Generalized Analytical Upper Bounds for. American Option Prices *

Transcription

1 Gnraliz Analyical Uppr ons for Amrican Opion Prics an-lin hn parmn of inanc Naional aiwan Univrsiy 85 cion 4 Roosvl Roa aipi 06 aiwan an Hsih-hn han parmn of inanc Naional nral Univrsiy 300 hna Roa hnli 30 aiwan his vrsion: anary 005 W wol lik o hank han-han han Hsan-hi hn Rn-Raw hn in-hi Lin an hih-ko Yh for hlpfl commns h ahors hank h Naional cinc oncil of aiwan for financial sppor Any commns plas arss o hn a chns@manw

2 Gnraliz Analyical Uppr ons for Amrican Opion Prics Asrac his papr nralizs an ihns h analyical ppr ons of hn an Yh (00 for Amrican opions nr sochasic inrs ras sochasic volailiy an mps whr Amrican opion prics ar ifficl o comp wih accracy W firs nraliz horm of hn an Yh (00 an apply i o riv a ihr ppr on for Amrican calls whn h inrs ra is rar han h ivin yil Or ppr ons ar no only ih also convrin o h accra Amrican call opion prics whn ivin yil or srik pric is small or whn volailiy is lar W hn propos a nral horm which can appli o riv ppr ons for Amrican opions whos payoffs pn on svral risky asss As a monsraion w apply or nral horm o riv ppr ons for Amrican xchan opions an Amrican maximm opions on mlipl risky asss

3 I Inrocion Amrican opions rir nmrical mhos sch as laic mhos o provi accra pric simas h valaion prolm is vry im-consmin an ifficl whn mlipl sa varials ar involv or xampl h opions nr sochasic inrs ras sochasic volailiy an mps involv for ranom facors an rir xpnsiv laic mols Anohr complx xampl is pricin an Amrican opion whos payoff pns on mor han on nrlyin ass or hs siaions ih analyical ppr ons can provi sfl nchmark vals an conrol varias for h corrcion of nmrical rrors Opion pricin ons ar sfl cas ( hy provi aliaiv propris of opions ( hy can s o scrn mark aa for mpirical rsarch (3 hy sh lih on hin an (4 hy ar nrally oain wih h las assmpions on h invsor s prfrncs an h isriions of h nrlyin ass prics Opion pricin ons may riv y ( liminain simpl ominanc amon iffrn porfolios (s Mron (973 ( applyin a linar prorammin approach (s Garman (976 Richkn (985 an Richkn an Ko (988 (3 sin som mahmaical inaliis sch as nsn s inaliy an achy s inaliy (s Lo (987 oyl an Lin (997 an hn an Yh (00 an (4 sin scon-orr sochasic ominanc (s Lvy (985 an onsaninis an Prrakis (00 Proaly o h ifficly in alin wih h arly xrcis prolm h work on Amrican opion pricin ons is limi arr arrow an Mynni (99 riv an ppr on for Amrican p opions nr h lack-chols conomy whil roai an mpl (996 vlop ppr an lowr ons sin h capp call opion pricin chni (i an Amrican call opion is a simpl ominan porfolio of h capp call Alhoh h ons provi y arr arrow an Mynni (99 an roai an mpl (996 ar nrally ih hir ppr ons ar no in analyical form (xcp nr h lack-chols conomy an rir nmrical chnis hn an Yh (00 provi analyical form ppr ons ha ar applical o nral Amrican opions Amrican calls on ivin payin socks Amrican calls on frs Amrican ps on ivin payin socks an Amrican ps on frs Morovr hir ppr ons rly nihr on h isriion of h sa varial nor o hy rly on coninos im rain Alhoh hn an Yh s (00 analyical form ppr ons ar vry nral hy 3

4 can appli only in h cas whr h inrs ra is rar han h ivin yil hir ppr ons may inaa for opions on svral nrlyin asss cas i is likly o happn ha som nrlyin ass hav a ivin yil larr han h risk-fr ra In conras his aricl provis wo nral horms which can s o riv ppr ons for Amrican opions nr nral siaions inclin h cas whr h ivin yil is larr han h risk-fr ra As a monsraion w apply or nral horms o riv ppr ons for Amrican calls whn h inrs ra is smallr han h ivin yil for Amrican xchan opions an for Amrican maximm opions on mlipl risky asss W conri o h lirar on opion pricin ons in svral ways irs prvios paprs on opion ons concnra on ropan opions wih a sinl nrlyin ass or a sinl sa varial In conras w provi ppr ons for Amrican opions whos pricin involvs svral risky asss an/or svral risk facors ( sochasic inrs ras sochasic volailiy an mps for ach ass pric procss conly or ppr ons ar no only ih also convrin o h accra Amrican call opion prics whn ivin yil or srik pric is small or whn volailiy is lar hirly w corrc ypos in hn an Yh (00 an provi nmrical rsls o invsia h ihnss of hir ppr ons an h ihnss of ors h nmrical rsls inica ha or ppr ons ar nrally ihr han hos of hn an Yh (00 h rs of his aricl procs as follows cion II provis a nral analysis for oainin ppr ons of Amrican opions In his scion wo nral horms for vlopin Amrican ppr ons ar inroc cion III iscsss ppr ons nr sochasic inrs ras sochasic volailiy an mps sin h invrsion orir mho his mho was s y Hson (993 co (997 akshi ao an hn (997 an hn an Yh (00 c W also riv ppr ons for Amrican xchan opions an Amrican maximm opions nr h lack-chols conomy in his scion cion IV provis nmrical rsls o analyz h ihnss of or ppr ons cion V concls h papr II Gnral Analysis horm of hn an Yh (00 shows ha an Amrican opion is on from aov y h risk-nral xpcaion of is mariy payoff if his xpcaion is rar han h inrinsic val a all ims his horm is vry nral an h only ollowin hn an Yh (00 all xpcaions ar akn nr h risk-nral masr hroho 4

