On the goodness of t of Kirk s formula for spread option prices

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1 On he goodne of of Kirk formula for pread opion price Elia Alò Dp. d Economia i Emprea Univeria Pompeu Fabra c/ramon ria Farga, Barcelona, Spain Jorge A. León y Conrol Auomáico CINVESAV-IPN Aparado Poal México, D.F., Mexico Abrac In hi paper we inveigae he goodne of of he Kirk approximaion formula for pread opion price in he correlaed lognormal framework. oward hi end, we ue he Malliavin calculu echnique o nd an expreion for he hor-ime implied volailiy kew of opion wih random rike. In paricular, we obain ha hi kew i very pronounced in he cae of pread opion wih exremely high correlaion, which canno be reproduced by a conan volailiy approximaion a in he Kirk formula. hi fac agree wih he empirical evidence. Numerical example are given. Keyword: Spread opion, Kirk formula, Malliavin calculu, Skorohod inegral. JEL code: G, G3 Mahemaical Subjec Clai caion: 9B8, 9B70, 60H07. Inroducion A pread opion i a derivaive wrien a he di erence of wo underlying ae. Namely, he payo of a call pread opion wih rike K wih ime o mauriy i (S S K) ; where S and S denoe wo ae price. I i well-known ha if S and S are wo geomeric Brownian moion (ha may be correlaed) and K = 0; he correponding opion price i given by he Margrabe fomula (ee Margrabe (978)), which can be deduced from he fac ha S =S i a log-normal proce. hu, in hi cae, he pread opion value can be expreed a he claical Black-Schole call price wih iniial ae price S 0; where we ake he rike Suppored by he gran ECO and MEC FEDER MM y Parially uppored by a CONACy gran

2 equal o he expeced value of S q and volailiy equal o 0 ( 0 ). Here and 0 are he volailiy parameer of S and S, repecively, and denoe he correlaion. In he cae K 6= 0 he fracion S = S K i no log-normal and hen he argumen ued in he deducion of Margrabe formula canno be applied anymore. One propoed oluion i o aume he diribuion of he pread i able o be approximaed by he normal diribuion. hi lead u o he Bachelier mehod (ee Shimko (994)). Even hough he approximaion of he pread diribuion by he normal one i poor, hi mehod can be precie in ome range of parameer (ee, for example, Carmona and Durrleman (003)). A more ucceful mehod i uggeed by Kirk (995), who applied he Margrabe formula, aproaching S = S K by a log-normal random proce. Nowaday Kirk formula i he mo popular opion pricing approximaion expreion for pread opion due o i accuracy and i impliciy. Recenly, ome oher auhor have propoed new formula and mehod o eimae pread opion price. Among hem, we can menion Alexander and Venkaramanan (0), Bjerkund and Senland (0), Borovkova e al. (007), Carmona and Durrleman (003), and Deng e al. (008). Numerical evidence ha hown ha hee approache may improve Kirk eimae, pecially for high correlaed cae (ee, for inance, Bjerkund and Senland (0)). One inereing ool in he developmen of an accurae approximaion formula i he knowledge of he properie of he correponding implied volailiy. For example, in he cae of vainilla opion wih ochaic volailiy, we know ha implied volailiie exhibi mile and kew (ee Renaul and ouzi (996)). hen, we have o ake ino accoun ha accurae approximaion hould reproduce hem and, moreover, we can underand why ome formula fail (for example, we expec ha a conan volailiy expreion will fail for in or ou-of-he-money opion). Up o our knowledge, here are no imilar analyical udie in he cae of opion wih random rike (a for example pread opion). he main goal of hi paper i o conider analyically he relaion beween he implied volailiy and he ock price S for opion wih random rike. By mean of he Malliavin calculu hecnique, we develop an exenion of he Margrabe formula ha allow u o nd an expreion for he hor-ime kew lope for pread opion. hi analyi of he implied volailiy kew evidence ha he dependence beween he implied volailiy and he ock price S increae rongly when he correlaion parameer i cloe o. Hence we can expec ha Kirk approximaion reduce i accuracy for highly correlaed ae due o he fac ha in hi approximaion he volailiy i conan a a funcion of he ock price S. hi agree wih he numerical evidence (ee, for example, Baeva (0)). he organizaion of hi paper i a follow. In Secion we inroduce he framework of hi paper. Secion 3 i devoed o preen a exended Margrabe formula for he cae K 6= 0. hi formula i ued in Secion 4 o gure up an expreion for he derivaive of he implied volailiy wih repec o he logock price. he hor ime behaviour of hi derivaive i analyzed in Secion

