AUTOCALLABLE STRUCTURED PRODUCTS

Transcription

1 AUTOCALLABLE STRUCTURED PRODUCTS Auhor : Trsa Gullaume Professoal le : Lecurer ad research fellow Malg address : Uversé de Cergy-Poose Laboraore Thema boulevard du por F-95 Cergy-Poose Cedex Frace E-mal address : rsa.gullaume@u-cergy.fr Telephoe umber : Fax umber : Absrac I hs arcle a geeral flexble form of auocallable oe s aalycally valued ad s payoff profle ad rsk maageme properes are dscussed. Two models are cosdered : a wo-asse Black-Scholes oe (Black ad Scholes 97) ad a Mero jump-dffuso framework (Mero 976) combed wh a Ho-Lee wo-facor sochasc yeld curve (Ho ad Lee 986). I he former seg he geeral auocallable srucure uder cosderao cludes may popular feaures : regular coupos reverse coverble provso dow-ad-ou Amerca barrer bes-of mechasm ad sowball effec. These feaures are more or less fully addressed accordg o he ealed valuao dffcules. Smpler oes are easly desged ad prced o he bass of hs geeral srucure. I he secod aalycal framework he auocallable payoff srucure s resrced o he followg feaures : regular coupos a reverse coverble provso ad possble parcpao he growh of a sgle uderlyg equy asse. The formulae provded hs paper ca be expeced o be a valuable ool for boh buyers ad ssuers erms of rsk maageme. Ideed hey eable vesors o assess her chaces of early redempo as well as her expeced reur o vesme as a fuco of he corac s specfcaos whle hey allow ssuers o aalyze ad compue her varous rsk exposures a accurae ad effce maer.. Iroduco Auocallables also kow as auo-rgger srucures or kck-ou plas are very popular he world of srucured producs. They have capured a large par of he marke share rece years. Produc provders use hem o offer hgher payoffs ha hose o srucured producs ha auomacally ru o a full erm. I s sadard form a auocallable s a oe ha s lked o a uderlyg rsky asse

2 (usually a sgle sock a baske of socks or a equy dex) ad ha has o fxed maury. Wha s referred o as he maury of he auocallable s acually he maxmum durao hs produc ca say alve usually ragg from o 5 years. Several observao daes wh he produc s lfe are prespecfed he corac ypcally o a aual or sem-aual bass. A each observao dae f he value of he uderlyg s a or above a prespecfed level usually called he auocall rgger level or auocall barrer he he prcpal amou s pad back by he ssuer o he holder of he oe alog wh a coupo rae. I s sad he ha he oe auocalls. The prespecfed auocall rgger level s ofe defed as he level of he uderlyg asse a he corac s cepo bu does o have o be. I may also vary me. If here s o early redempo he oe proceeds o he ex observao dae where here s aga he possbly of early redempo. A lo of plas kck ou year oe or wo leavg vesors wh he choce o reves rollover subsues from he same provder swch o aoher offer or op back o he markes. Aoher level ca be prespecfed for each observao dae below he auocall rgger level such ha f he oe does o auocall bu he uderlyg s above ha lower level usually called he coupo barrer he he oe pays a coupo rae. Some auocallable oes have a memory fuco embedded also called a sowball effec. Wh a memory fuco he produc wll pay ay coupos ha have o bee pad o prevous observao daes f o a subseque observao dae all prerequses are me. Several varas of he sadard srucure are raded o he markes. Oe way o allow for yeld ehaceme s o clude a reverse coverble mechasm : f he oe has o bee auocalled ad f he uderlyg asse has falle below a prespecfed coverso level a maury whch s a kd of safey hreshold from he po of vew of he vesor he he prcpal amou s o redeemed ad shares (or her equvale cash amou) are pad back sead whch eals a capal loss for he vesor. Ths amous o he sale of a ou-of-he-moey pu by he vesor o he ssuer. Ths s a suable feaure he curre marke evrome. Ideed low eres raes leave lle room for yeld ehaceme f vesors wa full capal proeco as provders mus use more of he al capal o guaraee s complee reur. Moreover hgh volaly markes volaly has o be sold order o geerae a hgher come. The aswer s he o roduce capal-a-rsk srucures ypcally losg moey f he uderlyg dex has falle 5% or furher from s al level. Aoher way o allow for yeld ehaceme s o make he coupo paymes a a gve observao dae coge o he uderlyg asse o havg crossed a prespecfed lower level sce he laes observao dae. Ths amous o he roduco of a dow-ad-ou Amerca barrer (.e. a couous barrer) a me erval bewee wo observao daes. Ivesors may also wa o be gve he opporuy o parcpae poeal creases he value of he uderlyg sead of recevg fxed coupos. More specfcally some coracs defe a prespecfed upper level such ha f he uderlyg s above ha level a a gve observao dae he he oe auocalls ad he coupo pad o he holder of he oe s a perceage of he spo value

3 of he uderlyg. Fally some auocallable oes clude bes-of (wors-of) feaures whch usually coss provdg vesors wh a perceage of he maxmum (mmum) bewee wo or more asses case he corac allows for a parcpao he marke upward poeal. Despe her wdespread use he markes here are o may academc sudes o auocallable producs. Whle Bouzoubaa ad Ossera [] descrbe a varey of payoffs ad aalyse rsk maageme ssues all he oher corbuos focus o umercal prcg schemes. Fres ad Josh [8] sudy a produc specfc varace reduco scheme for Moe Carlo smulao purposes. Deg e al. [] dscuss fe dfferece mehods a umercal paral dffereal equao framework. Kamchueg C. [] dscusses smoohg algorhms for he Moe Carlo smulao of he Greeks. Alm e al. [] develop a Moe Carlo algorhm ha allows sably wh respec o dffereao. No research arcle has ye come up wh a aalycal soluo o he valuao problem rased by he auocallable srucured producs raded he markes.e. auocallables wh dscree observao daes. The ma purpose of hs paper s o fll hs gap by provdg a closed form formula for a flexble auocallable srucure wh dscree observao daes. I a Black-Scholes framework (Black ad Scholes 97) our formula ca capure he followg feaures : regular coupos reverse coverble provso dow-ad-ou Amerca barrer bes-of mechasm ad sowball effec. The way whch all hese feaures ca be embedded o a aalycal formula s more or less resrcve accordg o he ealed valuao dffcules : whle regular coupos he reverse coverble provso ad he sowball effec ca be fully ackled he Amerca barrer ad bes-of feaures are oly parally covered he sese ha he Amerca barrer s o alve durg he ere produc lfe ad he bes-of mechasm s lmed o wo asses. The advaage of usg a Black- Scholes model s ha eables o value sophscaed payoffs ad provdes a smple hedgg mehod. However s well kow ha he model assumpos are flawed. I parcular has log bee kow ha equy prces are o purely couous ad exhb jumps. The laer are a way o accou for he skew observed he opos marke especally he seeper skew for shor expraos (Gaheral 6). Secodly he assumpo ha eres raes are cosa s parcularly spurous for he valuao of auocallable oes. Ideed he laer are a combao of fxed come ad equy compoes whch are usually log-daed. Moreover he correlao bewee equy ad eres rae sources of radomess has a sgfca mpac. Whe he sock marke goes up he durao of he auocallable srucure goes dow. If here s a posve correlao bewee equy ad eres rae sellers of he oe make losses whle hedgg her eres rae exposure wheher equy creases as hey have o sell loger-daed zero coupo bods ad buy more shor erm zero coupo bods uder hgher eres raes or decreases as hey eed o sell shor erm bods ad buy more loger-daed bods uder lower eres raes. Coversely f here s a egave correlao bewee equy ad eres rae sellers of he oes make a e prof whle hedgg her eres rae exposure wheher equy goes up ad dow because of he oppose drecos of equy ad eres rae. As a cosequece prcg models ha do o ake he correlao bewee equy ad eres rae o