5 assmpions rir ar ha (i h risk-nral masr xiss an (ii h nominal risk-fr ra is sricly posiiv W rsa horm of hn an Yh (00 as follows: horm of hn an Yh (00 An Amrican opion is on from aov y h risk-nral xpcaion of is mariy payoff if his xpcaion is rar han h inrinsic val a all ims horm of hn an Yh (00 can prsn in formal mahmaics as follows: L h mariy a of h Amrican opion an X ( h inrinsic val a im If [ X ( ] > X ( for all whr [ ] rprsns akin h xpcaion in h risk-nral worl a im hn [ X ( ] is an ppr on of h Amrican opion val his aricl will xn h ia of hn an Yh in a way ha i is no ncssary o s h mariy payoff ( of h Amrican opion o riv h ppr on Insa w rplac X ( X ( wih ohr fncions in or nraliz horm which can appli o riv ihr ppr ons No ha oh hir horm an or horm ar prov y a iscr approximaion similar o h laic approach h rsls will hol in coninos im as rachs a limi A h Gnraliz horm h Gnraliz horm of hn an Yh (00 L h mariy of h opion conrac fin Y ( h( X ( X ( is h inrinsic val of h opion a im an ( saisfis a h ( s δ ( s for any s from im o im s < whr ( s h ( h( s h( s for any s ( c h( If [ Y ( ] > X ( for all hn [ Y ( ] opion val a im whr h is any fncion which δ is h iscon facor is an ppr on of h Amrican Proof: h papr 5

6 ollowin hn an Yh (00 w will prov his horm sin a iscr approximaion similar o h laic approach A im consir h fncion Y fin in h nraliz horm I is r ha Y ( is larr han h iscon rminal val: [ Y ( ] [ h( X ( ] δ ( X ( > [ ] [ ( ] whr h scon inaliy coms from coniion a of h nraliz horm Y X y consrain [ ( ] is also larr han h inrinsic val ( ( an hs is an ppr on of h Amrican opion pric a im inc [ Y ( ] X ( [ ( ] is r i is r ha Y is rar han h coninaion val of h Amrican opion a im : [ Y ( ] [ h( X ( ] [ h( h( X ( ] [ h( [ h( X ( ] [ h( [ Y ( ] > [ δ ( [ Y ( ] > δ ( max{ [ δ ( X ( ] X ( } [ ] h scon lin follows from coniion an h fifh lin follows from coniion a of h nraliz horm y consrain [ Y ( ] is also rar han h inrinsic val of h Amrican opion an hs is an ppr on of h Amrican opion val a im y mahmaical incion i i s sraihforwar o show ha [ Y ( ] is an ppr on of h Amrican opion val h rsl will hol in coninos im whn approachs zro (Q No ha all xpcaions ar akn nr h risk-nral worl h main iffrnc wn hn an Yh s (00 horm an or nraliz horm is ha w h hrfor hir ppr mliply h mariy payoff fncion y a fncion ( h on is a spcial cas of ors whr ( appropria fncion ( nraliz horm hn [ Y ( ] hn an Yh s As lon as w can fin an h which is smallr han on an saisfis h criria in h I shol no ha whn h fncion ( [ Y ( ] [ [ Y ( s ] for any s ( s is an ppr on which is ihr han h is always smallr han on In his cas or ppr on is a 6

7 s-marinal procss In conras h ppr on of hn an Yh is a marinal procss Nvrhlss as wih h iscon Amrican opion prics h iscon procsss of oh ppr ons ar spr-marinal procsss i [ Y ( ] [ δ ( s s [ Y ( s ] [ X ( ] [ δ ( s [ X ( ]] s A rhr xnsion: horm h nraliz horm is acally sill rsriciv in h sns ha h ppr on is rla o h mariy payoff of h Amrican opion If w xn or oncp o allow Y o any ranom varials which saisfy similar (or sam c ( criria in h nraliz horm hn i is possil o riv ppr ons for nral yps of Amrican opions Nx w will firs salish or horm an iv hr applicaions lar on horm L h mariy of h opio n conrac fin Y ( im which saisfis a Y X as a ranom varial a ( ( [ Y ( ] [ δ ( Y ( ] for any [ 0 ] c [ Y ( ] X ( for all [ 0 ] X is h inrinsic val of h opion a im hs [ Y ( ] whr ( ppr on of h Amrican val is an Proof A im rminal val: [ ] i is r ha Y ( is larr han h iscon [ Y ( ] [ δ ( Y ( ] δ ( X ( [ ] whr h firs inaliy coms from coniion of horm rom coniion c [ Y ( ] is also larr han h inrinsic val an hnc is an ppr on rican opion pric a im Y X of h Am inc [ ( ] ( is r i is r ha [ Y ( ] r han h coninaion val of is ra h Amrican opion a im : 7

8 [ Y ( ] [ δ ( Y ( ] [ δ ( [ Y ( ] [ δ ( max{ [ δ ( X ( ] X ( }] y consrain [ Y ( ] is also rar han h inrinsic val of h Amrican opion an hs is an ppr on of h Amrican opion val a im y mahmaical incion i is sraihforwar o show ha Y is [ ( ] an ppr on of h Amrican opion val a im h rsl will hol in coninos im whn approachs zro (Q h ia of or horm is i iniiv an can rason as follows oniion a is h rminal coniion which has o flfill y any ppr on oniion implis ha h ppr on for his prio is larr han h iscon val of h ppr on in h nx prio oniion c is a ncssary coniion for an ppr on i an ppr on ms rar han h inrinsic val of h opion a all Y is an ppr ims ominin all hr coniions will aran ha [ ( ] on of h Amrican opion pric I is worh noin ha horm of hn an Yh (00 h nraliz horm an horm ar ssain vn if h payoff fncion X ral fncion ( Y pns on prics ( or or n of mlipl nrlyin asss Howvr h horm of hn an Yh is applical only whn ivin yils of all asss ar smallr han h risk-fr ra Now w will show som applicaions of h nraliz horm an horm in h followin ssci ons Applicaions of h Gnraliz horm an horm Amrican alls on ivin Payin ocks (whn r > Accorin o h nraliz horm if on can fin a sial Y saisfis or criria hn w say [ Y ( ] ( ha is an ppr on of h Amrican pion pric L ( r o Y max K 0 whr is h sock pric K is ( { } h srik pric r is h inrs ra an is h ivin yil of h sock 3 I is hn an Yh s ppr on may vry rsriciv for crain yps of opions on mlipl is an ppr on of h Amrican xchan opion pric nrlyin asss or xampl [ X ( ] only whn r wh r an ar h ivin yils of h firs an scon nrlyin asss rspcivly 3 No ha whn hr can no possil confsion sscrips ar somims omi for simpliciy 8