3 5. Finally in Secion 6 we obain our reul on he goodne of he of Kirk formula a an applicaion. Saemen of he model and noaion In hi paper we conider he following model for he log-price of a ock under a rik-neural probabiliy meaure Q: dx = r d (dw p db ); [0; ]: () Here, r i he inananeou inere rae, W and B are independen andard Brownian moion and ( ; ). For he ake of impliciy, we aume ha he volailiy proce i a quare-inegrable deerminiic funcion which i righ-coninuou. In he following we denoe by F W and F B he lraion generaed by W and B; repecively. We de ne F := F W _ F B : In hi paper we conider European call opion wih payo h(x ) := e X K ; where we allow he rike K o be random. More preciely, we aume K i a quare-inegrable, poiive, coninuou, bounded and F W - meaurable proce. Noice ha hi choice include ome popular clae of opion a bake one. I i well-known ha he price of an European call wih rike K i given by he formula V = e r( ) E (e X K ) jf : () In he equel, we will make ue of he following noaion: M := E K j F W : Oberve ha, by he maringale repreenaion heorem, M = E (K ) 0 m(; )dw ; for ome F W meaurable and adaped proce m (; ) : v := Y ; wih Y := R a d; where a d := d dhm ;Xi M dhm ;M i (M ). Noe ha a = m(; ) M m (; ) (M ) = m(; ) M i a poiive quaniy. Alhough he righ-hand-ide of he la equaliy depend on, we denoe i by a in order o implify he noaion. I i eay o ee ha for all ( ; ), a C () : 3

4 BS(; x; K; ) denoe he price of an European call opion under he claical Black-Schole model wih conan volailiy, curren log ock price x, ime o mauriy ; rike price K and inere rae r: Remember ha in hi cae: BS(; x; K; ) = e x N(d ) Ke r( ) N(d ); where N denoe he cumulaive probabiliy funcion of he andard normal law and wih ~x := ln K r( ): d := x ~x p p ; L BS and for he Black-Schole di erenial operaor, in he log variable, wih volailiy : L xx (r )@ x r I i well known ha L BS BS(; ; ) = 0: Now we decribe ome baic noaion ha i ued in hi aricle. For hi, we aume ha he reader i familiar wih he elemenary reul of he Malliavin calculu, a given for inance in Nualar (006). Le u conider a andard Brownian moion = f ; [0; ]g de ned on a complee probabiliy pace (; F; P ): he e D ; i he domain of he derivaive operaor D in he Malliavin calculu ene. D ; i a dene ube of L () and D i a cloed and unbounded operaor from L () ino L ([0; ] ): We alo conider he ieraed derivaive D ;n ; for n > ; whoe domain i denoed by D n; W : he adjoin of he derivaive operaor D, denoed by ; i an exenion of he Iô inegral in he ene ha he e L a([0; ] ) of quare inegrable and adaped procee (wih repec o he lraion generaed by ) i included in Dom and he operaor rericed o L a([0; ] ) coincide wih he Iô inegral. We make ue of he noaion (u) = R 0 u d : We recall ha L n; := L ([0; ]; D n; W ) i included in he domain of for all n : 3 A decompoiion reul Before proving an exenion of he Hull and Whie formula, we ae he following reul, which i nedeed in he remaining of he paper. 4