4 accou wll uderprce he auocallable srucure whe hs correlao s posve ad overprce whe ha correlao s egave. Tha s why a secod model s cosdered whch cosss of a combao of a Mero-ype jumpdffuso (Mero 976) ad a Ho-Lee wo-facor sochasc yeld curve (Ho ad Lee 986). As a resul of he creased sophscao of he model assumpos fewer payoff specfcaos ca be ackled amely : regular coupos a reverse coverble provso ad possble parcpao he growh of a sgle uderlyg equy asse. I boh models he way o acheve a aalycal soluo s o prce he srucure as a whole he form of a opo valuao formula ha s a fuco of he corac s specfcaos ad of fxed levels of volaly. The obaed formulae ca be expeced o be a valuable ool boh for vesors ad ssuers erms of rsk maageme as hey allow o aalyze he fluece of each varable as well as o sudy he global properes of he srucure a accurae ad effce maer. They also provde a sesble fas approxmao of he srucure s far prce before more geeral umercal schemes may be esed allowg for more sochasc facors. The arcle s orgazed as follows : Seco he auocallable srucure prced uder he Black- Scholes model s descrbed deal; Seco rsk maageme ssues are aalysed; Seco he formula uder he Black-Scholes model s saed; mahemacal proof of hs formula s o repored because s cumbersome; Seco 5 he formula uder he jump-dffuso model wh a sochasc yeld curve s saed ad he proof of hs formula s provded. A appedx dscusses he umercal mplemeao of he formulae.. A flexble auocallable srucure Le us beg by descrbg deal he payoff srucure ha wll be valued a Black-Scholes seg Seco. Several observao daes... T are se wh he produc lfe T where T s he maxmum durao of he produc (s maury ). Two rsky asses S ad S are pcked. A each observao dae m m pror o expry auocall may occur wo ways : () If he value of S a me deoed by m S les wh he rage D U m m m D U m m he he vesor s al capal or ooal N s redeemed a me m alog wh a coupo rae y. m Boh he rage D U m level m ad he coupo rae D s ypcally he value of he uderlyg a cepo.e. m y are prespecfed he corac. The auocall rgger m S.

5 () If S m s greaer ha he level U he oe auocalls oo bu sead of yeldg a m prespecfed coupo rae provdes he vesor wh a perceage S / S f hs rao s greaer ha he reur rao / m of S / S f he laer s greaer ha / m m m of he reur rao S S or wh a perceage S S. Thus he asse S was roduced he frs place o allow for a bes-of mechasm he eve of early redempo wh a parcpao rae. Besdes f m S les wh a rage C D m prespecfed coupo rae bee los because he coupo barrers m C D m m he he oe does o auocall bu a z s pad ou o he vesor; furhermore all pror coupos ha may have m C k m had o bee crossed are also pad ou o he k vesor a me m. Ths amous o embeddg a memory fuco or sowball effec o he oe. Thus here are hree dffere barrers a each observao dae parcpao deoed by coupo barrer whou auocall deoed by C D U wh D beg defed as m m m m m : he auocall rgger level whou D he auocall rgger level wh parcpao deoed by U ad he m m C. The relave posos of hese barrers are as follows: m S mos raded coracs. The followg able (Table ) recapulaes all possble payoffs a ay observao dae pror o expry. Table : Possble payoffs a each observao dae S Eve S C m m m pror o expry uder a Black- m Scholes model Oucome he oe proceeds o he ex observao dae D S C he oe pays ou a prespecfed coupo N z m m m m foregog mssed coupos are recovered ad he oe proceeds o he ex observao dae U S D early ex wh coupo : he vesor s al m m m U ad S S m m S m U ad S S m m m m m capal s fully redeemed foregog mssed coupos are recovered ad he oe pays ou a fal N y m prespecfed coupo early ex wh parcpao S : he vesor s al capal s fully redeemed foregog mssed coupos are recovered ad he oe pays ou a fal coupo equal o a prespecfed perceage of he rao S / S m early ex wh parcpao S : he vesor s al capal s fully redeemed foregog mssed 5

6 coupos are recovered ad he oe pays ou a fal coupo equal o a prespecfed perceage of he rao S / S m Noes : Ths able shows he possble payoffs provded by he auocallable srucure uder cosderao a a observao dae pror o expry Le us ow ur o he profle of he oe a expry or maxmum durao T. The safey barrer deoed by B s he Europea-ype barrer uder whch he ooal N wll o be redeemed a par ad he reverse coverble mechasm wll be acvaed.for he vesor o receve a fal coupo Y wo codos mus be me : S mus le he rage D U D U ad he mmum value reached by S a ay me bewee ad mus o be smaller ha a prespecfed Amerca-ype dow-ad-ou barrer H. The way whch he vesor may parcpae he growh of asses S ad S also dffers from prevous observao daes as poeal parcpao he growh of S s o coge o S beg above levelu. Raher a Europea-ype up-ad- barrer W s specfcally aached o S ad may rgger parcpao he growh of S eve f S les below U. The bes-of mechasm s hus less resrcve a expry ha a he prevous observao daes. Fally a Europea-ype barrer sowball effec ca be acvaed. Thus here are sx dffere barrers a expry or maxmum durao C deermes wheher he memory fuco or T : - he safey barrer deoed by B uder whch he reverse coverble mechasm s acvaed - a ex barrer deoed by D rggerg a fal coupo payme - a Amerca-ype dow-ad-ou barrer deoed by H ad moored durg me erval ha deermes he possbly of recevg a fal coupo payme bu o he possbly of recevg a fal payme hrough equy growh parcpao - a ex barrer deoed by U rggerg a fal payme coge o he value of S - a ex barrer deoed by W rggerg a fal payme coge o he value of S - a coupo barrer C rggerg he memory fuco of he oe The followg order holds : B C D U. The barrer W has o sad above B bu s poso ca be freely chose relave o barrers C D ad U depedg o how large he 6