9 asy o show ha ( ( r h saisfis hr criria in h nraliz [ ] X ( horm Moro vr w can vrify ha Y ( > for all : ( max > max{ K 0} ( r ( r ( r max K 0 { } K 0 h firs lin follows from nsn s inaliy an h scon lin hols sinc ( r Amrican call opion an [ ] r > hrfor Y ( is an ppr on of h inc r > is r or ppr on ( [ ] ( r { K 0} is ihr han hn an Yh s ( max{ K 0} rhrmor or ppr on convrs o h accra Amrican call opion pric nr som circmsancs or insanc i is wll known ha whn h ivin yil is zro h Amrican opion pric als h pric of is ropan conrpar (s Mron (973 Or ppr on also convrs o h ropan opion pric whn h ivin yil approachs zro Morovr whn h srik pric is vry small or whn volailiy is vry lar oh h accra Amrican call opion pric an or ppr on will convr o h crrn sock pric max rom or horm w can propos anohr ppr on for Amrican call opions Y follows: whr ( [ r ] r max Y ( K 0 Howvr his ppr on is no ihr han h aov on an hs is no s in h nmrical analysis lar on Amrican all Opions on ivin Payin ocks (whn r < hn an Yh s horm can appli o Amrican opions only whn h inrs 9

10 ra is larr han h ivin yil On h ohr han or horm is applical no mar whhr h inrs ra is larr or smallr han h ivin yil Whn h inrs ra is smallr han h ivin yil w fin a fncion Y for Amrican call opions as follows: ( ( Y ( max [ r ] K 0 irs of all Y ( max{ K 0} X ( Y ( his saisfis coniion a of horm conly also flfills coniion of horm as follows: [ Y ( ] max > δ ( max [ δ ( Y ( ] K 0 > max ( r K 0 (3 ( r ( r K 0 W finally will show h a Y ( any im : [ ] ar always rar han h inrinsic val a (4 max K 0 r max max{ K 0} ( r ( K 0 whr h firs lin follows from nsn s inaliy hs from horm w know ha ( r max K 0 is in an ppr on for Amrican call opions whn r < 3 Amrican xchan Opions An xchan opion is an opion o xchan on ass for anohr h payoff from his opion is whr an ( { 0} X max ar vals of ass on an ass wo a im rspcivly A clos-form solion for valin ropan xchan opion was firs proc y 0

11 Marra (978 nr h lack-chols conomy rom hn an Yh s horm i is asy o vrify ha [ X ( ] is an ppr on of h Amrican xchan opion whn r whr an ar h ivin yils of ass on an ass wo rspcivly Now w will riv anohr ppr on of h Amrican xchan opion onsir a fncion ( Y as follows: (5 Y ( max [ r ] [ min( r ] 0 Usin a similar procr w can asily show ha coniions a o c of horm ar saisfi y ( Y ( is an ppr on of h Amrican xchan Y hs [ ] opion pric I is no ifficl o show ha or ppr on is ihr han hn an Yh s spcially whn oh an ar small or whn is lar an is small rhrmor i is r ha or ppr on is applical for any r an 4 Amrican Maximn Opions on Mlipl Risky Asss Opions on h maximm or minimm of wo risky asss wr firs inroc y lz (98 lz (98 show ha many coninn claims for xampl opion-ons compnsaion plans risk sharin conracs c hav a payoff fncion which incls h payoff fncion of a p or a call opion on h maximm or minimm of wo risky asss h payoff of a ropan call opion on h maximm of wo risky asss is X ( max{ max{ } K 0} whr an ar vals of ass on an ass wo a im rspcivly A clos-form solion which involvs h ivaria cmlaiv sanar normal isriion fncions for valin his opion was riv in lz (98 nr h lack-chols conomy ollowin horm of hn an Yh (00 i is sraihforwar o show ha h xpc val of h mariy payoff a any arirary im is an ppr on of h Amrican maximm opion cas is val is always rar han h arly xrcis val i

12 [ max{ max{ } K 0} ] max{ [ max{ }] K 0} max{ max{ [ ] [ ]} K 0} > { max{ } 0} max K Howvr h aov ppr on is ssain only whn h ivin yils of all risky asss ar smallr han h risk-fr ra As a monsraion his papr will riv anohr ppr on for h Amrican maximm opions sin or horm If on follows or horm i is asy o prov ha h xpc val of h followin fncion is also an ppr on of h Amrican maximm opion: (6 Y ( cas max max{ max max{ [ r ] [ r ] [ min( r ] [ r ] [ r ] [ min( r ] } K max max max max [ > max { max{ } K 0} } K 0 [ r ] [ r ] [ min( r ] K 0 [ r ] [ r ] [ min( r ] ] [ ] K 0 0 hr ar many poins worh iscssin irs of all alhoh h followin fncion saisfis coniions a an c of horm i os no saisfy coniion of horm an hs i is no an ppr on of h Amrican maximm call opion: Y max max{ [ r ] [ ] ( r ' conly or ppr on is vali for any } K 0 r an as lon as r > min( r hirly or ppr on is no ncssarily smallr han ha riv from horm of hn an Yh (00 h rason is o h fac ha h srik pr ic in or ppr on ( K [ min( r ] is smallr han h srik pric in hn an Yh s ppr on ( K Howvr or ppr on is ihr han hn an Yh s ppr on whn h ivin yils of all risky asss ar small Whn oh ppr ons ar applical on can ak h minimm of oh ppr ons as h ppr on of h Amrican maximm opion pric inally or