5 Lemma Le K be bounded, 0 < and G := F _ F W. Aume ( ; ) :hen, for any n 0, here exi C = C(n; ) uch ha E (@ n )BS(; X ; M ; v )jg x n C d (n) : Proof: In order o how hi reul, we proceed a in he proof of Lemma 4. in Alò, León and Vive (007) and we ue he fac ha K i a bounded and F W meaurable and adaped proce o obain ha E (@ n )BS(; X ; M ; v )jg x n C ( ) d a d (n) : We know ha for all ( ; ), a C () for ome poiive conan C (). hen R a d C () R d; from where he reul follow. Now we are able o prove he main reul of hi ecion, he exended Hull and Whie formula. We will need he following hypohei: (H) he proce a L ; : heorem Conider he model () and aume ha hypohei (H) hold. hen i follow ha V = E BS(; X ; M ; v ) F E ( where W := e r( 3 xx BS(; X ; M ; v ) W d ) e r( x BS(; X ; M ; v ) W m(; )d F ; (3) h D W R i a (r)dr : Proof: hi proof i imilar o he one of he main heorem in Alò, León and Vive (007), o we only kech i. Noice ha BS(; X ; M ; v ) = V : hen, from (), we have e r V = E(e r BS(; X ; K ; v )jf ): Now, uing he Iô formula o he proce e r BS(; X ; M ; v ) 5

6 and proceeding a in Alò, León and Vive (007) (ee alo Alò and Nualar (998), Alò (006) or Nualar (006)), we can wrie e r BS(; X ; M ; v ) = e r BS(; X ; M ; v ) e r L BS (v )BS(; X ; M ; v )d e x BS(; X ; M ; v ) (dw p db ) e K BS(; X ; M ; v )dm e xkbs(; X ; M ; v )d M ; X e BS(; X ; M ; v ) v v ( a ) d e xbs(; X ; M W ; v ) v ( ) d e KBS(; X ; M ; v ) W m(; ) v ( ) d e x BS(; X ; M ; v ) v d e KKBS(; X ; M ; v )d M ; M : 6

7 Hence, he fac ha L BS (v)bs(; X ; M ; v ) = 0; muliplying by e r and aking condiional expecaion we can eablih E e r( ) BS(; X ; M ; v ) F ( = E BS(; X ; M ; v ) e r( e r( BS(; X ; M ; v ) v a v ( ) d e r( xbs(; X ; M W ; v ) v ( ) d e r( KBS(; X ; M ; v ) W m(; ) v ( ) d e r( xkbs(; X ; M ; v )d M ; X e r( x BS(; X ; M ; v ) v d KKBS(; X ; M ; v )d M ; M ) F : Conequenly, he claical relaionhip beween he x BS BS ( xkbs BS K( KKBS BS K ( ) give E e r( ) BS(; X ; M ; v ) F ( = E BS(; X ; M ; v ) e r( BS(; X ; M ; v ) M v ( e r( BS(; X ; M ; v ) v a v ( ) d e r( xbs(; X ; M W ; v ) v ( ) d e r( KBS(; X ; M ; v ) W m(; ) v ( ) d e r( BS(; X ; M ; v ) v v ( e r( BS(; X ; M ; v ) M v ( ) d ) d M ; M ) F : ) d M ; X 7

8 ha i, E e r( ) BS(; X ; M ; v ) F ( = E BS(; X ; M ; v ) " d M ; X M e r( BS(; X ; M ; v ) v ( ) v a d v d e r( xbs(; X ; M W ; v ) v ( ) d ) e r( KBS(; X ; M ; v ) W m(; ) v ( ) d F : d M ; M (M ) Since, a d := d dhm ;Xi dhm ;M i we obain M (M ) E e r( ) BS(; X ; M ; v ) F ( = E BS(; X ; M ; v ) e r( xbs(; X ; M W ; v ) v ( ) d ) e r( KBS(; X ; M ; v ) W m(; ) v ( ) d F ; a we waned o prove. Example 3 Aume he model () wih conan volailiy : We conider a call pread opion wih rike equal o K = S 0 K, where K i a non-negaive deerminiic conan and S 0 i anoher ock price of he form S 0 = exp (X) 0 ; where dx 0 ( 0 ) = r d 0 dw ; [0; ]; for ome poiive conan 0. hen we can eaily check ha m (; ) = exp(r( ))S 0 0 ; M = exp(r( ))S0 K; and hen a := 0 exp(r( ))S 0 exp(r( ))S 0 K (0 ) (exp(r( ))S 0 ) (exp(r( ))S 0 K) : Noice ha, if K = 0 and D W a := 0 ( 0 ) a () = 0: hen, he equaliy (3) reduce o V = BS ; X ; exp (r ( )) S; q 0 0 ( 0 ) ; from where we recover he well-known Margrabe formula (ee Margrabe (978)). # 8

