A Note on the Kou s Continuity Correction Formula

Transcription

1 A Noe o he Kou s Coiuiy Correcio Forua Tig Liu Chag Feg YaQiog Lu * Bei Yao * Schoo of Opica ad Eecroic Iforaio Huazhog Uiversiy of Sciece ad Techoogy Schoo of Maheaic ad Saisic Huazhog Uiversiy of Sciece ad Techoogy Asrac: This arice iroduces a hyper-expoeia up diffusio process ased o he coiuiy correcio for discree arrier opios uder he sadard B-S ode usig easure rasforaio ad soppig ie heory o prove he correcio hus roadeig he codiios of he coiuiy correcio of Kou Key words: B-S ode discree arrier opios hyper-expoeia up diffusio coiuiy correcio forua Exordiu I 3 SGKou [] geeraized a doue expoeia up diffusio ode io he pricig of coiuiy sige arrier opios They aayzed he oi disriuio of he fia asse price y focusig o firs passage ie ad coiig i wih he oiear reewa heory of he ie series aaysis The Lapace rasfors ad he o-eory propery of expoeia disriuio were aso used durig he aayzig process I his arice which was puished i 997 Kou [-4] pu forward a coiuiy correcio of B-S ode which coied he pricig ehod of coiuiy arrier opios wih ha of discree arrier opios I 3 DJu [5] geeraized he coiuiy correcio forua io he doue arrier opios i view of Kou s repor Aso i 3 C D Fuh [6] iroduced he doue expoeia up diffusio ode o he pricig of discree sige arrier opios ad ook-ack opios His arice wideed he codiios of he coiuiy correcio forua of Kou ad oaied a correcio forua of discree arrier opios ased o he doue expoeia up diffusio process This arice uses Kou s heoreica derivaio ehod for he firs passage ie ad CDFuh s hough of provig he correcio forua ased o doue expoeia up diffusio ode for referece I order o geeraize he ode io he pricig of oh he discree sige ad doue arrier opios o he asis of he hyper-expoeia up diffusio process we coie he correcio forua of discree sige arrier opios raised y Kou wih DJu s correcio of discree doue arrier opios uder he B-S ode Ahough foray he correcio forua i his arice sees o e sae as Kou s ad DJu s here are o resricios of srike price K ad arrier vaue H i he correcio forua i his passage which eas ha he scope of appicaio of Kou s ad DJu s correcio are wideed Addiioay copared wih he doue expoeia up diffusio ode of CDFuh he hyper-expoeia up diffusio ode i his arice is ore geera * Correspodig auhor (eai: @63co)

2 pricig o arrier opios coiuiy correcio ode I 997 Seve Kou pu forward he cocep of coiuiy correcio i his paper which coied he coiuiy arrier opios wih he discree arrier opios hrough he coiuiy correcio forua Deoe he soppig ie y which is aso caed he firs passage ie he defiiio goes as foows: or ( H) if{ R : S H} ( ( H) if{ R : S H}) () ( H ) if{ N : S H} ( ( H ) if{ N : S H}) () Here H sads for he arrier eve ad T N is he frequecy of oiorig Forua () ad () represe he siuaio of coiuousy oiored ad discreey oiored respecivey Apparey a up ode has S H ad a dow ode has S H For coveiece we assue he risk-eura ieres rae r is a cosa i his case Because of he exisece of up he arke here is o a copee arke which eas ha he risk euraiy easure is o uique (ore deais ca e foud i Kou [7] ) As a aer of coveiece we assue he risk euraiy easure Q us eeds o ee he preise of raioa expeced equiiriu Kou has poied ou i his arice ha whe is arge eough which eas forua () is weaky coverge o forua () However wih he icreasig of he covergece rae wi ecoe very sow ad he error wi aso icrease To sove his proe Kou ipeeed ehods of ie series aaysis which coud provide iiaios of K ad H ad esaished correcio forua as foow ased o he B-S ode o fase he covergece rae: V ( H) V ( He ) ( ) (3) If H S + wi e appied if H S - wi e appied Here ( ) 586 wih () sadig for Riea zea fucio which ca e cocreey wrie as () s s