13 ppr on will convr o h accra Amrican call opion pric nr som circmsancs whn h ivin yil or srik pric is vry small ohnson (987 an oyl an s (990 frhr xn h analysis of lz o h pricin of ropan maximm opions on n risky asss nr h lack-chols conomy h payoff of a ropan call opion on h maximm of n risky asss is X ( max{ max{ L n } K0} imilarly on can apply or horm o show ha h xpc val of h followin fncion is an ppr on of h Amrican maximm opion on n risky asss: Y ( [ ] [ ] [ ] [ min( ] max max{ r r n r } K n r L n K 0 No ha nr h lack-chols conomy whr inrs ras an ivin yils ar consan h aov ppr on is acally h pric of a ropan call opion on h maximm of n risky asss wih as iniial prics ( ( ( an K n an as srik pric K hrfor h analyical n n min( solions of ohnson (987 an h approxima solions of oyl an s (990 ar ircly applical o or ppr ons III Molin oh hn an Yn s ppr ons an or ppr ons hav analyical solions nr many ass pric mols In fac as lon as a ropan opion has an analyical solion nr a mol oh ppr ons also hav analyical solions nr h sam mol cas oh ppr ons can rar as ropan opions wih an as mariy payoff 4 or xampl a ropan opion has an analyical solion nr h sochasic volailiy mol of Hson (993 an so os or ppr on In orr o compar wih hn an Yh (00 w will riv analyical solions for or ppr ons nr sochasic inrs ras sochasic volailiy an mps (VI mol in h sinl ass cass Howvr w apply h lack-chols mol in h mlipl ass cass for simpliciy 4 or xampl hn an Yh s ppr on [ X ( ] r an as payoff X ( is acally h pric of a ropan opion wih 3

14 A inl Ass ass Amrican alls on ivin Payin ocks whn r > h lack-chols mol has n xn o an nvironmn nr h sochasic inrs ra sochasic volailiy an mps; s for xampl Hson (993 co (977 an akshi ao an hn (997 ollowin hn an Yh (00 w assm ha h sock pric procss is lo normal an h rif an iffsion of h sock pric procss follow h sar roo procsss y v W v (7 v ( β v γ v W α r ( y y W y a whr y r is h sock pric r is h inrs ra is h coninos ivin yil ν is h sock rrn varianc an α β γ a an ar v paramrs associa wih h procsss inally W W ρ an h inrs ra procss is assm o inpnn of h sock an h varianc procsss No ha h procsss v an y ar sricly po siiv cas hy follow h sar roo procsss h ppr on of h Amrican call opions is (8 whr U Π ( y KM max {( K 0} Π K ( y ( y ( Π Π ( M o solv for h wo proailiis an ppr on has o saisfy: ( y ( M w firs inify h P whr h 4

15 (9 whr x ln( ( U x vv v U xx yu x γ vu ργvu ( αβ ( α λ v U v ( a ( a l y U U yu 0 y yu an λ an l ar mark prics of risk associa wih v an y Plin (8 ino (9 w oain h followin Ps for h proailiis an M ( yy xv (0 ( an vπx vπ vπ xx γ vπ [ αβ ( α λ v] Π [ a ( a l y] Π ργvπ ργvπ Π 0 xx vπ yπ x x v γ vπ vv vv M M yπ yy yπ y [ αβ ( α λ v] Π [ a ( a l y] Π ργvπ Π 0 v y yπ yy y ( ym [ a ( a l y ] M M ym 0 i w will solv rs M ( as follows: w hr A ( an ( y x yπ y v yy xv xv from aion ( W riv is clos-form solion M ( A ( ( y ar shown in Appnix I can asily shown ha h characrisic fncio ns f an f for solvin Π an Π saisfy h sam Ps ix wih h onary coniio n a in f Wih his onary coniion w can riv h characrisic fncions as follows (3 whr ( ( f f ( ( v ( y ix ( ( ( v ( y ixln M ( ( ( an ( ( ar shown in Appnix Accorin h invrsion horm proailiis an characrisic fncions hav h followin rlaionship (4 Π π 0 R i ln K i f ( Amrican alls on ivin Payin ocks whn r < 5

16 o riv h ppr on of Amrican calls whn r < w assm h sam ass pric mol as in h prvios sscion xcp ha r (5 y a( y y W r whr y > 0 ollowin h similar procr on can riv h ppr on o f h Amrican call opions as: (6 U ( y max{ ( K 0} ( y Π KΠ Π KΠ h wo proailiis Π an Π in aion (6 follow h sam formla as aion (4 wih wo iffrn characrisic fncions as follows: (7 f ( ( ( v ( y ix whr ( ( an ( (whr ar shown in Appnix 3 mps I has n wll ocmn ha h mp componn is imporan for pricin sock an sock inx opions h mp-iffsion mol was firs inroc y Mron (976 an hn s y as (99 akshi ao an hn (997 an co (997 c ollowin akshi ao an hn (997 w assm h followin mp-iffsion procss: λ µ v W ξ ( y [ ] ~ N ln[ ] ln whr: λ is h frncy of mps pr yar; ( µ is h prcna mp siz (coniional on a mp occrrin ha is lonormally inically an inpnnly isri ovr im wih nconiional man µ h sanar viaion of ln [ ] is ; ξ is a Poisson mp conr wih innsiy λ ; ha is Pr { ξ } λ an Pr { ξ } λ 0 ; 6