9 Remark 4 Noice ha, in he conex of he previou example, when K i negaive, he call opion on he pread S S 0 i equivalen o he correponding pu opion on he pread S 0 S wih poiive rike K. hen, wihou lo of generaliy, we can aume ha he pread opion i wrien wih a poiive K. 4 Derivaive of he implied volailiy Le I (X ) denoe he implied volailiy proce, which ai e by de niion V = BS(; X ; M ; I (X )): In hi ecion we prove a formula for i a-hemoney derivaive ha we ue in Secion 5 o udy he hor-ime behavior of he implied volailiy. Propoiion 5 Aume ha he model () hold wih a L ; and ha, for R every xed [0; ) ; d <. hen i follow (x ) = E( R e r( ) (@ x F (; X ; M ; v ) F (; X ; M ; v BS(; x ; M ; I (x )) where X=x ; a.. F (; X ; M ; v ) 3 xx BS(; X ; M ; v ) x BS(; X ; M ; v ) W m(; ) and x = ln(m ) r( ): Proof: Uing heorem and he expreion V = BS(; X ; M ; I (X )) we x BS(; X ; M ; I (X BS(; X ; M ; I (X ): (4) V = E(BS(; X ; M ; v )jf ) E( e r( ) F (; X ; M ; v )djf ); which implie = E(@ x BS(; X ; M ; v )jf ) E( e r( x F (; X ; M ; v )djf (5) We can check ha he condiional expecaion E( R e r( x F (; X ; M ; v )djf )i R well de ned and nie a.. due o he fac ha d <. hu, (4) 9

10 and (5) imply (x ) BS(; x ; M ; I (x BS(; x ; M ; v )jf x BS(; x ; M ; I (x )) )) E( e r( x F (; X ; M ; v )djf )# : X=x Noice ha E(@ x BS(; x ; M ; v )jf ) x E(BS(; x; M ; v )jf ) x=x x BS(; x; M ; I 0 (x))j x=x ; (7) where, by he Hull and Whie formula, I 0 (X ) i he implied volailiy of call opion wih conan rike M, for a cerain ochaic volailiy model where = 0 and he volailiy proce i given by a. x (BS(; x; M ; I 0 (x)) x=x x BS(; x ; M ; I 0 (x BS(; x ; M ; I 0 (x ) : (8) From Renaul and ouzi (996) we (x ) = 0: hen, (6), (7) and (8) imply ha (x @ BS(; x ; M ; I (x BS(; x ; M ; I 0 (x x BS(; x ; M ; I (x )) )) E( e r( x F (; X ; M ; v )djf )# : X=x On he oher hand, raighforward calculaion lead u o and x BS(; x ; M ; ) = e x N( p ) BS(; x ; M ; ) = e x (N( p ) N( p x BS(; x ; M ; ) = (ex BS(; x ; M ; )); which yield 0

11 @ x BS(; x ; M ; I 0 (x x BS(; x ; M ; I (x )) = (BS(; x ; M ; I 0 (x )) BS(; x ; M ; I (x ))) = E(BS(; x ; M ; v ) V jf ) = E( e r( ) F (; X ; M ; v )djf ) hi, ogeher wih (9), implie ha he reul hold. 5 Shor-ime behaviour he pourpoe of hi ecion i o udy he limi (x ) a # : he following reul i par of he ool needed for our reul. : Lemma 6 Aume he model () i ai ed. hen I (x ) p : 0 a.. a Proof: Noice ha he fac ha K i a quare-inegrable and coninuou random proce and he dominaed convergence heorem lead o ge V j X=x = E(e r( ) (e X K ) jf ) X=x = E (e r( ) (e X X e r( ) M K ) jf ) X=x E ((e X X M K e r( ) ) jf ) X=x = E ((e X X e r( ) )M e r( ) (M K )) jf ) X=x E (je X X e r( ) jm jf ) X=x E (jm K je r( ) jf ) X=x M E (je X X e r( ) jjf ) X=x E (jm K je r( ) jf ) 0 a::; X=x a. Hence, aking ino accoun ha, in he a-he-money cae, V j X=x BS(; x ; M ; I (x )); we deduce ha = I(x BS(; x ; M ; I (x )) = M e N r( ) ) p 0 a::;

