JUST WHAT YOU NEED TO KNOW ABOUT VARIANCE SWAPS


 Stewart Dean
 3 years ago
 Views:
Transcription
1 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
2 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 LevyMarchal, 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:// 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:// 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 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.
3 able of Conens Overvew... able of Conens.... Varance waps 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. Markomarke and ensves 8 Markomarke 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: DelaHedgng and P&L PahDependency DelaHedgng P&L pahdependency.. ac Replcaon of Varance waps 4 Inerpreaon 6.3. Valuaon 6 3. heorecal Insghs Idealzed Defnon of Varance Hedgng raeges & Prcng 8 elffnancng 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
4 . 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 Exchangeraded Conrac 3
5 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 moneymarke rae, whle s omsson has he benef of makng he payoff perfecly addve (3monh varance 9monh 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 Volaly Realzed Volaly JU WH Readers wh a mahemacal background wll recall Jensen s nequaly: E( Varance E( Varance. 4
6 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 ahemoneyforward 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 loglnear skew, smlar echnques gve he rule of humb: K β 4 4 ( 3 var AMF β AMF AMF 5 where AMF s he ahemoneyforward 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 nonseep 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 delahedged 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 pahdependency.] 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
7 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 : { {.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 forwardsar 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 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 3monh DAX varance and shor 3monh 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
8 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 oneyear 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
9 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 veganeural a ncepon..3. Markomarke and ensves Markomarke ecause varance s addve n me dmenson he markomarke 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 oneyear varance swap ssued 3 monhs ago on a vega noonal of $,, sruck a. he 9monh zerorae s %, realzed volaly over he pas 3 monhs 8
10 was 5, and a 9monh varance swap would srke oday a 9. he markomarke of he oneyear varance swap would be:, 3 Varwap ( % 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 markomarke addvy: Varwap Vega Noonal ( mpled mpled Noe ha Vega s equal o a ncepon f he srke s far and he noonal s vegaadjused: 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 oneyear varance swap on a vega noonal of $,, sruck a 5. Ahemoneyforward 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 markomarke would be: Dvdend sensvy MM, $, 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 exdvdend annualzed varance should be JU WH 9
11 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 oneyear varance swap on a vega noonal of $, sruck a. he far srke exdvdend s and he annual dvdend yeld s 5%, pad semannually. he adjused srke s hus ( 5 /.5.3. Were he dvdend yeld o ncrease o 5.5% he change n markomarke would be: MM 5/, Dv Yeld skew sensvy var ( $, 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
12 . Valuaon and Hedgng n Pracce.. Vanlla Opons: DelaHedgng and P&L PahDependency DelaHedgng 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 delaneuralzed 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 delahedgng. 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 delaneural 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 hghorder 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 lackcholes, 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 delahedged opon poson s essenally drven by realzed and mpled volaly. JU WH 7 Noe ha n lackcholes volaly s assumed o reman consan hrough me. he concep of Vega s hus nconssen wh he heory, ye crcal n pracce.
13 We sar wh he wellknown 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 lackcholes: 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 oneday sock reurn. quared, can be nerpreed as he realzed oneday 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 delahedged 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 pahdependency 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 pahdependency 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
14 pahdependency 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.. Pahdependency 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 delaheged 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% % % ock Prce (Inal 4% % % 8% 6% 4% % % ock Prce 5 3 % radng P&L 9 5 rke 35 5day Realzed 9 Volaly % % Dollar Gamma , 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 chsquare dsrbuon, whle Allen Harrs nclude a bellshaped char of he dsrbuon of fnal P&Ls of a dscreely delahedged opon poson. Neglecng he gamma dependence, he cenrallm heorem ndeed shows ha he sum of n ndependen chsquare varables converges o a normal dsrbuon. 3