17 v ξ an ar ncorrla wih ach ohr or wih W W an r W h characrisic fncions for h mp componn ar shown y akshi ao an hn (997 an co (997 as: (8 f f ( [ ] λiµ ( ( [ ] λiµ ( i ( i / ( i ( xp λ ( µ ( ( µ i ( i / ( i ( xp λ ( ( µ akshi ao an hn (997 an co (997 show ha if mps occr inpnnly wih h sock pric lvl an inrs ras hn h characrisic fncion of h mp componn can comin wih h characrisic fncion of h iffsion componn hrfor h characrisic fncions ar rspcivly mlipli y h oriinal fncions of (4 or (7 o calcla ppr ons Mlipl Ass ass Amrican xchan Opions on ivin Payin ocks Marra (978 val an opion o xchan on ass for anohr which is call an xchan opion W sa his mol in his sscion an s his sp o riv a clos-form solion for h ppr on of h Amrican xchan opion ollowin Marra (978 w assm ha h wo ass prics follow: 5 i y W i i i i i i whr yi r i i is h ivin yil of ass i i is h sanar viaion of rrn of ass i an W i is h rownian moion of ass i or simpliciy w follow Marra (978 o assm ha ivin yils an h risk-fr ra ar consan Unr his simplifi assmpion Marra (978 shows ha h pric of h ropan xchan opion is: (9 w N( z N( z whr ln z ln 5 or simpliciy w assm ha inrs ra an ivin yil ar consan in h mlipl ass cas 7

18 z z ρ orr ρ ( N h cmlaiv nsiy fncion (cf of h sanar normal isriion imilar o h proof of Marra (978 w can riv ppr ons for Amrican ns nr h lack-c ols conomy Accorin o h rsls in h prvios scion on ppr on of an Amrican xchan opion is: 0 max min( s r s r Y ov aion an h va chan opion whr h iniial val of ass on is an h val iniial of ass wo is hrfor or ppr on has h followin clos-form solion: xchan opio h ( [ ] [ ] [ ] { } [ ] 0 max r ζ ζ whr ζ an ζ min( h ppr on in h a c rar as l of an ropan x ζ ζ min( (0 ( ( ( min( z N z N whr min( ln z z Amrican Opions on Maximm of wo Risky Asss nr h lack-chols conomy lz (98 riv h clos-form solion for a ropan call opion on h maximm of wo risky asss as follows: ( ( ( ( min max ρ ρ r K r K r K r K z U 8

19 whr ( ar lack-chols formla for ropan call opions an is pri ropan call opion on h minimm of wo risky asss which follows: ( min h c of a ( ρ ( ln ( ln ρ ρ ( min ρ N K N r whr r K N ln ln ρ r K r K As sa in h prvios scion or ppr on for h Amrican maximm call opion is acally h pric of a ropan maximm call opion wih as iniial sock prics an as srik pric hrfor i follows a similar formla i IV Nmrical Rsls inrs ras volailiy an mps mol ar no corrc h corrc formla ar shown in Appnix A h nmrical rsls in hn an Yh (00 ar hs wron ( min( max ρ r K A orrc Nmrical Rsls in hn an Yh (00 h pricin formla for hn an Yh s (00 ppr ons nr h sochasic W rpor h corrc nmrs in hir al an al o show h ihnss of hir ppr ons 9

20 al shows hir ppr ons of calls an ps nr a sochasic volailiy nvironmn (V mol h risk-fr ra an h ivin yil rmain consan an no mps occr in h sock pric hir paramrs ar 00 v 0 04 r α 5 β 0 04 γ 0 λ 0 ρ 0 5 an al shows ha h oriinal ppr ons ar vn smallr han h ropan opion vals hs confirmin ha hir formla ar ofl al shows h ppr ons of Amrican calls on ivin payin socks ps on non-ivin payin socks calls on frs ps on frs an hir conrpar ropan opion vals nr h sochasic volailiy sochasic inrs ras an mps (VI mol ollowin hn an Yh (00 w also comp h ropan opion val s sin fo rmla in akshi a o an hn (997 sinc hs nmrs srv as lowr ons of Amrican opion vals h paramr vals s in hn an Yh (00 ar clos o hos sima y akshi ao an hn (997 sin &P 500 inx opion aa hir paramrs ar v 004 r α 5 β 0 04 γ 0 λ 0 ρ 0 5 a (in Panls an l 0 µ 0 0 an λ 0 6 rom al w fin ha hn an Yh s ppr ons ar nrally i ih cas hir vals ar vry clos o ropan opion vals h iffrncs wn h ppr ons an h conrpar ropan opion vals ar wihin 5% of h conrpar ropan opion vals for mos cass h only xcpion is for Amrican ps on non-ivin payin socks whr h iffrncs may larr han 30% (s Panl an Panl of al Amrican alls on ivin Payin ocks (whn r > W now compar h ihnss of or ppr on wih ha of hn an Yh s for Amrican calls on ivin payin socks whn r > W s h VI mol an aop h sam paramr vals as al W also calcla ropan opion vals as h nchmark vals o invsia h ihnss of oh ppr ons rom al 3 w can s ha or ppr on of calls is in ihr han hn an Yh s ppr on whn r > On avra or ppr ons ar 304% larr han ropan opion vals whil hn an Yh s ppr ons ar 53% larr 6 No ha h nrlyin ass os no pay a ivin in Panls an hs w s 005 o mach h iniial risk-fr ra 0