12 and hi allow u o complee he proof. Henceforh we conider he following hypohee: (H ) a L ; and, moreover, here exi a poiive conan C uch ha, for all 0 < < < r < ; D W a r D W D W a r C: Noice ha hi hypohee implie ha (H) hold. (H) here exi wo poiive conan c ; c uch ha for all r [0; ] c r c : Noice ha, a for all ( ; ), a C () for ome poiive conan C () ; hi hypohei implie ha a i lower bounded. (H3) he proce m(; ) L ; and moreover, here exi a poiive F proce C uch ha for all > > r > ; E jm(; r)j D W F E m(; r) F C : adaped Propoiion 7 Aume ha he model () and Hypohee (H )-(H3) hold. Alo aume ha here i a conan c > 0 uch ha c < K ; for all [0; ]. BS(; x ; M ; I (x (x = xxx xx BS(; x ; M ; v x BS(; x ; M ; v ) W a. O( ): Proof: Propoiion 5 give u BS(; x ; M ; I (x ) W d m(; )d F = E e r( ) (@ x xx BS(; X ; M ; v ) W e r( ) (@ x )@ x BS(; X ; M ; v ) W m(; )d F =: : X =x Now he proof i decompoed ino wo ep. d

13 Sep. Here we ee ha = E L(; x ; M ; v ) W djf O ( ) ; (0) where L(; X ; M ; v ) 3 xx BS(; X ; M ; v ): In fac, applying Iô formula o e r L(; X ; M ; v )( r W r a in he proof of heorem and aking condiional expecaion wih repec o F ; we obain ha dr) E( e r( ) L(; X ; M ; v ) W djf ) = E L(; X ; M ; v )( W d)jf 4 E( e r( ) (@ xxx xx)l(; X ; M ; v ) W r W r dr djf ) 4 E( e r( K (@ x )L(; X ; M ; v ) W m(; ) E( W r e r( r dr djf ) (D W W r ) r dr x L(; X ; M ; v ) djf )j E( e r( K L(; X ; M ; v ) m(; ) (D W W r ) r dr = E L(; X ; M ; v )( S S S 3 S 4 : djf ) W d)jf 3

14 Uing he proof of Lemma and Hypohee (H ) and (H), we can wrie js j = 4 E( e r( ) E (@ xxx xx)l(; X ; M ; v ) G ( W r r dr) W djf ) 6X k 3 C E 4 a d j( W r r dr) W jdjf 5 k=4 " 6X # C E ( ) k j( W r r dr) W jdjf k=4 Hence, uing Hypohee (H ), (H), and (H3), we can wrie Similarly, we have js j = 4 E ( js j C herefore, he BS(; x; K; K 6X ( ) k 4 = O( ): k=4 e r( ) K (@ x )L(; X ; M ; v ) G W r r dr) W m(; )djf : = x; K; BS(; x; K; ) x ogheer wih he hypohee of he Propoiion, implie 6X js j C E ( ) 3 jm(; )j F = O( ): In a imilar way, k=3 js 3 j = E( C C 4X E k=3 4X k=3 e r( x L(; X ; M ; v ) (D W W r ) r dr djf )j X=x ( ) (D W W r ) r dr d F ( ) = O( ): 4

15 Finally, he ame argumen give u ha js 4 j = O( ): Sep. In order o nih he proof we only need o proceed a in Sep. Here we ee ha = E P (; x ; M ; v ) W m(; )djf O ( ) ; () where P (; X ; M ; v ) = (@ x )@ x BS(; X ; M ; v ): In fac, applying Iô formula o e r P (; X ; M ; v )( m(; r) W r a in he proof of heorem and aking condiional expecaion wih repec o F ; we obain ha E e r( ) P (; X ; M ; v ) W m(; )d F = E P (; X ; M ; v )( m(; ) W d)jf dr) 4 E( e r( ) (@ xxx xx)p (; X ; M ; v ) W m(; r) W r dr djf ) 4 E( e r( k (@ x )P (; X ; M ; v ) W m(; ) W r E( e r( m(; r)dr djf )j x P (; X ; M ; v ) D W W r m(; r) dr djf ) E( e r( K P (; X ; M ; v ) m(; ) D W W r m(; r) dr djf ): Now, following he ame argumen a in Sep he proof i complee. 5