7 fluece of he bes-of mechasm should be. The barrer H has o le below ca be freely chose relave o barrers B ad C. The followg able (Table ) recapulaes all possble payoffs a expry. D bu s poso Table : Possble payoffs a expry or maxmum durao S S T uder a Black-Scholes model Eve Oucome S B he reverse coverble mechasm s rggered : S / S of he vesor s B C B C S C D S S C D D U S D U S ad ad ad S ad S ad S ad S W W W W oly a fraco al capal s redeemed foregog mssed coupos are defely los ad o fal coupo payme s receved he vesor s al capal s fully redeemed bu foregog mssed coupos are defely los ad o fal coupo s receved foregog mssed coupos are defely los bu he vesor s al capal s fully redeemed ad he oe pays ou a fal coupo equal o a prespecfed perceage S / S of he rao he vesor s al capal s fully redeemed foregog mssed coupos are recovered bu o fal coupo payme s receved he vesor s al capal s fully redeemed foregog mssed coupos are recovered ad he oe pays ou a fal coupo equal o a prespecfed perceage S / S of he rao f S H he vesor s al capal s fully redeemed foregog mssed coupos are recovered ad he oe pays ou a prespecfed fal coupo N y f S H he vesor s al capal s fully redeemed foregog mssed coupos are recovered bu o fal U ad S S S U ad S S coupo s receved as he Amerca dow-ad-ou barrer has bee breached he vesor s al capal s fully redeemed foregog mssed coupos are recovered ad he oe pays ou a fal coupo equal o a prespecfed perceage S / S of he rao he vesor s al capal s fully redeemed foregog mssed coupos are recovered ad he oe pays ou a fal coupo equal o a prespecfed perceage S / S of he rao Noes : Ths able shows he possble payoffs provded by he auocallable srucure uder cosderao a expry or maxmum durao These mulple feaures are roduced order o spa a large varey of possble coracs. Despe s raher complex payoff fuco he far value of he geeral auocallable srucure uder cosderao ca be aalycally obaed. Forhcomg Proposo provdes a formula for ay umber of 7

8 observao daes whou sowball effec whle Proposo provdes a formula for four observao daes. The reaso why a formula s wre dow he specfc case of four observao daes s hreefold : frs s useful o provde a fully explc example of how Proposo s expaded case he laer mgh look somewha erse o he reader a frs sgh; secod a auocallable oe wh four observao daes wll serve as he bass for subseque umercal examples; las bu o leas Proposo cludes he prcg of he sowball effec. The reaso why he laer s o cluded Proposo s because seems mpossble o come up wh a compac formula for geeral.e. whou specfyg he umber of observao daes.. Rsk maageme ssues Smpler oes ca be desged o he bass of he above geeral auocallable srucure. Proposo ad Proposo es he valuao of all kds of smpler srucures by pug he suable pus o he formulae : - oes ha do o allow ex hrough equy parcpao are valued by seg he parameer A equal o oe ad he parameers A ad A equal o zero - oes ha allow ex hrough parcpao asse S oly are valued by seg he parameer A equal o oe ad he parameers A ad A equal o zero - he bes-of provso s acvaed by seg he parameer A equal o oe ad he parameers A ad he parameers A equal o zero - he sowball effec s acvaed by seg he parameer A equal o oe Proposo All oher adjusmes are farly obvous. For example f you do o wa a Amerca barrer o codo ex wh a coupo rae a expry paymes pror o expry jus se all he jus le H ed o zero; f you do o wa coupo z parameers equal o zero. Ivesors eed o be aware of he cos of beefg from addoal opporues compared o a sadard corac. Iroducg ermedary coupo paymes before expry creases he oe s value a relavely sraghforward maer as log as hey are fxed coupo paymes. I s less easy o acpae precsely he effec of ex hrough equy parcpao or he effec of a memory fuco or ha of a bes-of provso. I Table he prces of four dffere ypes of coracs wh a growg umber of opos are compared. The frs oe s a pla sgle-asse oe wh o auocall hrough parcpao he growh of S ad o memory fuco. The secod oe adds o he frs oe he possbly of auocall hrough parcpao he growh of S. The hrd oe adds o he secod oe a memory fuco. The fourh oe adds o he hrd oe a bes-of provso. 8

9 I all hree volaly segs of Table he sharpes crease value comes from addg he possbly of auocall hrough parcpao he growh of asse S. Ths effec s more ad more proouced as volaly rses. The smalles crease value resuls from addg a memory fuco bu s o eglgble ad would be greaer f he coupo barrers were hgher. I he low volaly seg s oly slghly smaller ha he creases value resulg from addg oher opos. Ths overall behavor s esseally a refleco of he probably of early auocall as wll be see shorly. The mpac of he bes-of provso s subsaal especally he hgh volaly seg. Overall he roduco of o-sadard feaures hus creases he value of he auocallable oe o a large exe : averagg across all hree volaly segs he prce of corac s.5% hgher ha ha of corac. There s a major dfferece bewee coracs ad o he oe had ad corac o he oher had as regards he effec of volaly : he value of he former s a decreasg fuco of volaly whereas he value of he laer grows wh volaly. We wll come back more deal o he ssue of prce sesvy laer hs Seco. Table : Prces of varous ypes of -year maury auocallable oes Low volaly Medum volaly Hgh volaly % 5% 8% 5% 5% 6% Value of corac Value of corac Value of corac Value of corac Noes : Ths able compares he values of four ypes of auocallable oes wh maxmum expry years ad aual observao daes. Corac s a sgle-asse oe lked o S wh o early ex hrough parcpao ad o memory fuco. Corac cludes he possbly of early ex hrough parcpao he growh of S. Corac cludes a memory fuco. Corac cludes a bes-of provso. All repored prces are obaed usg forhcomg Proposo wh he followg pus : S S r.5% 5% D D D D U U U U 5 C C C C 9 B 75 W H 8 y y y y 8% z z z z 5% Corac s valued by seg ad. Corac s valued by seg ad. Corac s valued by seg. Corac s valued by seg. Whaever he umber of opos avalable a auocallable corac he ssue of he expeced auocall dae s fudameal. Ideed poeal auomac early redempo s he specfc propery of hese producs relave o oher dex-lked oes ad mos vesors who choose o pu her moey auocallable srucures hope o ge her capal back a a early sage. The vesed ooal ca be 9

10 redeemed as soo as o he frs observao dae. I hs respec moderaely bullsh vesors would be well advsed o se a hgh value for y he coupo rae offered upo ex whou parcpao eher S or S a me. They wll hus be a poso o beef from hgher reurs ha f hey ves sadard fxed-come. O aoher had srogly bullsh vesors had beer rade off a hgh value of y for hgh levels of parcpao he growh of asses S ad S. They wll hus be able o beef from a amplfcao of upsde marke movemes relave o a log poso he socks hemselves whle beg proeced from dowsde rsk. Overall erms of corac specfcaos he chaces o beef from auocall as soo as he frs or maybe he secod observao dae deped mosly o he choce of he auocall rgger level. Ivesors who coemplae lowerg ha level mus be aware ha hs wll cause he cos of he oe o rse. For a gve prce hs meas ha he coupo raes ad he parcpao raes wll be lower. The auocall rgger levels may crease or decrease durg he oe s lfe. The former case s ofe referred o as a sep-up auocall he markes whle he laer s called a sep-dow auocall. Whe auocall levels vary hey usually do so a sep-dow paer so ha he vesor s more ad more lkely o kck ou as me passes. Whe dealg wh oes ha provde possble early ex wh parcpao he growh of rsky asses s obvous ha lowerg he U s wll crease he lkelhood of early redempo wh parcpao whch may be called he probably effec. Bu vesors should be aware ha hs posve probably effec may be offse by he lower expeced reur receved whe he oe auocalls whch may be called he reur effec so ha he overall effec o he value of he oe s ambguous. Roughly speakg he hgher he volaly he more he reur effec eds o preval over he probably effec as he lkelhood of reachg relavely hgh U s creases. Table repors he maury breakdow o a four-year auocallable oe cludg equy parcpao ad a bes-of provso as a fuco of volaly; for smplcy volaly s supposed o be fla.e. he skew ad he erm srucure are o ake o accou. Table shows a couple of oceable resuls. Frs he mos lkely oucome s o auocall o he frs observao dae wheher volaly s low or hgh. The probably of auocall occurrg as early as he frs or he secod observao dae creases wh volaly. Ths crease s o lear acceleraes whe volaly s shfed from he medum volaly caegory o he hgh volaly oe. Whe volaly s low he chaces of auocall occurrg a he fourh ad fal observao dae are que subsaal bu hey go dow as volaly ges hgher. Aga hs decrease s o lear cosderably acceleraes as volaly moves o he hgh volaly regme a a faser pace ha he observed crease he probably of auocall a he frs observao dae. Ieresgly auocall a he hrd observao dae s always he leas lkely oucome whaever he volaly pu ad s probably of occurrece chages more or less learly ulke he probably of auocall a he frs or a he las observao dae.

11 Table : Maury breakdow of a -year auocallable oe as a fuco of volaly uder rsk euraly Low volaly Medum volaly Hgh volaly % 5% 8% 5% 5% 6% Maury = year Maury = years Maury = years Maury = years Noes : Ths able repors he rsk-eural probables ha he acual maury o a -year auocallable oe provdg he payoff descrbed Table ad Table wll be eher or years. For smplcy he volales of S ad S are assumed o be fla. The repored probables are obaed usg forhcomg Proposo wh he followg pus : S S r.5% 5% D D D D U U U U 5 Oher pus o Proposo have o relevace here. The probably ha he acual maury s year s obaed by addg up. The probably ha he acual maury s years s obaed by addg up. The 5 6 probably ha he acual maury s years s obaed by addg up maury s years s equal o oe mus he probably ha he acual maury s or years. 9. The probably ha he acual For rsk maageme purposes he probables of auocall eed o be compued uder he physcal measure. I Table 5 he rsky asses S ad S clude rsk prema. Overall he srucure of he maury breakdow remas he same as uder he rsk-eural measure. There s lle dfferece bewee he probables of early ex year ad year Table ad Table 5. However uder he rsk-averse physcal measure he lkelhood ha he oe wll exed o s maxmum maury of four years s eve lower whle he probably ha auocall wll occur as early as year s eve hgher. Uder he hgh volaly regme ex a maxmum expry becomes he leas lkely oucome Table 5. Ieresgly he probably of auocall year does o crease mooocally wh volaly as s lower he medum volaly regme ha he low volaly oe. Ths observao pos ou he eed o accuraely compue he sesvy of he probably of early ex a ay observao dae wh respec o volaly. Ths s easly doe by mere dffereao haks o he formulae provded hs paper. Table 5 : Maury breakdow of a -year auocallable oe as a fuco of volaly uder rsk averso Low volaly Medum volaly Hgh volaly % 5% 8% 5% 5% 6% Maury = year

12 Maury = years Maury = years Maury = years Noes : Ths able repors he probables ha he acual maury o a -year auocallable oe provdg he payoff descrbed Table ad Table wll be eher or years. The dfferece wh Table s ha he reurs o asses S ad S ow clude rsk prema of % ad % respecvely. Fally wo oher feaures should be bore md by vesors. Frs mus be oced ha he besof provso plays a more mpora role a expry coge o S beg above he ha a prevous observao daes as s o U barrer. Ideed as log as S rades above he depede barrer W he bes-of provso ca be acvaed eve f S rades a levels below D uless he reverse coverble mechasm s acvaed (.e. he reverse coverble provso domaes he bes-of oe). Oe should also pay aeo o he fac ha he ma role of he Amerca dow-adou barrer s o deerme wheher a fal coupo payme s receved case he oe does o auocall before expry so ha hs barrer wll o have a sgfca mpac o he value of he oe uless a suffcely hgh value for compesao o he vesor whe early redempo has o ake place. y s agreed o he corac whch makes sese as a Now from he perspecve of raders who sell hese producs he rsks assocaed wh he varous compoes of he auocallable srucure are very smlar o hose assocaed wh dgals. These compoes have posve Dela ad sellers of he opo wll have o buy Dela of he uderlyg meag ha hey wll be log dvdeds shor eres raes ad log borrow coss of he uderlyg. To ackle he dscouy rsk aroud he coupo barrers a barrer shf wll usually be appled so ha he resulg shfed payoff over-replcaes he al payoff by he leas amou whle makg he Greeks of he ew payoff maageable ear he barrer. The poso volaly depeds o he coupo barrer levels ad o he forward prce of he uderlyg asse S. The Vega of he auocallable s compoes s posve f he uderlyg s forward prce s lower ha he rgger level; oherwse Vega s egave. Hece he Vega of he geeral auocallable srucure uder cosderao hs paper wll deped o he respecve weghs of he dowsde ad upsde compoes wh he overall srucure. Typcally wh he D ' s equal o he sarg spo prce S ad he C ' s beg smaller ha he D ' s he compoes of he auocallable srucure provdg fxed coupo paymes wll have a egave Vega. The reverse holds for he compoes ha provde parcpag paymes. I vew of he poeally offseg effecs from he varous compoes oe should always check