21 Amrican alls on ivin Payin ocks (whn r < hn an Yh s ppr ons ar no availal for Amrican calls on ivin payin socks whn r < On h ohr han or ppr ons ar sill availal in his cas o monsra h ihnss of or ppr ons in his cas w also s h VI mol an aop h sam paramr vals as al xcp ha w s r 003 an 0 05 h rsls ar prsn in al 4 W fin ha whn h ivin yil is larr han h risk-fr ra or ppr ons ar no as ih as h cas whr h ivin yil is smallr han h risk-fr ra h rsls ar xpc cas or ppr ons work s (i no rror for Amrican calls whn h ivin yil is zro Amrican xchan Opions Pricin Amrican xchan opions is a wo-imnsional sochasic prolm nr h lack-chols conomy rksn an nslan (993 prov ha h aov wo-imnsional sochasic prolm can simplifi o a on-imnsional sochasic r ρ no h pric of an Amrican prolm L ( xchan opion on wo asss wih iniial sock prics an risk-fr ra r risk-as rif rms an 7 volailiis an corrlaion cofficin ρ an im o ma riy rksn an nslan (993 show ha h followin rlaionship hols: ( ( r ρ ( r 0 00 whr ρ No ha oh h rif rm an h volailiy paramr rla o h scon ass is zro hrfor h lf-han si of h aov aion corrspons o h pric of an Amrican call whr h nrlyin ass has crrn val risk-as rif an volailiy h xrcis pric an h ma riy of h call ar an rspcivly an whr h risk-fr ra is r o analyz h ihnss of or ppr on w n nchmark vals of Amrican xchan opions W apply h aapiv msh mol of ilwski an Gao (999 o calcla h Amrican xchan opion pric sin h on-imnsional solion of 7 h risk-as rms ( ar al o r ( r in h lack an chols conomy

22 rksn an nslan (993 h nmr of im sps in h aapiv msh mol is 0000 h paramr vals ar aop from hn hn an Yan (00 as follows: ρ 0 5 r an an ar vari rom Panl A of al 5 i is clar ha or ppr ons ar vry ih i vry clos o h accra prics of h Amrican xchan opions Or ppr ons ar ao % larr han h accra prics in nral I is also r ha or ppr ons ar ihr han hn an Yh s (00 In Panl whr h ivin yil of ass on is larr han h risk-fr ra hn an Yh s ppr on is no availal whil ors is sill workal Howvr or ppr ons ar no as ih as h cas in Panl A h rsls ar consisn wih or armn ha or ppr ons ar ihr whn ivin yils of h nrlyin asss ar small Amrican Maximm all Opions W finally invsia h ihnss of or ppr ons for Amrican maximm call opions on wo ivin payin socks W s h laic mol of hn hn an Yan (00 o calcla h accra pric of Amrican maximm call opions h nmr of im sps in h laic is limi o 000 cas i is a wo-imnsional laic o corrc h nmrical rrors o h chosn mim nmr of im sps w mploy h conrol varia chni of Hll an Whi (988 o oain h accra pric of Amrican maximm call opions as follows: P ~ ( P P ~ P A A (3 max max max max whr A P ~ max an P ~ max ar prics of Amrican an ropan maximm call opions calcla from h laic mol of hn hn an Yan (00 wih 000 im sps rspcivly an P max is h clos-form solion of h ropan maximm call opion h paramr vals s hr ar also from hn hn an Yan (00 as follows: ρ 0 5 r an K an ar vari rom Panl A of al 6 w fin ha or ppr ons ar also i ih for in-h-mony Amrican maximm opions Or ppr ons ar ao % larr han h accra p rics in his cas As h opions com o-of-h-mony or ppr ons ar loosr han h prvios cas h rror is sill smallr han 6%

23 I is also inrsin o know ha or ppr ons work wll for in-h-mony cass whil hn an Yh s (00 work wll for o-of-h-mony cass Morovr hn an Yh s ppr on is no availal whil ors is sill workal in Panl whr h ivin yil of ass on is larr han h risk-fr ra Gnrally spakin h rsls in al 6 confirm or prvios claim ha or ppr ons ar ihr whn ivin yils of h nrlyin asss ar small V onclsion ollowin h framwork of hn an Yh (00 w riv ppr ons of Amrican opion prics hs ppr ons ar spcially sfl whn hr ar svral sa varials involv in h pricin mol Or ppr ons ar clos form whn h conrpar ropan opion has a clos-form solion Or ppr ons ar vry nral in h sns ha hy o no rly on isriion assmpions or coninos rain Morovr or ppr ons ar no only ih also convrin o h accra Amrican call opion prics whn ivin yil or srik pric ar small or whn volailiy is lar h only rir inps o implmn or ppr ons ar h risk-nral procsss of h sa varials his is no a prolm cas of h rcn avancs in mpirical rivaivs rsarch or xampl on can apply h impli inomial r approach of Rinsin (994 an is many xnsions sch as rman Kani an hriss (996 ackwrh (997 rin-ons an Nrr (000 c o oain h risk-nral procss of h sock pric ackwrh (999 for an xclln rviw on opion-impli risk-nral isriions an procsss Or ppr on is sill fasil vn in h xrmly complica cas whr h pricin of Amrican opions pns on mlipl risky asss an mlipl risk facors ( sochasic inrs ras sochasic volailiy an mps for ach ass pric procss In his cas on can apply sanar Mon arlo simlaions o calcla h xpc val of or ppr on which is compaionally mor fficin (an may ihr han ohr ppr ons nra y ohr complica Mon arlo mhos 8 his iss is lf for fr rsarch In fr rsarch w wol lik o mpirically compar h ihnss of or ppr 8 Mon arlo mhos can provi ias hih simas of h Amrican opion prics sin h forsih ias (s roai an Glassrman (997 approach or h aliy approach (s Hah an Koan (00 an Rors (00 h compaion is nrally im consmin cas i rirs h simlanos rminaion of h opimal xrcis onary 3

24 on wih hos nra y ohr approachs arr arrow an Mynni (99 an roai an mpl (996 Alhoh h ppr ons vlop y hm ar as on h lack-chols conomy hy can xn o nral isriions wih slih moificaions inally w lik o know if h risk-nral procsss of sa varials ar impli y h ropan opion prics an how ofn h Amrican opion prics may viola or ppr ons h rsls will sh lih on h fficincy of opion marks 4