15 .. ac Replcaon of Varance waps In he prevous paragraph we saw ha a vanlla opon rader followng a delahedgng 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 ( nongammaweghed 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 lowsrke opons o he aggregae gamma s small compared o hghsrke opons. herefore, a naural dea s o ncrease he weghs of lowsrke opons and decrease he weghs of hghsrke 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 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 closedform 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 nonconsan (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 JU WH 9 Recall ha dollar gamma s defned as he secondorder 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
16 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 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 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
17 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 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 secondorder 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 logasse (a dervave whch pays off he logprce 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 ouofhe money pus and N calls ouofhemoney calls respecvely, a quck proxy for he far value of a varance swap of maury s gven as: 6
18 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, For a srke of 6.65 volaly pons, and a year presen value facor of , he fxed leg s worh,7, 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., % Y 7.6%.4% 6.53% X5E P,935., % Y 6.4%.8% 3.89% X5E P,935.,76. 6% Y 5.%.5%.83% X5E P,935., % 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., % Y 8.7%.79% 6.7% X5E P,935.,64.5 9% Y 7.3%.67% 5.54% X5E P,935., % 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, % Y.%.8% 3.47% X5E C,935. 3,5. % Y.9%.35% 3.% X5E C,935. 3, % Y.8%.5%.96% X5E C,935. 3,85.5 3% Y.9%.6%.74% X5E C,935. 3, % Y.%.3%.55% X5E C,935. 4,9. 4% Y.5%.%.38% X5E C,935. 4, % Y.9%.%.% X5E C,935. 4,4.5 5% Y 3.4%.% ource: JPMorgan. 7
19 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 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 nondscouned 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 selffnancng 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 selffnancng because s markomarke value V V ϕ ψ verfes: u 8
20 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 markomarke P&L: change n asse prce mulpled by he quany held a me. here s no addon or whdrawal of wealh. elffnancng sraegy One can verfy ha he followng choce for (V, φ, ψ s selffnancng: u u d V ψ ϕ Le us pon ou a few mporan hngs: he selffnancng 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 selffnancng 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 selffnancng sraegy we can proceed o prce a varance swap by akng he rskneural expecaon of π / : ln ],ln [ln u u d E E E π snce s assumed o be marngale under he rskneural 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
Methodology of the CBOE S&P 500 PutWrite Index (PUT SM ) (with supplemental information regarding the CBOE S&P 500 PutWrite TW Index (PWT SM ))
ehodology of he CBOE S&P 500 PuWre Index (PUT S ) (wh supplemenal nformaon regardng he CBOE S&P 500 PuWre TW Index (PWT S )) The CBOE S&P 500 PuWre Index (cker symbol PUT ) racks he value of a passve
More informationPricing and Valuation of Forward and Futures
Prcng and Valuaon of orward and uures. Cashandcarry arbrage he prce of he forward conrac s relaed o he spo prce of he underlyng asse, he rskfree rae, he dae of expraon, and any expeced cash dsrbuons
More informationA New Approach For Modelling & Pricing Correlation Swaps in Equity Derivatives
9 A MAY ew 006 Approach For Modellng & Prcng Correlaon Swaps n Equy Dervaves A ew Approach For Modellng & Prcng Correlaon Swaps n Equy Dervaves GLOBAL DERIVATIVES TRADIG & RISK MAAGEMET 006 ICBI h Annual
More informationGUIDANCE STATEMENT ON CALCULATION METHODOLOGY
GUIDANCE STATEMENT ON CALCULATION METHODOLOGY Adopon Dae: 9/28/0 Effecve Dae: //20 Reroacve Applcaon: No Requred www.gpssandards.org 204 CFA Insue Gudance Saemen on Calculaon Mehodology GIPS GUIDANCE STATEMENT
More informationMORE ON TVM, "SIX FUNCTIONS OF A DOLLAR", FINANCIAL MECHANICS. Copyright 2004, S. Malpezzi
MORE ON VM, "SIX FUNCIONS OF A DOLLAR", FINANCIAL MECHANICS Copyrgh 2004, S. Malpezz I wan everyone o be very clear on boh he "rees" (our basc fnancal funcons) and he "fores" (he dea of he cash flow model).
More information12/7/2011. Procedures to be Covered. Time Series Analysis Using Statgraphics Centurion. Time Series Analysis. Example #1 U.S.
Tme Seres Analyss Usng Sagraphcs Cenuron Nel W. Polhemus, CTO, SaPon Technologes, Inc. Procedures o be Covered Descrpve Mehods (me sequence plos, auocorrelaon funcons, perodograms) Smoohng Seasonal Decomposon
More informationSelected Financial Formulae. Basic Time Value Formulae PV A FV A. FV Ad
Basc Tme Value e Fuure Value of a Sngle Sum PV( + Presen Value of a Sngle Sum PV  ( + Solve for for a Sngle Sum ln  PV  ln( + Solve for for a Sngle Sum 
More informationFinance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C.
Fnance and Economcs Dscusson Seres Dvsons of Research & Sascs and Moneary Affars Federal Reserve Board, Washngon, D.C. Prcng Counerpary Rs a he Trade Level and CVA Allocaons Mchael Pyhn and Dan Rosen 2000
More informationIndex Mathematics Methodology
Index Mahemacs Mehodology S&P Dow Jones Indces: Index Mehodology Ocober 2015 Table of Conens Inroducon 4 Dfferen Varees of Indces 4 The Index Dvsor 5 Capalzaon Weghed Indces 6 Defnon 6 Adjusmens o Share
More informationGuidelines and Specification for the Construction and Maintenance of the. NASDAQ OMX Credit SEK Indexes
Gudelnes and Specfcaon for he Consrucon and Manenance of he NASDAQ OMX Cred SEK Indexes Verson as of Aprl 7h 2014 Conens Rules for he Consrucon and Manenance of he NASDAQ OMX Cred SEK Index seres... 3
More informationProt sharing: a stochastic control approach.