3 3 This ehod of aduse is caed coiuiy correcio which has wo asic hypoheses: (a) he arrier-crossig proaiiy of discree price is ower ha he coiuous price; () here wi e overshoo whe oecs are discreey oiored (Fig -) Fig- Accordig o he correcio forua we coud fid ha he arrier-crossig proaiiy wi e ower afer he aduse of discreey oiored arrier The aou of he aduse is S H H(e ) which is he overshoo i Fig- To soe degree his kid of aduse is he expeced vaue of a overshoo hyper-expoeia up diffusio ode We assue ha uder he risk euraiy easure Q he asse price S wi oey he foowig hyper-expoeia up diffusio geoeric Browia oio ode: J ds = ( r -d- ) Sd SdW Sd( e ) (4) N u i J J i i J J (5) f ( J) p e q e i d Here W is a sadard Browia oio N is a Poisso process wih iesiy S sads for he uderyig asse price a previous oe ad cosa rd represe he risk-free ieres rae divided rae (excep he exi righ siuaio) ad risk voaiiy respecivey Ji : i is a sequece of idepede ideicay

4 4 disriued rado variaes which oeys a asyerica hyper-expoeia disriuio wih he coo desiy fucio f( J ) where p i u ; q d represe he proaiiy of i i upig upward ad dowward ad he up vaue respecivey Accordig o forua (5) we coud kow ha he up is icudig upward up of u cass ad dowward up of d cass he upward up proaiiy of cass i is up proaiiy of cass is p i he dowward p For coveiece we assue a he foowig rado processes : N : W : J : i i ad u d p q i i are idepede Wha s ore wihou oss of geeraiy we coud cocude fro i u he i ea of every up upward ad he ea of every up dowward d ha: Q p q E ( e ) J u i i d i= i Wha we eed o pay aeio o is ha whe u=d= he hyper-expoeia up diffusio ode wi degeerae o he sadard B-S ode whie whe u=d=he hyper-expoeia up diffusio ode wi degeerae o he doue expoeia up diffusio ode which eas ha B-S ode ad he doue expoeia up diffusio ode are he specia fors of he hyper-expoeia up diffusio ode Usig Io ea ad heories of cacuaio of sochasic paria differeia equaios he souio of ode (4) uder coiuousy oiored siuaio is: S exp S X (6) X W M (7) where N ad M J M r d

5 5 Now we cosider he discreey oiored siuaio Assue he discreey oiored ie ierva is for coveiece we ake aoher assupio ha he oiored ie sep is equa Therefore he discreey oiored asse price is S S S exp( X ) X a he h ie of oiorig As a resu X ( Z M ) (8) i i d where Z ~ N () i i d i i d N ~ ( ) ~ M J N P J J Accordig o forer assupios a rado variaes here are idepede Now we suppose F ~ Sk k is he - r agera cosequey e S F wi si e a Q-arigae V( H) sads for he price of coiuousy oiored opios whie V ( ) H is he price of discreey oiored opios ad he oher paraeers are supposed o e he sae Take he coiuous up-i-pu opios for exape he opio reur is ( K- S ) H S T ( H ) T ad he correspodig discree siuaio is ( K- S T ) We use X ( H) T he ogarih uderyig asse price o repace for ore expici expressio Therefore equaios () () coud e wrie as: ( H) if( R : X ) ( X ); ( H) if( Z : X ) ( X ) (9) where og( H S) Usig heories of easure rasforaio here are foruas of coiuousy ad discreey oiored opios as foows: S rt S T V ( H) e KQ ( XT a ( X ) T) Q ( XT a ( X ) T) rt S T V ( H) e KQ ( X a ( X ) T) Q ( X a ( X ) T) () Where a og( K S) Now we ry o fid he reaioship ewee his wo oi disriuio desiies ( X ) ad ( X ) For (7) ad (8) defie i Cosequey he characerisic fucio ca e expressed as T ix ix ( ) E[ e ] ( ) E[ e ] I Lévy process accordig o he Lévy- Khichie forua for ( ) ( ) e

6 6 where ( ) is he expoeia characerisic fucio of X i Lévy process Siiary for = ( ) ( ) e Le () e he correspodig characerisic fucio of up size J ad he J X cuuaive geeraig fucio is G( ) ( i ) ( aey G( ) E( e ) ) Therefore G( ) [ J ( i ) ] () u pi i d q J ( i ) i i I N Cai [8] for G( ) he uer of rea roos are u+d+ ad ee: d + d d d u u u u More specific proof ad properies of he cuuaive geeraig fucio G ca e foud i NCai [9] For coveiece we use i ad o repace i i u ad d respecivey Deoe ( X ) ad ow we provide he correspodig Lapace rasfor Proposiio [9] For hyper-expoeia up diffusio ode (4) ad (5) whe G( ) for he Lapace operaor of soppig ie ( R ) ca e wrie as foows: ( ) Ee ( ) x x [ ] u - e x ()

7 7 where : ( ) T is uiquey deeried y iear equaios AB J (here J is oy a sig which is differe fro J ) A is a ( u) ( u ) fu rak arix: u A u u u u u u u u B is a diagoa arix i a for of ( u) ( u ) ad ca e expressed as B Diag e e e u J ( e e e ) { u }Wha s ore T I view of proposiio we ca oai he proaiiy disriuio of is iy iy e ( iy) e ( iy) p( ) dy (3) iy