25 Appnix A orrc h ypos of hn an Yh (00 A rivaion of h rs Pric Gss h frs pric as follows: ( y( v( A y Io s lmma w oain h followin P for : vv v vγ y ργv y [ αβ ( α λ v] [ a ( a l y] 0 Pl in h ss solion for h frs pric an oain a sysm of hr Os: v yy v y γ ( α λ ργ A ( a l αβ a Wih onary coniions ( ( ( 0 A w can show ha 0 whr ( ( a l ( a l ( a l ( a l ( a l ( a l an A a { ( [ ( a l ] ln ln ( a l ( a l ( a l ( a l } ( A rivaion of h haracrisic ncions of alls Gss h ppr on of calls as follows: [ max{ K0} ] [ ] Π KΠ Π h Ps for h proailiis com: U KΠ 5

26 v Π an v Π xx γ v Π vv y Π [ αβ ( α λ ργ v] xx Π yy v ργvπ v y Π [ a ( a l y y] Π Π 0 γ v Π vv [ αβ ( α λ v] y [ a ( a l y] Π Π 0 xv y Π yy ργvπ v Πv y Π x y xv Gss h followin form for h characrisic fncions: f f ( ( v ( y ix ln( / ( ( ( v ( y ix ( Pl in h ss solion for h characrisic fncions an oain a sris of Os as follows: γ ργ x [( α λ ( i ] ( i ( a l ( i αβ a an γ αβ a [( α λ ργi] ( i ( a l i h solions o an ( ar: γ ( ( 6

27 ( ( αβ γ a ( ( ( ( ln ln ( ( ln ln whr [( λ ργ ( i ] ( i α γ ( λ ργ ( i α ( a l ( i a l [( λ ργ i] ( i α γ ( α λ ργ i ( a l i A3 rivaion of h haracrisic ncions of Ps Gss h ppr on of ps as follows: U P K K y ( Π ( Π whr is h sam as calls an is h proailiy oain in h forwar Π masr h P for h v Π xx v y Π Π γ v Π Π coms: y Π [ αβ ( α λ ργ v] x vv yy ργvπ Π v xv [ a ( a l y] Π y Π 0 Gss h followin form for h characrisic fncion: 7

28 ( ( ( ( ix y f ν Pl in h ss solion for h characrisic fncions an oain a sris of Os as follows: ( ( [ ] ( i i ργ λ α γ ( i l a a αβ h solions o an ar: ( ( ( ( ( ( ( ( ln ln ln ln a γ αβ whr ( ( [ ] ( i i γ ργ λ α ( ( i ργ λ α ( i l a l a A4 rivaion of h haracrisic ncions of rs Opions Gss h ppr on of frs calls as follows { } [ ] 0 max s s K K U Π Π h ppr on of frs ps is as follows: { } [ ] ( ( 0 max s s P K K U Π Π 8

29 h Ps for h proailiis com: an whr v Π v Π x ln xx γ v Π vv ργvπ xv [ αβ ( α λ ργ v] Π Π Π 0 xx γ v Π vv ργvπ v v v x [ αβ ( α λ v] Π Π Π 0 ( f v xv x Gss h followin form for h characrisic fncions: ( ( ( ν ix Pl in h ss solion for h characrisic fncions an oain a sris of Os as follows: γ ργ αβ [( α λ ( i ] ( i γ αβ [( α λ ργi] ( i h solions o an ( ar: whr αβ γ ( ( ln [( λ ργ ( i ] ( i α γ ( λ ργ ( i α [( λ ργ i] ( i α γ ( α λ ργ i ( ln 9

30 rivaion of an h haracrisic ncions of Or Uppr on of alls Whn ( M r > Gss h followin form for : ( M ( ( ( y A M Pl in h ss solion for ( an oain a sris of Os: ( l a A a Wih onary coniion ( ( A w can show ha: ( ( whr ( l a l a an ( [ ] ( } ln ln { a A Nx w will show h characrisic fncions of or ppr on Gss h followin form for h characrisic fncions: ( ( ( ( ix y v f ( ( ( ( ( ( M ix y v f ln Pl in h ss solion for h characrisic fncions an oain a sris of Os: ( ( [ ] ( i i ργ λ α γ ( i l a a αβ an ( [ ] ( i i ργ λ α γ ( ( i l a a αβ h solions o an ( ar: 30

31 γ ( ( whr ( ( αβ γ a ( ( ( ( ln ln [( λ ργ ( i ] ( i α γ ( λ ργ ( i α ( ( ln ln ( a l i a l [( λ ργ i] ( i α γ ( α λ ργ i ( a l ( i rivaion of h haracrisic ncions of Uppr on of alls Whn r < Gss h followin form for h characrisic fncions: f ( ( ( v ( y ix Pl in h ss solion for h characrisic fncions an oain a sris of Os: ργ γ αβ a [( α λ ( i ] ( i ( a l i 3

32 an ( [ ] ( i i ργ λ α γ ( i l a a αβ h solions o an ( ar: ( ( γ ( ( ( ( ( ( ( ( ln ln ln ln a γ αβ whr ( ( [ ] ( i i γ ργ λ α ( ( i ργ λ α ( i l a l a ( [ ] ( i i γ ργ λ α ( ργ i λ α ( ( i l a 3

33 AL orrc nmrical rsls in hn an Yh s al all Opion P Opion K Oriinal Uppr on orrc Uppr on ropan Opion Val Oriinal Uppr on orrc Uppr on ropan Opion Val PU(sc h sock pric is 00 h iniial volailiy is 004 h risk-fr ra is 005 h ivin is 003 h im o mariy is on yar an ohr paramrs ar: α 5 β 0 04 γ 0 λ 0 ρ 05 an 33

34 AL orrc nmrical rsls in hn an Yh s al ρ K Panl A alls on ivin Payin ocks Panl Ps on Non-ivin Payin ocks Panl alls on rs Panl Ps on rs (conin on nx pa 34