Pro sharng: a sochasc conrol approach. Donaen Hanau Aprl 2, 2009 ESC Rennes. 35065 Rennes, France. Absrac A majory of lfe nsurance conracs encompass a guaraneed neres rae and a parcpaon o earnngs of he
More informationPrices of Credit Default Swaps and the Term Structure of Credit Risk
Prces of Cred Defaul Swaps and he Term Srucure of Cred Rsk by Mary Elzabeh Desrosers A Professonal Maser s Projec Submed o he Faculy of he WORCESTER POLYTECHNIC INSTITUTE n paral fulfllmen of he requremens
More informationThe Rules of the Settlement Guarantee Fund. 1. These Rules, hereinafter referred to as "the Rules", define the procedures for the formation
Vald as of May 31, 2010 The Rules of he Selemen Guaranee Fund 1 1. These Rules, herenafer referred o as "he Rules", defne he procedures for he formaon and use of he Selemen Guaranee Fund, as defned n Arcle
More informationEstimating intrinsic currency values
Cung edge Foregn exchange Esmang nrnsc currency values Forex marke praconers consanly alk abou he srenghenng or weakenng of ndvdual currences. In hs arcle, Jan Chen and Paul Dous presen a new mehodology
More informationGround rules. Guide to the calculation methods of the FTSE Actuaries UK Gilts Index Series v1.9
Ground rules Gude o he calculaon mehods of he FTSE Acuares UK Gls Index Seres v1.9 fserussell.com Ocober 2015 Conens 1.0 Inroducon... 4 1.1 Scope... 4 1.2 FTSE Russell... 5 1.3 Overvew of he calculaons...
More informationPricing Rainbow Options
Prcng Ranbow Opons Peer Ouwehand, Deparmen of Mahemacs and Appled Mahemacs, Unversy of Cape Town, Souh Afrca Emal address: peer@mahs.uc.ac.za Graeme Wes, School of Compuaonal & Appled Mahemacs, Unversy
More informationThe Joint Cross Section of Stocks and Options *
The Jon Cross Secon of Socks and Opons * Andrew Ang Columba Unversy and NBER Turan G. Bal Baruch College, CUNY Nusre Cakc Fordham Unversy Ths Verson: 1 March 2010 Keywords: mpled volaly, rsk premums, reurn
More informationSpline. Computer Graphics. Bsplines. BSplines (for basis splines) Generating a curve. Basis Functions. Lecture 14 Curves and Surfaces II
Lecure 4 Curves and Surfaces II Splne A long flexble srps of meal used by drafspersons o lay ou he surfaces of arplanes, cars and shps Ducks weghs aached o he splnes were used o pull he splne n dfferen
More informationTHE USE IN BANKS OF VALUE AT RISK METHOD IN MARKET RISK MANAGEMENT. Ioan TRENCA *
ANALELE ŞTIINłIFICE ALE UNIVERSITĂłII ALEXANDRU IOAN CUZA DIN IAŞI Tomul LVI ŞnŃe Economce 009 THE USE IN BANKS OF VALUE AT RISK METHOD IN MARKET RISK MANAGEMENT Ioan TRENCA * Absrac In sophscaed marke
More informationThe US Dollar Index Futures Contract
The S Dollar Inde uures Conrac I. Inroducon The S Dollar Inde uures Conrac Redfeld (986 and Eyan, Harpaz, and Krull (988 presen descrpons and prcng models for he S dollar nde (SDX fuures conrac. Ths arcle
More informationEvaluation of the Stochastic Modelling on Options
Zhjuan Mao, Zhan Lang, Jnguo Lan, Hongkun Zhang / Inernaonal Journal of Engneerng Research and Applcaons (IJERA) ISSN: 4896 www.jera.com Vol., Issue 3, MayJun 0, pp.463473 Evaluaon of he Sochasc Modellng
More informationTesting techniques and forecasting ability of FX Options Implied Risk Neutral Densities. Oren Tapiero
Tesng echnques and forecasng ably of FX Opons Impled Rsk Neural Denses Oren Tapero 1 Table of Conens Absrac 3 Inroducon 4 I. The Daa 7 1. Opon Selecon Crerons 7. Use of mpled spo raes nsead of quoed spo
More informationManaging gap risks in icppi for life insurance companies: a risk return cost analysis
Insurance Mares and Companes: Analyses and Acuaral Compuaons, Volume 5, Issue 2, 204 Aymerc Kalfe (France), Ludovc Goudenege (France), aad Mou (France) Managng gap rss n CPPI for lfe nsurance companes:
More informationFixed Income Attribution. Remco van Eeuwijk, Managing Director Wilshire Associates Incorporated 15 February 2006
Fxed Incoe Arbuon eco van Eeuwk Managng Drecor Wlshre Assocaes Incorporaed 5 February 2006 Agenda Inroducon Goal of Perforance Arbuon Invesen Processes and Arbuon Mehodologes Facorbased Perforance Arbuon
More informationMathematical Model of Data Backup and Recovery
6 IJCSNS Inernaonal Journal of Compuer Scence and Nework Secury, VOL.4 No.7, July 04 Mahemacal Model of Daa Backup and Recovery Karel Burda The Faculy of Elecrcal Engneerng and Communcaon Brno Unversy
More informationInsurance. By Mark Dorfman, Alexander Kling, and Jochen Russ. Abstract
he Impac Of Deflaon On Insurance Companes Offerng Parcpang fe Insurance y Mar Dorfman, lexander Klng, and Jochen Russ bsrac We presen a smple model n whch he mpac of a deflaonary economy on lfe nsurers
More informationAnalyzing Energy Use with Decomposition Methods
nalyzng nergy Use wh Decomposon Mehods eve HNN nergy Technology Polcy Dvson eve.henen@ea.org nergy Tranng Week Pars 1 h prl 213 OCD/ 213 Dscusson nergy consumpon and energy effcency? How can energy consumpon
More informationBoth human traders and algorithmic
Shuhao Chen s a Ph.D. canddae n sascs a Rugers Unversy n Pscaaway, NJ. bhmchen@sa.rugers.edu Rong Chen s a professor of Rugers Unversy n Pscaaway, NJ and Peng Unversy, n Bejng, Chna. rongchen@sa.rugers.edu
More informationHow Much Life Insurance is Enough?