8 8 3 Pricig o discree sige arrier opios I his secio we wi have furher discussio o he overshoo which has ee eioed efore ased o he discree ode (8) ad oai he ai cocusio of his arice Firsy we ake a aduse o he discree ode X deoe X W he W As a resu M W Z V ( ) ( X ) ( W ) where Therefore he overshoo ca e udersood as X ( W ) R The preparaio has ee doe however wo preparaory eas wi e iroduced efore fiay exhii he ai cocusio Lea For coiuousy oiored soppig ie discreey oiored soppig ie (forua 9) auriy T ad oiorig ie ierva T whe he oiorig ies he foowig equaio is saisfied for arirary cosa : P( ( X ) T) P( ( X ) T) o( ) Proof: Assue (4) is a rue saee i view of () ad (3) whe E[ e ] E[ e ] o( ) ( ) ( ) X X (4) Fro proposiio he equaio aove wi e rue if he foowig equaio is proved:

9 9 E[ e ] e o( ) (5) u ( ) ( ) where : To prove his forua for a arirary cosa > ad a very sa cosa h> defie a fucio u() which ca e wrie as : ux x h ( ) u ( ) (6) h x e e x h Wha we eed o pay aeio o is ha u herefore he coiuiy of u() is esured The ased o i i u i is o difficu o fid ha for x u x Ad ecause X h ( ) u ( X ) h e e X h (7) ux fro which we ca cocude ha he overshoo ca e represeed as he su of he up ad discree reae I order o oai equaio (5) we aso eed o ook for he correspodig discree sequece fucio of fucio u which is defied i (6) Here we sove his proe y adusig Kou s ehod of he firs passage ie uder he doue expoeia up diffusio process Noicig ha u( x) Lu( x) x h (8) where L sads for he ifiiesia geeraor Lu x u x u x u x y u x f dy '' ' ( ) ( ) ( ) [ ( ) ( )] Y( y) (9) Here u x is supposed o e wice coiuousy differeiae For e u( X ) : we use forua Io o cosruc a series of fucio u ( x) : u ( x) possesses he foowig properies: (a) u ( x ) is hree ies coiuousy differeiae;

10 () whe x( h] [ h ) u( x) u( x) ; (c) x( h h ) u( x) [] ad here are cosa k k k 3 ha ca ee he foowig iequaiies: u ( x) k u ( x) k u ( x) k ' '' ''' 3 As a resu for x R here are u ( x) u( x) The coiig (8) ad (9) for a x h '' ' Lu( x) u( x) u( x) [ ( ) ( )] ( ) u x y u x fy y dy '' ' u ( x) u ( x) u( x) u( x y) fy( y) dy As u u for a x h hx Y u( x) [ u ( x y) u( x y)] f ( y) dy hx M u Lu ( x) ( ) () where M u p i i Ad ecause u is a cosa we ca oai ha M is aso i a cosa as a resu a Focusig o for u ( ) ( ) Lu x () e u X : we ipee forua Io ad coie() ( ) ( ) s : ( ) ( ( ) ( )) s s M e u X e u X Lu X ds () is he oca arigae ad ( ) M u u are a series of soppig ie { ; } () () Accordig o he defiiio here T ad p T i as we Therefore ( ) ( ) M for every { M T: } is a arigae As M we ca ( ) M oai ha M T ad accordig o Leesgue s doiaed covergece heore

11 ore precisey ad i view of () ( ) s [ ( ) ( ( ) ( )) ] s s E e u X e u X Lu X ds EM i EM i EM u() ( ) ( ) ( ) T T ( ) ( ) i E[ e u( X )] E[ e u( X )] s i E[ e ( u ( X ) Lu ( X )) ds)] s i E[ e ( u ( X ) Lu ( X )) ds)] s s If we iegrae he hree foruas aove for a u E e u X ( ) () [ ( )] s s ( ) ( ) E[ e u( X ) { }] E[ e u( X ) { }] ad whe accordig o he oudedess of u u() E[ e u( X ) { }] Now we are goig o deiver he derivaio of correcio forua of discreey oiored siuaio For coveiece X k is used o repace X k Firsy discreize () he we oai k ( ) ( k ) k : ( k ) [ ( ) ( ) s M e u X e u X Lu X [ k ] ( k ) k e u ( X ) e [ u ( X ) Lu ( X ) k e [ u ( X ) Lu ( X ) [ k ] Based o Kuiius [] e 5 k whe accordig o he defiiio of u u E e u X ( ) () i [ ( )] ( k ) i{ E[ e u( X )] o( )} E e u X o ( k ) [ ( k )] ( )