35 AL (conin orrc nmrical rsls in hn an Yh s al K Panl alls on ivin Payin ocks Panl Ps on Non-ivin Payin ocks Panl G alls on rs Panl H Ps on rs h op nmrs ar ppr on vals an h oom nmrs ar ropan vals h iniial vals for h sa varials ar 00 for h sock pric 004 for h iniial volailiy val an 005 for h inrs ra lvl h paramrs for h varianc procss ar α 5 β 0 04 γ 0 0 ρ 0 5 an λ 0 h paramrs for h inrs ra procss ar a (in Panls an an l 0 h paramrs for h mp procss ar µ 0 0 an λ 06 h ivin yil is assm o a consan of 003 λ 35

36 AL 3 Uppr on of h Amrican calls on ivin payin socks whn ρ K r > K λ h op nmrs ar hn an Yh s ppr on vals h scon nmrs ar or ppr on vals h oom nmrs ar ropan opion vals h paramr vals ar h sam as al 36

37 AL 4 Uppr on of h Amrican calls on ivin payin socks whn ρ K r < K h op nmrs ar ppr on vals an h oom nmrs ar ropan vals h paramr vals ar h sam as al xcp r 0 03 an 0 05 λ 37

38 AL 5 Uppr ons of h Amrican xchan opions on ivin payin socks Panl A: r > > hn an Yh Or ppr on Amrican vals ropan vals h paramr vals ar Panl : > r > hn an Yh Or ppr on Amrican vals ropan vals 30 NA NA NA NA NA h paramr vals ar AL 6 Uppr ons of h Amrican maximm call opions on wo ivin payin socks Panl A: r > > K hn an Yh Or ppr on Amrican vals ropan vals h paramr vals ar an 003 Panl : > r > K hn an Yh Or ppr on Amrican vals ropan vals 30 NA NA NA NA NA h paramr vals ar 40 3 ρ 05 r an

39 Rfrncs akshi G ao an Z hn 997 "mpirical Prformanc of Alrnaiv Opion Mols" ornal of inanc as 99 "h rash of 87: Was i xpc? h vinc from Opions Marks" ornal of inanc rksn P an nslan G 993 "Amrican xchan Opions an a P-all ransformaion: A No" ornal of sinss inanc an Acconin oyl P P an Y K s 990 "An Alorihm for ompin Vals of Opions on h Maximm or Minimm of vral Asss" ornal of inancial an Qaniaiv Analysis oyl P P an X Lin 997 "ons on oninn laims as on vral Asss" ornal of inancial conomics rin-ons M an A Nrr 000 "Opions Prics Impli Pric Procsss an ochasic Volailiy" ornal of inanc roai M an mpl 996 "Amrican Opion Valaion: Nw ons Approximaions an a omparison of xisin Mhos" Rviw of inancial is 9-50 roai M an P Glassrman 997 "Pricin Amrican-yl criis Usin imlaion" ornal of conomic ynamics an onrol arr P R arrow an R Mynni 99 "Alrnaiv haracrizaions of Amrican P Opions" Mahmaical inanc hn R R L hn an Yan 00 "Opion Pricin in a Mli-ass ompl Mark conomy" ornal of inancial an Qaniaiv Analysis hn R R an K Yh 00 "Analyical Uppr ons for Amrican Opion Prics" ornal of inancial an Qaniaiv Analysis

40 onsaninis G M an Prrakis 00 "ochasic ominanc ons on rivaivs Prics in a Mliprio conomy wih Proporional ransacion oss" ornal of conomic ynamics an onrol rman I Kani an N hriss 996 "Impli rinomial rs of h Volailiy mil" ornal of rivaivs 7- ilwski an Gao 999 "h Aapiv Msh Mol: A Nw Approach o fficin Opion Pricin" ornal of inancial conomics Garman M 976 "An Alra for valain H Porfolios" ornal of inancial conomics Hah M an L Koan 00 "Approximain Pricin an xrcisin of Hih-imnsional Amrican Opions: A aliy Approach" workin papr MI Hson 993 "A los orm olion for Opions wih ochasic Volailiy wih Applicaions o on an rrncy Opions" Rviw of inancial is Hll an A Whi 988 "h Us of h onrol Varia chni in Opion Pricin" ornal of inancial an Qaniaiv Analysis ackwrh 999 "Opion-Impli Risk-Nral isriions an Impli inomial rs: A Lirar Rviw" ornal of rivaivs 66-8 ohnson H 987 "Opions on h Maximm or h Minimm of vral Asss" ornal of inancial an Qaniaiv Analysis Lvy H 985 "Uppr an Lowr ons of P an all Opion Val: ochasic ominanc Approach" ornal of inanc Lo A W 987 "mi-paramric Uppr ons for Opion Prics an xpc Payoffs" ornal of inancial conomics Marra M 978 "h Val of an Opion o xchan On Ass for Anohr" ornal of inanc

41 Mron R 973 "h hory of Raional Opion Pricin" h ll ornal of conomics an Manamn cinc Mron R 976 "Opion Pricin Whn Unrlyin ock Rrns Ar isconinos" ornal of inancial conomics Richkn P 985 "On Opion Pricin ons" ornal of inanc Richkn P an Ko 988 Opion ons wih ini Rvision Opporniis ornal of inanc Rors 00 "Mon arlo Valaion of Amrican Opions" workin papr Univrsiy of ah Rinsin M 994 "Impli inomial rs" ornal of inanc co L 987 "Opion Pricin whn h Varianc hans Ranomly: hory simaion an an Applicaion" ornal of inancial an Qaniaiv Analysis co L 997 "Pricin ock Opions in a mp-iffsion Mol wih ochasic Volailiy an Inrs Ras: Applicaions of orir Invrsion Mhos" Mahmaical inanc lz R M 98 "Opions on h Minimm or h Maximm of wo Risky Asss" ornal of inancial conomics