How Much Lfe Insurance s Enough? UlyBased pproach By LJ Rossouw BSTRCT The paper ams o nvesgae how much lfe nsurance proecon cover a uly maxmsng ndvdual should buy. Ths queson s relevan n he nsurance
More informationIMES DISCUSSION PAPER SERIES
IMS DISCUSSION PPR SRIS Rsk Managemen for quy Porfolos of Japanese Banks kra ID and Toshkazu OHB Dscusson Paper No. 989 INSTITUT FOR MONTRY ND CONOMIC STUDIS BNK OF JPN C.P.O BOX 23 TOKYO 1863 JPN NOT:
More informationKalman filtering as a performance monitoring technique for a propensity scorecard
Kalman flerng as a performance monorng echnque for a propensy scorecard Kaarzyna Bjak * Unversy of Souhampon, Souhampon, UK, and Buro Informacj Kredyowej S.A., Warsaw, Poland Absrac Propensy scorecards
More informationThe Feedback from Stock Prices to Credit Spreads
Appled Fnance Projec Ka Fa Law (Keh) The Feedback from Sock Prces o Cred Spreads Maser n Fnancal Engneerng Program BA 3N Appled Fnance Projec Ka Fa Law (Keh) Appled Fnance Projec Ka Fa Law (Keh). Inroducon
More informationCapacity Planning. Operations Planning
Operaons Plannng Capacy Plannng Sales and Operaons Plannng Forecasng Capacy plannng Invenory opmzaon How much capacy assgned o each producon un? Realsc capacy esmaes Sraegc level Moderaely long me horzon
More informationBest estimate calculations of saving contracts by closed formulas Application to the ORSA
Bes esmae calculaons of savng conracs by closed formulas Applcaon o he ORSA  Franços BONNIN (Ala)  Frédérc LANCHE (Unversé Lyon 1, Laboraore SAF)  Marc JUILLARD (Wner & Assocés) 01.5 (verson modfée
More informationINTERNATIONAL JOURNAL OF STRATEGIC MANAGEMENT
IJSM, Volume, Number, 0 ISSN: 5554 INTERNATIONAL JOURNAL OF STRATEGIC MANAGEMENT SPONSORED BY: Angelo Sae Unversy San Angelo, Texas, USA www.angelo.edu Managng Edors: Professor Alan S. Khade, Ph.D. Calforna
More informationStructural jumpdiffusion model for pricing collateralized debt obligations tranches
Appl. Mah. J. Chnese Unv. 010, 54): 4048 Srucural jumpdffuson model for prcng collaeralzed deb oblgaons ranches YANG Rucheng Absrac. Ths paper consders he prcng problem of collaeralzed deb oblgaons
More informationRevision: June 12, 2010 215 E Main Suite D Pullman, WA 99163 (509) 334 6306 Voice and Fax
.3: Inucors Reson: June, 5 E Man Sue D Pullman, WA 9963 59 334 636 Voce an Fax Oerew We connue our suy of energy sorage elemens wh a scusson of nucors. Inucors, lke ressors an capacors, are passe woermnal
More informationfinish line dr 1 L 1 v b v r finish line.
Answer, Key { Homewor { Rubn H Landau 1 Ths prnou should have 1 quesons. Chec ha s complee before leavng he prner. Also, mulplechoce quesons may connue on he nex column or page: nd all choces before
More informationP R = P 0. The system is shown on the next figure:
TPG460 Reservor Smulaon 06 page of INTRODUCTION TO RESERVOIR SIMULATION Analycal and numercal soluons of smple onedmensonal, onephase flow equaons As an nroducon o reservor smulaon, we wll revew he smples
More informationPerformance Measurement for Traditional Investment
E D H E C I S K A N D A S S E T M A N A G E M E N T E S E A C H C E N T E erformance Measuremen for Tradonal Invesmen Leraure Survey January 007 Véronque Le Sourd Senor esearch Engneer a he EDHEC sk and
More informationThe Cause of ShortTerm Momentum Strategies in Stock Market: Evidence from Taiwan
he Cause of Shorerm Momenum Sraeges n Sock Marke: Evdence from awan HungChh Wang 1, Y. Angela Lu 2, and ChunHua Susan Ln 3+ 1 B. A. Dep.,C C U, and B. A. Dep., awan Shoufu Unversy, awan (.O.C. 2 Dep.