12 Furherore e e k u() E[ e u( X ) { }] o( ) u h ( X ) E[ e e e ] P( X h) E[ e ] P( X h) o( ) I II o( ) As / ad X are asypoicay idepede whe u h ( X ) E( I) E[ e ] e E[ e ] u h E[ e ] e [ E[ R ]] o( ) (3) E[ e ] o( ) which oeys Tayor expasio of he firs order ad ER Here we use Kou s reae of R for referece ad defie rado wak i i d B Z Z N E R E B E B o [ ] [ ] [ ] ( ) Whe h accordig o u forua (3) ca e wrie as u h E( I) E[ e ] e [ ] o( ) E[ e ] o( ) Therefore

13 3 u() { E[ e ] o( )} P(X ) { E[ e ] o( )} P(X ) ( ) o E[ e ][ P(X ) P(X )] o( ) E[ e ] o( ) Lea ca e proved y susiuig (4) ad (6) io (5) (4) Lea 3 For discree arrier opios wih ies of oiorig ad vaue of arrier H here are soppig ie ( ) B (for coveiece deoe ( ) ( ) ( ) X B B X ) ad heir correspodig ogarih uderyig asse B prices X X (forua 8) For (forua ) he oi B disriuio ( X ( ) ) ad ( ) ee he foowig equaio: B X B P( X ( ) ( ) ) ( ) ( ) x P X x B B o (5) where B og( H S) Proof: ( ) X ( ) ( ) X ( ) ( ) X ( ) { ( ) } { ( ) } X { X ( ) } E[ e ] E[ e ] E[ e ] ( ) ( X ( ) ) ( ) ( X ( ) ) { X } e E e { X } e E[ e ] [ ] e ( I II) O he asis of proposiio i Kou [] () ad () are asypoicay idepede he we ca oai ha X ( ) ( ( ) ) X [ { X } ] [ ] I E e E e ( ) e E[ e { X } ] o( ) Deoe y x / for c whe c accordig o Zhag []

14 4 Therefore ( ) ( X ( ) X X ) II E[ e ] { X } ( ) ( ( ) ( )) X [ { X } ] [ ] E e E e e e d c E e e e e d o ( ) [ { X } ] ( ) E[ e ] e E[ e ] e E[ e ] o( ) ( ) X ( ) B ( ) ( ) { ( ) } B { } X { X } B B e E[ e ] e E[ e ] o( ) B B { X B} { X B} The ea 3 ca e proved y usig coroary 33 i Kou [] (6) Theore Le V( H ) deoe he price of coiuousy oiored opios wih arrier vaue H accordigy V ( H ) is he price of discreey oiored opios wih he oiorig frequecy Cosequey for a arirary discree arrier opio wih auriy T T ad whe we oai he foowig correcio forua: V ( H) V ( He ) o( ) (7) where ( ) 586 wih () he Riea zea fucio The proof of he heore ca e derived direcy fro ea ad 3 Wha we eed o iusrae is ha ahough he cocusio here sees o e siiar o Kou [] s here are o resricio of K H (upward) ad K H (dowward) for he srikig price K ad arrier H Referece [] SGKouFirs passage ies of a up diffusio process[j]advaces i Appied Proaiiy335():54-53 [] SkouA coiuiy correcio for discree arrier opios[j]maheaica Fiace 9977(4) : [3] MBroadiePGasseraSGKouCoecig discree ad coiuous pah-depede opios[j]fiace Sochasic9993:55 8 [4] S G Kou O pricig of discree arrier opios[j] Saisic Siica 3 3: [5] DJuCoiuiy correcio for discree arrier opios wih wo arriers[j]

15 Joura of Copuaioa ad Appied Maheaics 3 37: 5 58 [6] CDFuhS FLuoJFYePricig discree pah-depede opios uder a doue expoeia up diffusio ode[j]joura of Bakig & Fiace 3 37: 7 73 [7] SGKouA Jup-Diffusio Mode for Opio Pricig[J]Maagee Sciece 48(8):86 [8] NCaiSGKouOpio pricig uder a hyper-expoeia up diffusio ode[j] Prepri 7 [9] NCaiO firs passage ies of a hyper-expoeia up diffusio process[j] Operaios Research Leers937 :7-34 [] NThakoorDYTagaMBhuruhEfficie ad high accuracy pricig of arrier opios uder he CEV diffusio[j]joura of Copuaioa ad Appied Maheaics459:8 93 [] CHZhagA oiear reewa heory[j]aas of Proaiiy9886: