JU WH he quany ( s he squared daly reurn on he underlyng sock; akng expecaon gves he sock varance over one day:. (Remember ha mpled volaly s gven on an annual bass. Replacng he expeced squared reurn by s expresson and dvdng boh sdes of Equaon by fnally yelds: y he books Θ Γ. Consder he lack-choles-meron paral dfferenal equaon: rf f f f r (Eq. where f(, s he value of he dervave a me when he sock prce s, and r s he shor-erm neres rae. Equaon ( holds for all dervaves of he same underlyng sock, and by lneary of dfferenaon any poroflo Π of such dervaves. Idenfyng he Greek leer correspondng o each paral dervave, we can rewre Equaon as: rπ Θ r Γ In he case of a dela-hedged porfolo, we have, whence: Θ rπ Γ ecause he shor-erm rae s ypcally of he order of a few percenage pons, he frs erm on he rgh-hand sde s ofen neglgble, and we have he approxmae relaonshp: Θ Γ. 3 Here we acually deal wh condonal expecaon upon he nformaon avalable a a ceran pon n me. 7
Appendx C Peak Dollar Gamma When he neres rae s zero, he dollar gamma of a vanlla opon wh srke K, maury and mpled volaly s gven n funcon of he underlyng level as: Γ $ (, K (ln( / K.5 exp π In Exhb C below we can see ha he dollar gamma has a bell-shaped curve whch peaks slghly afer he srke. I can ndeed be shown ha he peak s reached when s equal o: * Ke / Exhb C Gamma and Dollar gamma of an a-he-money European vanlla Gamma A YOU NEED O KNOW AOU VARIANCE WAP * Dollar Gamma 5 5 5 JU WH 8
References & blography Allen, Harrs (, Volaly Vehcles, JPMorgan Equy Dervaves raegy Produc Noe. Demeerf, Derman, Kamal, Zou (999, More han You Ever Waned o Know Abou Volaly waps, Goldman achs Quanave raeges Research Noes. Carr, Madan (998, owards a heory of Volaly radng n VOLAILIY, R.A. Jarrow Rsk ooks. Gaheral (, Case udes n Fnancal Modellng Fall, NYU Couran Insue. Hull (, Opons, Fuures & Oher Dervaves 4h edon, Prence Hall. Musela, Rukowsk (997, Marngale Mehods n Fnancal Modellng, prnger. Vallan (, A egnner's Gude o Cred Dervaves, Workng Paper, Nomura Inernaonal. Wlmo (998, Dscree Hedgng n he heory and Pracce of Fnancal Engneerng, Wley. JU WHA YOU NEED O KNOW AOU VARIANCE WAP 9