More informationA New Method to Evaluate EquityLinked Life Insurance
Coneporary Manageen Research Pages , Vol. 0, No., March 04 do:0.790/cr.097 A New Mehod o Evaluae EquyLnked Lfe Insurance MngShann Tsa Naonal Unversy of Kaohsung EMal: ssa@nuk.edu.w ShhCheng Lee YuanZe
More informationLinear Extension Cube Attack on Stream Ciphers Abstract: Keywords: 1. Introduction
Lnear Exenson Cube Aack on Sream Cphers Lren Dng Yongjuan Wang Zhufeng L (Language Engneerng Deparmen, Luo yang Unversy for Foregn Language, Luo yang cy, He nan Provnce, 47003, P. R. Chna) Absrac: Basng
More informationWho are the sentiment traders? Evidence from the crosssection of stock returns and demand. April 26, 2014. Luke DeVault. Richard Sias.
Who are he senmen raders? Evdence from he crosssecon of sock reurns and demand Aprl 26 2014 Luke DeVaul Rchard Sas and Laura Sarks ABSTRACT Recen work suggess ha senmen raders shf from less volale o speculave
More informationLecture 40 Induction. Review Inductors Selfinduction RL circuits Energy stored in a Magnetic Field
ecure 4 nducon evew nducors Selfnducon crcus nergy sored n a Magnec Feld 1 evew nducon end nergy Transfers mf Bv Mechancal energy ransform n elecrc and hen n hermal energy P Fv B v evew eformulaon of
More informationTHOMSON REUTERS/CORECOMMODITY CRB INDEX CALCULATION SUPPLEMENT
THOMSON REUTERS/CORECOMMODITY CRB INDEX CALCULATION SUPPLEMENT SEPTEMBER 2013 Thomson Reuers/CoreCommody CRB Index Calculaon Supplemen Ths supplemen conans he rules for calculang he Thomson Reuers/CoreCommody
More informationA GENERALIZED FRAMEWORK FOR CREDIT RISK PORTFOLIO MODELS
A GENERALIZED FRAMEWORK FOR CREDIT RISK PORTFOLIO MODELS H. UGUR KOYLUOGLU ANDREW HICKMAN Olver, Wyman & Company CSFP Capal, Inc. * 666 Ffh Avenue Eleven Madson Avenue New Yor, New Yor 10103 New Yor, New
More informationStatic replication of barrier options: some general results
Arcle 7/6/ 3: pm Page ac replcaon of barrer opons: some general resuls Lef B. G. Andersen Managng Drecor, Banc of Amerca ecures, 9 Wes 57h ree, New York, NY 9, UA Jesper Andreasen Drecor, Nordea Markes,
More informationFINANCIAL CONSTRAINTS, THE USER COST OF CAPITAL AND CORPORATE INVESTMENT IN AUSTRALIA
FINANCIAL CONSTRAINTS THE USER COST OF CAPITAL AND CORPORATE INVESTMENT IN AUSTRALIA Gann La Cava Research Dscusson Paper 20052 December 2005 Economc Analyss Reserve Bank of Ausrala The auhor would lke
More informationAn Architecture to Support Distributed Data Mining Services in ECommerce Environments
An Archecure o Suppor Dsrbued Daa Mnng Servces n ECommerce Envronmens S. Krshnaswamy 1, A. Zaslavsky 1, S.W. Loke 2 School of Compuer Scence & Sofware Engneerng, Monash Unversy 1 900 Dandenong Road, Caulfeld
More informationSearching for a Common Factor. in Public and Private Real Estate Returns
Searchng for a Common Facor n Publc and Prvae Real Esae Reurns Andrew Ang, * Nel Nabar, and Samuel Wald Absrac We nroduce a mehodology o esmae common real esae reurns and cycles across publc and prvae
More informationDEPARTMENT OF ECONOMETRICS AND BUSINESS STATISTICS. Exponential Smoothing for Inventory Control: Means and Variances of LeadTime Demand
ISSN 44077X ISBN 0 736 094 X AUSTRALIA DEPARTMENT OF ECONOMETRICS AND BUSINESS STATISTICS Exponenal Smoohng for Invenory Conrol: Means and Varances of LeadTme Demand Ralph D. Snyder, Anne B. Koehler,
More informationGround rules. FTSE Global Bonds Index Series v1.7
Ground rules FTSE Global Bonds Index Seres v.7 fserussell.com Ocober 205 Conens.0 Inroducon... 3 2.0 Managemen responsbles... 7 3.0 Elgble of secures... 9 4.0 rce sources... 5.0 erodc Change o he orfolos...
More informationCombining Mean Reversion and Momentum Trading Strategies in. Foreign Exchange Markets
Combnng Mean Reverson and Momenum Tradng Sraeges n Foregn Exchange Markes Alna F. Serban * Deparmen of Economcs, Wes Vrgna Unversy Morganown WV, 26506 November 2009 Absrac The leraure on equy markes documens
More informationThe performance of imbalancebased trading strategy on tender offer announcement day
Invesmen Managemen and Fnancal Innovaons, Volume, Issue 2, 24 HanChng Huang (awan), YongChern Su (awan), YChun Lu (awan) he performance of mbalancebased radng sraegy on ender offer announcemen day
More informationY2K* Stephanie SchmittGrohé. Rutgers Uni ersity, 75 Hamilton Street, New Brunswick, New Jersey 08901 Email: grohe@econ.rutgers.edu.