16 Remark 8 hi proof only need ome inegrabiliy and regulariy condiion. So, depending on he coe cien of he model () and he proce K, Hypohee (H )-(H3) can be ubiued by appropiae inegrabiliy condiion. Now we can ae he main reul of hi paper. oward hi end, we need o ae he following aumpion: (H4) Aume ha m(; ) ha coninou pah and ha, for each [0; ] xed, up E a r a <^^r< ~a F 0 a ; a.. and m(; ) up E m(; )a r a <^^r< ~a F 0 a ; a.. where, by convenion, ~a := m(; ) K m (; ) K (H5) here exi a F -meaurable random variable D a uch ha, for every xed > 0; up E D W a r D a F 0; a.. <<r< a : heorem 9 Conider he model () and uppoe ha Hypohee (H )-(H5) hold and here exi a poiive conan c uch ha c < K: (x ) = m(; ) K D a : () ~a Proof: We can BS(; x ; M ; I (x )) = M e r( p p ; ) e I(x ) ( ) x 3 xx BS(; x ; M ; v ) = M e r( ) = M e r( ) p e x BS(; x ; M ; v ) ( ) 8 v ( ) 6 p e v ( ) M v : ( ) 8 v 3 ( ) 3

17 hen we can wrie, due o Lemma 6 and Propoiion (x ) e I(x ) ( ) 8 ( ) E(e v ( ) 8 v 3 e I (x ) ( ) 8 ( ) E e v ( ) 8 v W djf ) M v ( ) W m(; )djf O( ) = : S S O( ) : By Lemma 6, we know ha I (x ) ( ) 0 a.. a : hen, " lim S = # lim ( ) E(e v ( ) 8 v 3 W djf ) and lim S = lim Now, le u ee ha In fac, we can eablih where and lim ( ) E(e v W m(; )djf ) ( ) 8 v # M v ( ) : (3) lim S ~a D a = 0 a..: (4) S ~a D a = lim E v A := exp ( ) 8 A B ~a D a F B := v ( ) a r D W a r drd: Conequenly lim E A B ~a D a F = lim E A B F lim E B D a ~a ~a ~a F = lim U ~a lim U : 7 v

18 Applying Schwarz inequaliy for condiional expecaion, i follow ha U " E A # ~a F E B F From he dominaed convergence heorem and (H), i i eay o ee ha E A ~a F end o zero a.. a ; and a imple calculaion give u ha (H ) and (H) imply ha E B F i bounded, from where we deduce ha lim U = 0: Oberve ha we alo have, ju j = ( ) E a r v D W a r D a drd ~a F C ( ) E a r v D W a r drd ~a F C ( ) E D W a r D a F drd = : ju ; j ju ; j : Uing Hypohee (H ) and (H) we obain ha C ju ; j ( ) E a r v ~a drd F C ( ) E a r v ~a drd F C = ( ) E a r a ~a ( ) d drd F C ( ) 3 E a r a ~a F ddrd; which end o zero, a.. a, becaue of Hypohei (H4). Similarly, C ju ; j ( ) E D W a r D a F drd ; which end o zero by Hypohei (H5). hu we have proved (4) i rue. On he oher hand, by (3) we can wrie m(; ) lim S K ~a D m(; ) a = lim E A B K ~a D a F : 8

19 bu now and v A := exp ( ) 8 M v B := v ( ) a r m(; )D W a r drd: Finally, proceeding imilarly a before, we have (3) yield ha S converge o m(;) K ~a D a, which, ogeher wih (4), implie ha () i ai ed. 6 Applicaion o he udy of pread opion hi ecion i devoed o apply he previou reul o udy he implied volailiy behaviour for pread opion. hi udy allow u o predic when he Kirk approximaion formula for pread opion may fail. Moreover, we will ee how hi analyi give u a ool o improve Kirk formula. 6. Shor-ime behaviour of he implied volailiy for pread opion Conider an pread opion wih K = S 0 K a in Example 3. For he ake of impliciy we will aume he inere rae r = 0. hen i i eay o ee ha We can eaily check ha, for < a := 0 S 0 S 0 K (0 ) (S 0 ) (S 0 K) : D W a = 0 K (S 0 K) (0 ) S 0 S 0 K = ( 0 ) S 0 0 Hence, we deduce ha S 0 K S 0 K (S 0 K) : K (S 0 K) 0 S 0 q D W a = D W a = DW r a p a S = p 0 0 a S 0 ( 0 ) SK 0 K (S 0 K) : 9