13 wheher he Vega of he overall srucure s posve or egave. Oce edowed wh he formulae provded hs paper hs becomes a easy ask usg aalycal or umercal dffereao. Oe of he dffcules assocaed wh hedgg he sale of a auocallable oe s ha hs produc s sesve o he skew. Traders ca hedge he compoes provdg fxed paymes by mplemeg a smple roll-over sraegy as follows : a each observao dae f here s o auocall ad he oe proceeds o he ex observao dae hey wll ake a log poso call spreads exprg a me. These call spreads wll be more or less gh or wde depedg o how coservave he rader s bearg md ha he gher he call spread he larger Gamma ca become ear he coupo barrer. The cach s ha he moeyess of he volved fuure call opos cao be kow a cepo. As a resul he forward mpled volaly pus ha mgh be used o prce he srucure a me wll geerally be wrog. Moreover all compoes wh expry greaer ha are codoal o o havg auocalled a a prevous observao dae whch roduces a pah-depede eleme he valuao problem. I hs respec he usefuless of he formulae provded hs arcle s wofold : frs hey are a fuco of he skew a me whch by defo s observable.e. hey do o volve prcg a combao of forward sar dgals ha would evably requre he pu of he forward mpled volaly surface; secod by gvg he rsk-eural codoal probables of auocallg a each observao dae hese formulae provde raders wh a smple way o accuraely wegh he call spreads pu place for each maury. Table 6 compares he prces of four-year maury auocallable oes wh ad whou he possbly of ex hrough parcpao uder hree sylzed regmes of skew. For he sake of smplcy he a-he-moey volaly remas fla up ul maxmum durao. I le wh mos observed daa o equy he slope of he upper par of he skew curve (.e. he par o he lef of he a-he-moey level) s greaer ha he slope of he lower par of he skew curve (.e. he par o he rgh of he a-he-moey level); besdes he small skew decreases more or less learly wh me whle he decrease he large skew wh respec o me dsplays more curvaure. As he dowsde compoes of he auocallable srucure have egave Vega ad he upsde compoes have posve Vega he effec of he skew s o lower he value of he auocallable srucure as he skew creases he volaly of dowsde compoes ad decreases he volaly of he upsde compoes wh respec o he a-he-moey le. The effec of he skew o oparcpag auocallable oes s moderae bu o-eglgble ad would obvously be bgger uder more exreme skew curves ha hose chose here. The effec of he skew o parcpag auocallable oes s of grea magude so ha prcg he auocallable oe usg fla a-he-moey volaly wll largely overesmae he produc s value. I mus be oced ha for some compoes of he auocallable srucure s o obvous o kow whch po of he mpled volaly surface o use due o he pah-depede aure of he produc ad o he may dffere coupo ad early redempo barrers. The rule followed hs paper s for every compoe volved o pck he

14 po he mpled volaly surface ha maches he las barrer ad he observao dae he dgal; for example he dgal provdg a coupo equal o y upo early ex a me whe S D U wh coordaes D he S volaly surface. codoal o auocall o havg occurred pror o s prced usg he po For mpled volaly curves ha smle.e. ha dsplay hgher mpled volales o he lef bu also o he rgh of he a-he-moey level he prevous aalyss does o hold aymore. Ideed he hgher volaly o he ou-of-he-moey compoes of he auocallable srucure rases her prces sce hey are Vega posve so ha he e effec of he smle s ambguous ad depeds o he wegh of he upsde dgals wh respec o he dowsde dgals wegh meag her respecve corbuo o he oal oe s value. Clearly oes whou he possbly o auocall hrough upward marke parcpao are skew egave wheher he mpled volaly curve s skew- or smle-shaped. Table 6 : Prces of wo dffere kds of auocallable oes uder dffere regmes of skew Corac : o possbly of ex hrough parcpao Corac : possbly of ex hrough parcpao cludg a bes-of provso No skew Small skew Large skew Noes : Ths able compares he prces of four-year maury auocallable oes wh ad whou he possbly of early ex hrough parcpao uder hree sylzed regmes of skew. The repored prces are compued usg Proposo. The small skew ad large skew regmes are defed by he mpled volaly surface dsplayed Table 7. The o skew regme assumes a decally fla volaly a 6% ad %. The erm-srucure of he rsk-free rae s gve by : -year a.5% -year a.75% -year a % ad -year a.5%. The corac specfcaos are as follows: S S 5% C C C C 9 D D D D U U U U 5 B 75 W H 8 z z z 5% A A A A for corac y y y y 8% A A A A for corac Table 7 : Impled volaly surface used o compue he prces Table 6 Coordaes C C C C Small skew S volaly surface Small skew S volaly surface Large skew S volaly surface Large skew S volaly surface

15 D D D D U U U U B W Noes : Ths able repors he mpled volaly pus used o defe he small skew ad he large skew Table 6. Oly he pos ha are eeded o compue he auocallable oe s value are repored.. Valuao formulae uder a Black-Scholes model Ths seco cludes he wo formulae deoed by Proposo ad Proposo Seco ad. Proposo Le S ad B ad he dyamcs of S be wo geomerc Browa moos drve by sadard Browa moos B wh correlao coeffce. Uder he rsk-eural measure deoed by S ad S are gve by : ds r S S db ds r S S db RN () () where r s he rsk-free rae ad are wo cosa couous dvded raes ad are wo volaly pus assumed o be exraced from a mpled volaly surface across a couous rage of srkes ad maures for boh asses S ad S. be defed as Le he fuco... ;... follows : for ad s he sadard gaussa cumulave dsrbuo fuco ; for s gve by he followg specal case of he mulvarae sadard gaussa cumulave dsrbuo fuco : 5

16 x x x exp... dx dx... dx x x x / () The umercal compuao of he fuco Le he followg oaos hold : - d D l S - B b l S - U u l S H h l c r / - s explaed he Appedx. C l S... W S w l S S x l S r / () (5) / (6) (7) - ˆ / ˆ r / ˆ r / (8) - s he dcaor fuco akg value f he argume sde he braces s rue ad value zero. oherwse s a parameer ha akes value oe f he auocall oe does o allow ex hrough parcpao - equy growh ad zero oherwse s a parameer ha akes value oe f he auocall oe allows ex hrough parcpao asse - S ad S oly ad zero oherwse - s a parameer ha akes value oe f he auocall oe allows ex hrough parcpao asse S or asse S.e. f cludes a bes-of provso ad zero oherwse - he symbol a ; j... k deoes he sequece of real umbers a where rages from j o k The he o-arbrage value V AUTOCALL of a auocallable oe wh observao daes ul maxmum expry cludg all he feaures defed Table ad Table excep for he sowball effec ha s excep for he possbly of recoverg foregog mssed coupos a ay observao dae s gve by he followg formula : exp AUTOCALL (9) V r N z r N y exp 6

17 exp exp N N 5 N exp 6 exp N r N 7 8 exp exp r N N 9 r N y exp r N exp exp exp N N 5 6 exp N where : 7 s he probably uder he moey marke umerare ha he coupo N z s pad ou o he vesor a me. I s gve by : () d j j j j c d d ; j... ; ; j... ; j j j j ; j... ; j... j j s he probably uder he moey marke umerare of auocall a me whe he oe does o allow ex hrough equy parcpao. I s gve by : d j j d ; j... ; j j ; j... j () s he probably uder he moey marke umerare of auocall a me whou equy parcpao whe he oe allows equy parcpao. I s gve by : () 7

18 d j j j j d d u ; j... ; ; j... ; j j j j ; j... ; j... j j s he probably uder he S umerare of auocall a me wh parcpao he growh of S he absece of a bes-of provso. I s gve by : d j j u ; j... ; j j ; j... j () s he probably uder he S umerare of auocall a me 5 wh parcpao he growh of S whe he corac cludes a bes-of provso. I s gve by : d j j u x ; j... ; j 5 j ; j... j () s he probably uder he S umerare of auocall a me 6 wh parcpao he growh of S whe he corac cludes a bes-of provso. I s gve by : d ˆ j j u ˆ x ˆ ; j... ; j 6 j ; j... j (5) s he probably uder he S umerare of ex a maxmum expry 7 coverble provso. I s gve by : d b ;... ; ;... 7 uder he reverse (6) 8

19 s he probably uder he moey marke umerare of ex a maxmum expry 8 whou a fal coupo because he auocall barrer has o bee reached he absece of a bes-of provso. I s gve by : 8 d d b ;... ; ;... d d d ;... ; ;... (7) s he probably uder he moey marke umerare of ex a maxmum expry 9 whou a fal coupo because he auocall barrer has o bee reached whe he corac cludes a bes-of provso. I s gve by : d d b w ;... ; 9 ;... d d d w ;... ; ;... (8) s he probably uder he S umerare of ex a maxmum expry wh parcpao he growh of S whe he corac cludes a bes-of provso ad whe S s below he parcpao barrer U a me. I s gve by : 9

20 d ˆ d ˆ b ˆ w ˆ ;... ; ;... d ˆ d ˆ d ˆ w ˆ ;... ; ;... (9) s he probably uder he moey marke umerare of ex a maxmum expry wh a fal coupo N y whe he oe does o allow ex hrough equy parcpao. I s gve by : d h d ;... ; ;... d h d h ;... ; h e ;... d d d ;... ; ;... d d d h ;... ; h e ;... () s he probably uder he moey marke umerare of ex a maxmum expry wh a fal coupo N y whe he oe allows equy parcpao. I s gve by :

21 d h u ;... ; ;... d h u h ;... ; h e ;... d d u ;... ; ;... d d u h ;... ; h e ;... () s he probably uder he moey marke umerare of ex a maxmum expry whou recevg a fal coupo N y because he dow-ad-ou barrer has bee crossed he me erval whe he oe does o allow ex hrough equy parcpao. I s gve by : d d d ;... ; ;... () s he probably uder he moey marke umerare of ex a maxmum expry whou recevg a fal coupo N y because he dow-ad-ou barrer has bee crossed he me erval whe he oe allows ex hrough equy parcpao. I s gve by :

22 d d d ;... ; ;... d d u ;... ; ;... () s he probably uder he S umerare of ex a maxmum expry 5 he growh of S he absece of a bes-of provso. I s gve by : wh parcpao 5 d d u ;... ; ;... () s he probably uder he S umerare of ex a maxmum expry 6 he growh of S whe he corac cludes a bes-of provso. I s gve by : wh parcpao d d u x ;... ; ;... 6 (5) s he probably uder he S umerare of ex a maxmum expry 7 wh parcpao he growh of S whe he corac cludes a bes-of provso ad whe S s above he parcpao barrer U a me. I s gve by : d ˆ d ˆ u ˆ x ˆ ;... ; ;... 7 (6)

23 Ed of Proposo Alhough Proposo looks bulky may acually be regarded as que compac gve ha accomodaes ay umber of observao daes ad ha ess a large varey of possble payoffs. However as meoed earler Proposo does o ackle he memory fuco embedded o some auocallable oes. So le us ow roduce Proposo whch expads Proposo he specal case hus provdg a easy way for readers o make sure hey fully udersad how o expad Proposo for ay eger ; furhermore Proposo prces he sowball effec. Proposo Le he assumpos of Proposo hold. The he o-arbrage value V AUTOCALL of a auocallable oe wh observao daes ul expry edowed wh all he feaures descrbed Table ad Table cludg he sowball effec s gve by he followg formula : AUTOCALL exp r N y exp N exp N exp N 7 r N y 8 9 exp N exp N exp N exp r N z r N y 5 6 exp N exp N exp N exp exp exp exp exp N r N y 6 7 V r N z (7) exp 5 6 exp r N z exp exp r N z exp r N z z r N z N r N r N 5 exp exp r N 8 9

24 exp exp exp N N exp N exp r N z z z exp r N z z r N z 5 The parameer oherwse. The full expaso of all akes value oe f he auocall oe cludes a memory fuco ad value zero erms m...5 m s gve Appedx. There are 6 ou of he 5 erms hs formula ha are o a drec expaso of Proposo for m he role of whch s o value he coupo-relaed memory fuco; hey are ad 5. The meag of hese 6 erms ca be uvely defed as follows : exp r N z s he rsk-eural value of he dgal allowg o recover a me () he coupo ha was o receved a me ; () exp r N z z s he rsk-eural value of he dgal allowg o recover a me he coupo ha was o receved a me ad he coupo ha was o receved a me ; exp r N z s he rsk-eural value of he dgal allowg o recover a me () he coupo ha was o receved a me ; (v) exp r N z z z s he rsk-eural value of he dgal allowg o recover a me he coupo ha was o receved a me ad he coupo ha was o receved a me ad he coupo ha was o receved a me ; (v) exp r N z z s he rsk-eural value of he dgal allowg o recover a me he coupo ha was o receved a me ad he coupo ha was o receved a me ; exp r N z s he rsk-eural value of he dgal allowg o recover a me (v) 5 he coupo ha was o receved a me The full expaso of he 5 m erms Proposo s as follows :

25 c d d d u u u x u ˆ x ˆ ; ; 5 6 d c d d ; ; 7 d d ; 8 d d d u ; ; 9 d u ; d u x ; d ˆ u ˆ x ˆ ; c c ; d d c ; d d d ; d d d ; 5 d d d ; 6 d d u ; 5

26 d d u ; 7 d d u x ; 8 d ˆ d ˆ u ˆ x ˆ ; 9 c c c ; c c c ; d c c ; d d d b ; d d d b ; d d d d ; d d d b w ; 5 d d d d w ; 5 d ˆ d ˆ d ˆ b ˆ w ˆ ; 5 5 d ˆ d ˆ d ˆ d ˆ w ˆ ; 5 d d h d ; 6 h d d h d h ; e 6

27 d d d d ; h d d d d h ; e d d h u ; 7 6 h d d h u h ; e d d d u ; h d d d u h ; e d d d d ; 8 6 d d d d ; 9 d d d u ; 7 d d d u ; d d d u x ; 5 d ˆ d ˆ d ˆ u ˆ x ˆ ; 5 c c c c ; c c c c ; d c c c ; 7

28 c d c d c c ; 5 where he fuco ; s he same as eq. () excep for he bouds of he x egral whch are chaged from x (lower boud) ad x (upper boud) o x ad x respecvely ad for he bouds of he x egral whch are chaged from x (lower boud) ad x (upper boud) o x ad x respecvely. 5. Aalycal formula for he value of a geeral auocallable oe uder a jump-dffuso equy model correlaed wh a wo-facor sochasc eres rae process The auocallable payoff srucure cosdered hs seco s descrbed he followg Table 8 ad Table 9. Table 8 : Possble payoffs a each observao dae m m pror o expry uder he jump-dffuso sochasc eres rae model Eve Cosequece S m Cm he oe proceeds o he ex observao dae Dm S m Cm he oe pays ou a prespecfed coupo M z m ad he oe proceeds o he ex observao dae Um S m Dm early ex wh coupo : he vesor s al capal N s fully redeemed ad he oe pays ou a fal prespecfed coupo M ym S m Um early ex wh parcpao S : he vesor s al capal M s fully redeemed ad he oe pays ou a fal coupo equal o a prespecfed perceage of he rao S / S m 8

29 Table 9 : Possble payoffs a expry or maxmum durao T sochasc eres rae model uder he jump-dffuso Eve Cosequece S H he reverse coverble mechasm s S / S of he rggered : a fraco vesor s al capal M s redeemed H S D he vesor s al capal M s fully redeemed U S D he vesor s al capal M s fully redeemed ad he oe pays ou a prespecfed coupo M y S U he vesor s al capal M s fully redeemed ad he oe pays ou a fal coupo equal o a prespecfed perceage of he rao S / S The ew modelg framework s ow roduced. Le W W ad W be hree correlaed Browa moos whose cosa parwse correlao coeffces are deoed by.. ad.. The defaul-free eres rae process r s drve by : dr d dw dw (8) where ad are wo posve cosas ad s a o-radom pecewse couous fuco sasfyg a lear growh codo. Eq. (8) s a wo-facor Ho-Lee model (986). The reaso ha was chose s wo-fold : () allows calbrao o he observed daa by fg a suable fe-dmesoal dsrbuos of r ; () he are mulvarae ormal whch eables o preserve he aalycal racably of he full model as wll be see laer he proof of forhcomg Proposo. The laer po should be emphaszed as here are oher classcal eres rae models ha are uvarae ormal ad ha ca be appropraely calbraed bu whose fe-dmesoal dsrbuos are o mulvarae ormal such as he wo-facor Hull ad Whe model (99) ad s geeralzao kow as he G++ model. The reader wshg deals abou eres rae models ad how hey ca be fed o marke daa may refer o Brgo ad Mercuro (6). The uderlyg equy asse process ds S S s drve by : r d dw I dn (9) 9

30 where : () () s a posve pecewse couous o-radom fuco such ha N s a Posso process of cosa esy. () I U where f N ad U depede decally dsrbued radom varables akg values Le J l U s ds s a sequece of be ormally dsrbued wh mea ad varace ; he he parameer s defed by : exp /. I s assumed ha all radom processes are defed o a suable probably space whch s he smalles algebra geeraed by he radom varables U N s : s W s N s ad. I s recalled ha a Browa moo ad a Posso process relave o he same flrao mus be depede (Shreve ). Thus he sochasc dffereal equao for he equy asse prce s a jump-dffuso process exedg he semal Mero model (Mero 976) by egrag a sochasc eres rae process drve by eq. (8). The reaso for ha exeso s ha s esseal o facor he correlao bewee sources of radomess he equy world ad he fxed come world as explaed he roduco of hs arcle. The valuao formula ca ow be saed uder he headg Proposo. Proposo Uder he model assumpos of Seco 5 he far value V a me of he auocallable coge clam whose payoff s defed by Table 8 ad Table 9 wh four observao daes s gve by : exp () V M!!!! y z 5 y 6 7 z 9 8 y z y

31 U S D S S U D S C S l / l / l / l / l / l / 5 l D / S l U / S 6 ; l D / S l D / S 7 ; l D / S l S / U 8 ; l D / S l D / S 9 ; l D / S l C / S ; l D / S l D / S U S ; l D / S l D / S l D / S ; l D / S l D / S l S / U ;

32 l D / S l D / S l D / S ; 5 l D / S l D / S l C / S ; l D / S l D / S l D / S l U / S ; l D / S l D / S l D / S l D / S ; l D / S l D / S l D / S l S / U ; l D / S l D / S l D / S l D / S ; l D / S l D / S l D / S l H / S ;

33 l D / S l D / S l D / S l H / S ; where he followg defos hold : u. exp r s dsdu () 6 exp s ds k s ds k k k () u r s dsdu s ds k (). / /. k. s ds. s ds k s ds /. k / () (5) s ds k k / (6) / /... s ds j / / /. s ds j (7)

34 j /.. s ds. s ds / /. s ds. s ds s ds. k k (9) The fucos b ad b b; are equal o he uvarae ad bvarae sadard ormal cumulave dsrbuo fucos respecvely. For he fuco b... b b ;... s defed by eq. (). (8) Ed of Proposo. j We ow proceed o he proof of Proposo. The proof reles o hree lemmas called Lemma Lemma ad Lemma. Uless saed oherwse he oaos used here have already bee defed (8) (9). Lemma Le r ad posve umber. The gve S S N we have : b S l / b Proof of Lemma : Gve N le be drve by eq. (8) ad eq. (9) respecvely. Le b be a l / l / P S b S S b S For he soluo of eq. (9) yelds codoal o N : N S l N r s ds s ds s dw s J S where each deoed as follows : J ; () J s a ormal radom varable wh expecao ad varace whch wll be Iegrag eq. (8) yelds : ()

35 r r s ds W W Le us deoe by L he Hlber space of radom varables wh fe varace o F. The gve W s well kow ha W adms he followg orhogoal decomposo L (see e.g. Shreve ) : W W B. where B s a sadard Browa moo defed o he same probably space as W ad W ad () () depede of W. Furhermore for ay gve Browa moo suable probably space ad ay fxed we have: W s s defed o a W s ds s dw s () ; whch mples ha for ay gve x ad fxed we have : W s ds x W x Thus as far as he compuao of P s cocered he egral follows : u. r s dsdu W B Nex gve L as follows : W ad W he radom varable. W r s ds (5) ca be expaded as (6) adms a orhogoal decomposo W W a B a B (7) where a ad a are real-valued scalars ad B s a sadard Browa moo defed o he same probably space as he processes W W ad W ad depede of he processes W W ad B. Noe ha a mus be posve sce by defo of he mulvarae ormal radom vecor W W W a s he sadard devao of W codoal o From he defo of lear correlao ad he bleary of covarace we oba : cov W W. cov.w.w cov B ab W ad W (8) 5

36 a.... The real umber. s he paral correlao bewee W ad W codoal o Nex from he sadard devao of ormal radom varables we oba : W..... W. ad he sably uder addo of a se of ucorrelaed a a (9) Now as he fuco s o-radom he somery propery of he Io egral mples ha he radom varable s dw s s ormally dsrbued wh mea zero ad varace. Hece wh regard o he compuao of P he egral s dw s s ds W. B. B. Sce he radom varables s ds ca be expaded as : J are parwse depede ad are also depede of ad W he followg equaly hus holds dsrbuo for ay gve : S l S N W (5) W u /. r s dsdu s ds N Z. s ds Z s ds Z s ds N Z /.. (5) where Z Z Z ad Z are four ucorrelaed sadard ormal radom varables Cosequely gve by N he radom varable ad sadard devao gve by. S l S s ormally dsrbued wh mea gve 6

37 Lemma Le r ad S be drve by eq. (8) ad eq. (9) respecvely. Le be four o-radom mes such ha : posve umbers. The S b S b S b S b. Le b b b ad b be four exp!!!! l b / S l b / S l b / S l b / S ; (5) Proof of Lemma : Gve N le P S b S b S b S b. The frs sep s o shf o log coordaes as he proof of Lemma. Each radom varable l / s couous. Hece oe ca express P as he followg quadruple egral : P l b / S l b / S l b / S l b / S dx dx dx dx S S S S l l l l S S S S dx dx dx dx The by codog ad usg he weak Markov propery of he process P l b / S l b / S l b / S l b / S dx dx dx dx S S S l dx l dx l dx S S S S S l S S dx S l dx l dx S S dx l S S S (5) S we have : Usg (5) s sraghforward o show ha he correlao bewee l S / S j j devao of s equal o he sadard devao of (5) l S / S ad l S / S dvded by he sadard l S / S. Thus from he defo of he bvarae ormal dsrbuo we have : j 7

38 S S l dx S j dxj l S (55) xj j x exp / / j j j j j Hece (5) becomes : P l b / S l b / S l b / S l b / S dx dx dx dx x x / x exp / / Summg over all possble values of he process N each me erval ad mulplyg by her respecve probables of occurrece yelds Lemma. The smaller-dmesoal case of obvous maer. Lemma S b S b S b (56) s esed by Lemma a Le Y be a ormal radom varable wh mea quadrvarae ormal radom vecor such ha : X x X x X x X x Y ad varace Y. Le X X X X be a x x x x X X X X ; X. X X. X X. X X X X X (57) where X ad X are he mea ad he sadard devao of X respecvely ad. X X j s he correlao coeffce bewee X ad j Y X X X X s mulvarae ormal we have : E Y exp X x X x X x X x X j. The f he radom vecor 8

39 x X X. Y X Y x X X. Y X Y X X x Y X X. Y X Y x X X. Y X Y Y X X X. X X. X X. X exp ; (58) where X. Y s he correlao coeffce bewee X ad Y Proof of Lemma : Le f y x x x x deoe he jo desy of he sadardzed radom vecor Y X X X X Y X X X X where Y Y Y The E Y exp X x X x X x X x Y X X ad X. X (59) x x x x X X X X X X X X y z z z z exp y f y z z z z dydz dz dz dz Y X X X X Followg he same approach as he proof of Lemma a lle algebra shows ha he ormal radom varables X adm he followg orhogoal decomposos : X Y Z X. Y X Y X Y Z Z X. Y X. X Y X Y. X (6) (6) X Y Z Z Z (6) X. Y X. X Y X. X Y. X X Y. X. X X Y Z Z Z Z (6) X. Y X. X Y X. X Y. X X. X Y. X. X X Y. X. X. X where he Z ' s are parwse depede sadard ormal radom varables ha are all depede of Y ad where he followg defos hold : (6) Xa. Xb Xa. Y Xb. Y X Y X..... a Y X XY Y X X Y X Y X b a b X Y a a b b a a X. X Y. X b c a X. X X. Y X. Y X. X Y X. X Y b c b c a b a c X Y. X b a (65) X Y. X. X X. Y X. X Y X. X Y. X (66) c a b b a b b c a 9

40 (67) X. X Y. X. X X. X X. Y X. Y X. X Y X. X Y X. X Y. X X. X Y. X X Y. X. X c d a b c d c d a d a c b d a b c a c a b X Y. X. X. X X. Y X. X Y X. X Y. X X. X Y. X. X (68) d a b c b a b b c a c d a b Thus we have : X Y X. YY; X Y X Y X. Y... X Y X Y X X ; YY X X Y X Y X X Y X X (7) (7) (69) X X. YY X. X Y. X X X. YY X. YY X. X Y X X. YY X. X Y ; X Y X Y. X X Y X Y. X. X X Y X X X (7) X X. YY X. X Y. X X X. YY X. YY X. X Y X X. YY X. X Y X Y X Y. X X Y X. X Y. X. X X X... YY X X Y X X X. YY X X. YY X. X Y X X. YY X. X Y ; X Y. X. X. X Y X Y X X Y X Y. X. X. X From (69) - (7) oe ca derve deses : f y z z z z Y X X X X f y z z z z as he followg produc of codoal Y X X X X f y f z y f z y z f z y z z f z y z z z Y X Y X Y X X Y X X X Y X X X Subsug (7) o (59) ad performg he ecessary calculaos he yelds Lemma. (7) The proof of Proposo ca ow esue. Accordg o he o-arbrage heory of valuao a complee marke (Harrso ad Plska [98] ) he prce of he coge clam uder cosderao s equal o he expecao of s dscoued payoff uder he equvale margale measure. Bu he marke here s complee due o he roduco of jumps. Ths rases he queso of he choce of a releva equvale margale measure. A ecessary codo for he dscoued prce process exp r s ds S o be a margale s o se he drf coeffce he dyamcs of

41 S equal o r rae process r. Ths ca be show exacly he same way as whe he rskless s cosa (see e.g. Lambero ad Lapeyre 997) so he deals are omed. Accordg o he classcal argume by Mero (976) ha jump rsk s dversfable ad herefore o rewardable wh excess reur hs codo should be boh ecessary ad suffce. If we cosder hs argume o be rue he he dyamcs of S ca oly be gve by eq. (9). However oe mus bear md ha hs argume s debaable as emprcal evdece suggess ha here are dusry wde shocks ad eve coury wde shocks ha are o easly dversfable so ha he possbly of perfec hedgg remas heorecal. The leraure o mea-varace hedgg (Schwezer 99) ad quale hedgg (Föllmer ad Leuker 999) ca be cosuled for alerave approaches. Deog by he probably measure uder whch eq. (9) holds ad assumg ha hs s he correc prcg measure he he auocallable payoff defed Table 8 ad Table 9 mples ha a each fxg dae compued : j j pror o maxmum expry he followg expecaos mus be j exp r s ds M yj S D... S j Dj Dj S j U N j j (7) j S j exp r s ds M S D... S j Dj S j U N j j S (75) j exp r s ds M zj S D... S j Dj C j S j D N j j (76) A maxmum expry he followg expecao s also requred : S exp r s ds M S D S D S D S H N (77) S I s clear ha he valuao problem comes dow o compug wo kds of expecaos deoed by E ad E j j : j E exp r s ds S c N j (78) j S j E exp r s ds S c N j (79) S