Revew of Economc Dynamcs 2, 850856 Ž 1999. Arcle ID redy.1999.0065, avalable onlne a hp:www.dealbrary.com on Y2K* Sephane SchmGrohé Rugers Unersy, 75 Hamlon Sree, New Brunswc, New Jersey 08901 Emal:
More informationInformationbased trading, price impact of trades, and trade autocorrelation
Informaonbased radng, prce mpac of rades, and rade auocorrelaon Kee H. Chung a,, Mngsheng L b, Thomas H. McInsh c a Sae Unversy of New York (SUNY) a Buffalo, Buffalo, NY 426, USA b Unversy of Lousana
More informationExpirationday effects, settlement mechanism, and market structure: an empirical examination of Taiwan futures exchange
Invesmen Managemen and Fnancal Innovaons, Volume 8, Issue 1, 2011 ChaCheng Chen (Tawan), SuWen Kuo (Tawan), ChnSheng Huang (Tawan) Expraonday effecs, selemen mechansm, and marke srucure: an emprcal
More informationPayout Policy Choices and Shareholder Investment Horizons
Payou Polcy Choces and Shareholder Invesmen Horzons JoséMguel Gaspar* Massmo Massa** Pedro Maos*** Rajdeep Pagr Zahd Rehman Absrac Ths paper examnes how shareholder nvesmen horzons nfluence payou polcy
More informationPedro M. Castro Iiro Harjunkoski Ignacio E. Grossmann. Lisbon, Portugal Ladenburg, Germany Pittsburgh, USA
Pedro M. Casro Iro Harjunkosk Ignaco E. Grossmann Lsbon Porugal Ladenburg Germany Psburgh USA 1 Process operaons are ofen subjec o energy consrans Heang and coolng ules elecrcal power Avalably Prce Challengng
More informationInternational Portfolio Equilibrium and the Current Account
Inernaonal Porfolo Equlbrum and he Curren Accoun Rober Kollmann (*) ECARE Free Unversy of Brussels Unversy of Pars XII Cenre for Economc Polcy Research UK Ocober 006 Ths paper analyzes he deermnans of
More informationAttribution Strategies and Return on Keyword Investment in Paid Search Advertising
Arbuon Sraeges and Reurn on Keyword Invesmen n Pad Search Adversng by Hongshuang (Alce) L, P. K. Kannan, Sva Vswanahan and Abhshek Pan * December 15, 2015 * Honshuang (Alce) L s Asssan Professor of Markeng,
More informationNo. 322009. David Büttner and Bernd Hayo. Determinants of European Stock Market Integration
MAGKS Aachen Segen Marburg Geßen Göngen Kassel Jon Dscusson Paper Seres n Economcs by he Unverses of Aachen Geßen Göngen Kassel Marburg Segen ISSN 18673678 No. 322009 Davd Büner and Bernd Hayo Deermnans
More informationPricing of Arithmetic Asian QuantoBasket Options
Helsnk Unversy of Technology Faculy of Inforaon and Naural cences Deparen of Maheacs and yses Analyss Ma.8 Independen research proecs n appled aheacs Prcng of Arhec Asan QuanoBaske Opons Tana Eronen
More informationAPPLICATION OF CHAOS THEORY TO ANALYSIS OF COMPUTER NETWORK TRAFFIC Liudvikas Kaklauskas, Leonidas Sakalauskas
The XIII Inernaonal Conference Appled Sochasc Models and Daa Analyss (ASMDA2009) June 30July 3 2009 Vlnus LITHUANIA ISBN 9789955284635 L. Sakalauskas C. Skadas and E. K. Zavadskas (Eds.): ASMDA2009
More informationIntegrating credit and interest rate risk: A theoretical framework and an application to banks' balance sheets
Inegrang cred and neres rae rsk: A heorecal framework and an applcaon o banks' balance shees Mahas Drehmann* Seffen Sorensen** Marco Srnga*** Frs draf: Aprl 26 Ths draf: June 26 Cred and neres rae rsk
More informationSocial security, education, retirement and growth*
Hacenda P úblca Espa ñola / Revsa de Econom ía P úblca, 198(3/2011): 936 2011, Insuo de Esudos Fscales Socal secury, educaon, reremen and growh* CRUZ A. ECHEVARR ÍA AMAIA IZA** Unversdad del Pa ís Vasco
More informationIndex Mathematics Methodology
Index Mahemacs Mehodology S&P Dow Jones Indces: Index Mehodology Augus 2016 Table of Conens Inroducon 4 Dfferen Varees of Indces 4 The Index Dvsor 4 Capalzaon Weghed Indces 6 Defnon 6 Adjusmens o Share
More informationHEAT CONDUCTION PROBLEM IN A TWOLAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD
Journal of Appled Mahemacs and Compuaonal Mechancs 3, (), 455 HEAT CONDUCTION PROBLEM IN A TWOLAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD Sansław Kukla, Urszula Sedlecka Insue of Mahemacs,
More informationWhat Explains Superior Retail Performance?