20 hen, from heorem 9, we ge = lim (x m(; ) K 0 S 0 S 0 K D a ~a p 3 ( 0 ) SK 0 a (S 0 K) : (5) Remark 0 Noice ha he above quaniy i allway poiive. In he following example we will udy i behaviour a a funcion of K and Example In Figure we plo (x ) a a funcion of K for = 0:9 (olid) and = (dah), and wih S = 00; = 0:5 and 0 = 0:4: We oberve he limi kew (x ) i zero in he cae K = 0. hi wa expeced from Example 3, where we found ha in hi cae he implied volailiy i conan, = 0: Noice alo ha, even hi kew increae wih K, hi incremen eem o be clearly bigger in he compleely correlaed cae = : implied volailiy kew Figure (x ) a a funcion of K for = 0:9 (olid) and = (dah). Here = 0:5; 0 = 0:4: Example In Figure we plo (x ) a a funcion of for K = 5 (olid) and K = 0 (dah) and for he ame parameer value a in Fig.. can oberve he limi kew (x ) ha i maximum in he compleely correlaed cae = : Noice ha hi mean ha he conan volailiy approximaion given by Kirk formula i expeced o be le accurae in hi cae. hi fac i conien numerical empirical evidence (ee for example Baeva (0) and Borovkova (007)). 0

21 iimplied volailiy kew Figure (x ) a a funcion of for K = 5 (olid) and K = 0 (dah). Here = 0:5; 0 = 0:4: Example 3 In Figure 3 we (x ) a a funcion of and K for he ame parameer value a in Fig. and Fig.. Noice ha hi limi kew i ubanially bigger near he cae = : ro z K correlaion Figure (x ) a a funcion of and K: 6. Applicaion o he udy of he accuracy of Kirk formula Wih he above noaion, he Kirk approximaion for an pread opion can be wrien a: q BS(; X ; M ; a ): I i well-known ha Kirk formula i a very accurae approximaion for pread opion given i impliciy (ee for example Baeva (0), Bjerkund and Senland (0) or Carmona and Durrleman (0)). Neverhele, i i well-known i may fail for highly correlaed ae (ee for example Baeva (0)). he reul in he above ecion give an analyical reaon for hi phenomenon.

22 In fac, noice ha p a (he volailiy parameer in he Kirk formula) i a proce ha doe no depend on X nor on he ime o mauriry : hen, Kirk formula may no reproduce he hor-ime volailiy kew ha we have een appear in he highly correlaed cae ( cloe o ) and we can expec i can fail when i near one. In he following example, we compare Kirk approximaion price wih he one obained wih a Mone-Carlo imulaion procedure wih 00,000 rial.we ue hee imulaion reul a he benchmark for he rue pread opion value Example 4 In he following able we can compare he price given by Kirk formula and by he Mone-Carlo imulaion, for di eren value for K and : Here X = ln(00); S 0 = 00; r = 0; = 0:5; = 0:5 and 0 = 0:4. he di erence i given in % of he Mone-Carlo price. A expeced from our analyical udy, he error increae rongly when he correlaion i cloe o. K= 0:60 0:98 0:99 0:999 5 Mone-Carlo Kirk Error 9; ; ; 40 % ; 890 ; 59 ; 30 % ; 8386 ; 8775 ; 7 % 0 Mone-Carlo Kirk Error 7; ; ; 545 % ; 74 ; ; 84 % ; 007 ; 05 7 ; 93 % ; 50 ; 540 ; 75 % 0; ; 8848 ; 56 % Noice ha our udy of he volailiy kew give no only a heoreical underanding of i goodne-of- of Kirk approximaion, bu alo give u ome hin o improve i. In fac, from (5) and by uing aylor expanion we can expec ha for mall ime o mauriie and for near-he-money opion, he expreion q ^I (X ) := a S 0 0 S 0 K p 3 ( 0 ) SK 0 a (S 0 K) (X x ) can be a reaonable approximaion for he implied volailiy I (X ). In fac, le u conider he modi ed Kirk approximaion given by BS(; X ; M ; ^I (X )): In he following example we will check numerically he goodne-of- of hi approximaion. Example 5 In he following able we can compare he price given by he modi ed Kirk formula and by he Mone-Carlo imulaion, for di eren value for K and and for he ame parameer a in Example 4. Noice ha we have reduced igni caively he error of approximaion wih repec o he reul in hi la example, pecially in he cae of highly correlaed ae.