Wha Explans Superor Real Performance? Vshal Gaur, Marshall Fsher, Ananh Raman The Wharon School, Unversy of Pennsylvana vshal@grace.wharon.upenn.edu fsher@wharon.upenn.edu Harvard Busness School araman@hbs.edu
More informationReturn Persistence, Risk Dynamics and Momentum Exposures of Equity and Bond Mutual Funds
Reurn Perssence, Rsk Dynamcs and Momenum Exposures of Equy and Bond Muual Funds Joop Hu, Marn Marens, and Therry Pos Ths Verson: 2222008 Absrac To analyze perssence n muual fund performance, s common
More informationNetwork Effects on Standard Software Markets: A Simulation Model to examine Pricing Strategies
Nework Effecs on Sandard Sofware Markes Page Nework Effecs on Sandard Sofware Markes: A Smulaon Model o examne Prcng Sraeges Peer Buxmann Absrac Ths paper examnes sraeges of sandard sofware vendors, n
More informationThe Definition and Measurement of Productivity* Mark Rogers
The Defnon and Measuremen of Producvy* Mark Rogers Melbourne Insue of Appled Economc and Socal Research The Unversy of Melbourne Melbourne Insue Workng Paper No. 9/98 ISSN 13284991 ISBN 0 7325 0912 6
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationThe Virtual Machine Resource Allocation based on Service Features in Cloud Computing Environment
Send Orders for Reprns o reprns@benhamscence.ae The Open Cybernecs & Sysemcs Journal, 2015, 9, 639647 639 Open Access The Vrual Machne Resource Allocaon based on Servce Feaures n Cloud Compung Envronmen
More informationA STUDY ON THE CAUSAL RELATIONSHIP BETWEEN RELATIVE EQUITY PERFORMANCE AND THE EXCHANGE RATE
A STUDY ON THE CAUSAL RELATIONSHIP BETWEEN RELATIVE EQUITY PERFORMANCE AND THE EXCHANGE RATE The Swedsh Case Phlp Barsk* and Magnus Cederlöf Maser s Thess n Inernaonal Economcs Sockholm School of Economcs
More informationNikkei Stock Average Volatility Index Realtime Version Index Guidebook
Nikkei Sock Average Volailiy Index Realime Version Index Guidebook Nikkei Inc. Wih he modificaion of he mehodology of he Nikkei Sock Average Volailiy Index as Nikkei Inc. (Nikkei) sars calculaing and
More informationJournal of Econometrics
Journal of Economercs 7 ( 7 4 Conens lss avalable a ScVerse ScenceDrec Journal of Economercs ournal homepage: www.elsever.com/locae/econom Inernaonal mare lns and volaly ransmsson Valenna Corrad a,, Waler
More informationMarketWide ShortSelling Restrictions
MarkeWde ShorSellng Resrcons Anchada Charoenrook and Hazem Daouk + Ths verson: Augus 2005 Absrac In hs paper we examne he effec of markewde shorsale resrcons on skewness volaly probably of marke crashes
More informationModèles financiers en temps continu
Modèles fnancers en emps connu Inroducon o dervave prcng by Mone Carlo 204204 Dervave Prcng by Mone Carlo We consder a conngen clam of maury T e.g. an equy opon on one or several underlyng asses, whn
More informationMATURITY AND VOLATILITY EFFECTS ON SMILES
5// MATURITY AND VOLATILITY EFFECTS ON SMILES Or Dyng Smlng? João L. C. Dqe Unversdade Técnca de Lsboa  Inso Speror de Economa e Gesão Ra Mgel Lp,, LISBOA, PORTUGAL Paríca Texera Lopes Unversdade do Poro
More informationProceedings of the Annual Meeting of the American Statistical Association, August 59, 2001
Proceedngs of he Annual Meeng of he Amercan Sascal Assocaon, Augus 59, 00 SHELF LIFE ESTIMATION FOR DRUG PRODUCTS WITH TWO COMPONENTS Annpey Pong and Damaraju Raghavarao Bosascs, Novars Pharmaceucals
More informationPavel V. Shevchenko Quantitative Risk Management. CSIRO Mathematical & Information Sciences. Bridging to Finance
Pavel V. Shevchenko Quanave Rsk Managemen CSIRO Mahemacal & Informaon Scences Brdgng o Fnance Conference Quanave Mehods n Invesmen and Rsk Managemen: sourcng new approaches from mahemacal heory and he
More informationSwiss National Bank Working Papers
0110 Swss Naonal Bank Workng Papers Global and counryspecfc busness cycle rsk n mevaryng excess reurns on asse markes Thomas Nschka The vews expressed n hs paper are hose of he auhor(s and do no necessarly
More informationDeveloping a Risk Adjusted Pool Price in Ireland s New Gross Mandatory Pool Electricity Market
1 Developng a Rsk Adjused Pool Prce n Ireland s New Gross Mandaory Pool Elecrcy Marke Déaglán Ó Dónáll and Paul Conlon Absrac The Sngle Elecrcy Marke (SEM) Programme, whch esablshed for he frs me a gross
More informationLong Run Underperformance of Seasoned Equity Offerings: Fact or an Illusion?