23 K= 0:60 0:98 0:99 0:999 5 Mone-Carlo Modi ed Kirk Error 9; ; ; 37 % ; 890 ; ; 809 % ; 8386 ; ; 804 % 0 Mone-Carlo Modi ed Kirk Error 7 Concluion 7; ; ; 45 % ; 74 ; 888 ; 368 % ; 007 ; 0367 ; 660 % ; 50 ; ; 44 % 0; ; 80 ; 400 % We have ued he Malliavin calculu echnique o nd an expreion for he hor-ime implied volailiy kew of opion wih random rike. In paricular, we have een ha hi analyical udy give u a key o underand why ome approximaion formula may fail for ome e of parameer. A an applicaion, we have een ha hi kew i very pronounced for pread opion in he high correlaion cae, wich explain why a conan volailiy approximaion a Kirk formula canno be accurae in hi cae. Finally, our approach give u ome hin o improve hi eimae, by inroducing a kew in he approximaion of he correponding implied volailiy. Our preliminar numerical analyi how ha hi can be a naural and e cien way o improve Kirk formula. A precie developemen of hi improvemen i lef for fuure reearch. Reference [] C. Alexander and A. Venkaramanan: Cloed form approximaion for pread opion. Applied Mahemaical Finance 8 (5), (0) [] E. Alò: A generalizaion of he Hull and Whie formula wih applicaion o opion pricing approximaion. Finance Soch. 0, (006). [3] E. Alò, J.A. León and J. Vive: On he hor-ime behavior of he implied volailiy for jump-di uion model wih ochaic volailiy. Finance Soch., (007). [4] E. Alò and D. Nualar: An exenion of Iô formula for anicipaing procee. J. heor. Probab., (998) [5]. Baeva: On he pricing and eniiviy of pread opion on wo correlaed ae. Preprin (0). [6] P. Bjerkund and G. Senland: Cloed form pread opion valuaion. Quaniaive Finance ifir, -0 (0). [7] S. Borovkova, F. J. Permana and H. v.d.weide: A cloed form approach o he valuaion and hedging of bake and pread opion. he Journal of Derivaive 4 (4), 8-4 (007). 3

24 [8] R. Carmona andv. Durrleman: Pricing and hedging pread opion. SIAM Rev. 45 (4), (003). [9] S. Deng, M. Li and J. hou: Cloed-form approximaion for pread opion price and greek. Preprin (008) [0] J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna: Mauriy cycle in implied volailiy. Finance Soch. 8 (4), (004). [] J. Jacod and P. Proer: Rik-neural compaibiliy wih opion price. Finance Soch. 4 (), (00). [] E. Kirk: Correlaion in he energy marke. In Managing Energy Price Rik (Fir Ediion). London: Rik Publicaion and Enron, pp (995) [3] W. Margrabe: he value of an opion o exchange one ae for anoher. he Journal of Finance 33 (), (978). [4] A. Medvedev and O. Scaille: Approximaion and calibraion of hor-erm implied volailiie under jump-di uion ochaic volailiy. Rev. Finance Sud. 0 (), (007). [5] D. Nualar: he Malliavin Calculu and Relaed opic. Second Ediion. Springer-Verlag, Berlin (006). [6] E. Renaul and N. ouzi: Opion hedging and implied volailiie in a ochaic volailiy model. Mah. Finance 6 (3) (996). [7] D. Shimko: Opion on fuure pread: hedging, peculaion and valuaion. Journal of Fuure Marke 4 (), 83 3 (994). 4