Long Run Underperformance of Seasoned Equy Offerngs: Fac or an Illuson? 1 2 Allen D.E. and V. Souck 1 Edh Cowan Unversy, 2 Unversy of Wesern Ausrala, EMal: d.allen@ecu.edu.au Keywords: Seasoned Equy Issues,
More informationDividend Modeling. Forschungsseminar Stochastische Analysis und Stochastik der Finanzmärkte Humboldt University, Technical University Berlin
J.P. Morgan Inroducon no Quanave esearch vdend Modelng Forschungssemnar ochassche Analyss und ochas der Fnanzmäre Humbold Unversy, Techncal Unversy Berln ecember 6, 2 Hans Buehler, JP Morgan Eques Q Q
More informationPolicies Convertible Bonds and Stock Liquidity
ISSN 18368123 Polces Converble Bonds and Sock Lqudy Jason Wes No. 201103 Seres Edor: Dr. Alexandr Akmov Copyrgh 2011 by auhor(s). No par of hs paper may be reproduced n any form, or sored n a rereval
More informationA Heuristic Solution Method to a Stochastic Vehicle Routing Problem
A Heursc Soluon Mehod o a Sochasc Vehcle Roung Problem Lars M. Hvaum Unversy of Bergen, Bergen, Norway. larsmh@.ub.no Arne Løkkeangen Molde Unversy College, 6411 Molde, Norway. Arne.Lokkeangen@hmolde.no
More informationThe Performance of Seasoned Equity Issues in a Risk Adjusted Environment?
The Performance of Seasoned Equy Issues n a Rsk Adjused Envronmen? Allen, D.E., and V. Souck 2 Deparmen of Accounng, Fnance and Economcs, Edh Cowan Unversy, W.A. 2 Erdeon Group, Sngapore Emal: d.allen@ecu.edu.au
More informationAn empirical analysis of the dynamic relationship between investmentgrade bonds and credit default swaps
An emprcal analyss of he dynamc relaonshp beween nvesmengrade bonds and cred defaul swaps Robero Blanco * Smon Brennan ** Ian W Marsh *** Workng Paper no. 211 * ** *** Banco de España. Emal: rblanco@bde.es
More informationWhat influences the growth of household debt?
Wha nfluences he growh of household deb? Dag Hennng Jacobsen, economs n he Secures Markes Deparmen, and Bjørn E. Naug, senor economs n he Research Deparmen 1 Household deb has ncreased by 10 11 per cen
More informationHEDGING METHODOLOGIES IN EQUITYLINKED LIFE INSURANCE. Alexander Melnikov University of Alberta, Edmonton email: melnikov@ualberta.
HDGING MHODOLOGI IN QUIYLINKD LIF INURANC Aleander Melnkov Unversy of Alera dmonon emal: melnkov@ualera.ca. Formulaon of he Prolem and Inroducory Remarks. he conracs we are gong o sudy have wo yes of
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More informationt φρ ls l ), l = o, w, g,
Reservor Smulaon Lecure noe 6 Page 1 of 12 OILWATER SIMULATION  IMPES SOLUTION We have prevously lsed he mulphase flow equaons for onedmensonal, horzonal flow n a layer of consan cross seconal area
More informationManaging banks exposure to the property market in a low interest rate environment: Hong Kong s experience
BOE/CCBS Chef Economss Workshop Managng banks exposure o he propery marke n a low neres rae envronmen: Hong Kong s experence Dong He Hong Kong Moneary Auhory 18 May 211 1 Hong Kong dollar neres raes have
More informationTHE IMPACT OF UNSECURED DEBT ON FINANCIAL DISTRESS AMONG BRITISH HOUSEHOLDS. Ana del Río and Garry Young. Documentos de Trabajo N.
THE IMPACT OF UNSECURED DEBT ON FINANCIAL DISTRESS AMONG BRITISH HOUSEHOLDS 2005 Ana del Río and Garry Young Documenos de Trabajo N.º 0512 THE IMPACT OF UNSECURED DEBT ON FINANCIAL DISTRESS AMONG BRITISH
More informationAustralian dollar and Yen carry trade regimes and their determinants
Ausralan dollar and Yen carry rade regmes and her deermnans SukJoong Km* Dscplne of Fnance The Unversy of Sydney Busness School The Unversy of Sydney 2006 NSW Ausrala January 2015 Absrac: Ths paper nvesgaes
More informationCONTROLLER PERFORMANCE MONITORING AND DIAGNOSIS. INDUSTRIAL PERSPECTIVE
Copyrgh IFAC 5h Trennal World Congress, Barcelona, Span CONTROLLER PERFORMANCE MONITORING AND DIAGNOSIS. INDUSTRIAL PERSPECTIVE Derrck J. Kozub Shell Global Soluons USA Inc. Weshollow Technology Cener,
More informationMarketClearing Electricity Prices and Energy Uplift
MarkeClearng Elecrcy Prces and Energy Uplf Paul R. Grbk, Wllam W. Hogan, and Susan L. Pope December 31, 2007 Elecrcy marke models requre energy prces for balancng, spo and shorerm forward ransacons.
More information