Calibration of stochastic volatility models via second order approximation: the Heston model case

Transcription

1 Calibraion of ochaic volailiy model via econd order approximaion: he Heon model cae Elia Alò Deparamen d Economia i Emprea Univeria Pompeu Fabra and Barcelona Graduae School of Economic eliaalo@upfedu Rafael De Saniago Deparmen of Managerial Deciion Science IESE Buine School raniago@ieeedu Joep Vive Deparamen de Probabilia, Lògica i Eadíica and Iniu de Maemàica de la Univeria de Barcelona IMUB Univeria de Barcelona joepvive@ubedu Ocober, Abrac Uing a uiable Hull and Whie ype formula we develop a mehodology o obain a econd order approximaion o he implied volailiy for very hor mauriie Uing hi approximaion we accuraely calibrae he full e of parameer of he Heon model One of he reaon ha make our calibraion for hor mauriie o accurae i ha we alo ake ino accoun he erm-rucure for large mauriie We may ay ha calibraion i no memoryle, in he ene ha he opion behavior far away from mauriy doe influence calibraion when he opion ge cloe o expiraion Our reul provide a way o perform a quick calibraion of a cloed-form approximaion o vanilla opion ha can hen be ued o price exoic derivaive he mehodology i imple, accurae, fa, and i require a minimal compuaional co JEL Claificaion: G3 Mahemaic Subjec Claificaion : 9B8, 9B7 Inroducion Alhough he aumpion of conan volailiy lend robune o he Black-Schole model ee El Karoui, Jeanblanc-Pique and Shreve 998, in he la decade he need for more general non-conan volailiy model ha been he driving force behind numerou work in financial mahemaic One of he reaon behind hi driving force i he fac ha price of exoic derivaive baed upon he Black-Schole formula are ofen inaccurae, a exoic conrac are ypically more eniive o he volailiy han vanilla opion hu, he need wa fel for model ha could accoun for he mile or kew ha were ofen oberved in he marke Suppored by gran ECO and MEC FEDER MM Suppored by gran ECO9-83-E Suppored by gran MEC FEDER MM and 9-73

2 One approach o olving hi problem ha been o le he volailiy of he underlying randomly flucuae according o one or more correlaed Brownian moion hi approach, ared by Hull and Whie 987, Wiggin 987, Sein and Sein 99 and Heon 993, ha proved o be ucceful and ha evolved over ime ino a variey of reearch line A drawback of ochaic volailiy model i heir increaed mahemaical complexiy, which ranlae ino he difficuly of obaining cloed-form oluion hi, in urn, make calibraion compuaionally inenive and low A recen rend in he lieraure ha been he developmen of cloed-form approximaion formula for opion price ee Fouque, Papanicolau and Sircar, Fouque, Papanicolau, Sircar and Sølna 3, Hagan, Kumar, Leniewki and Woodward 8, DeSaniago, Fouque and Sølna 8, Anonelli and Scarlai 8, Benhamou, Gobe and Miri 9, a and b, Fouque, Papanicolau, Sircar and Sølna, or Alò he main advanage of cloed-form approximaion i ha hey allow for fa calibraion and provide a beer underanding of he role of model parameer Anoher drawback of ochaic volailiy model i ha, while hey are able o explain volailiy mile and kew for inermediae and long mauriie hree monh and more, calibraion when he opion are cloe o mauriy remain unaifacory See, for example, Janek, Kluge, Weron and Wyup he main purpoe of hi paper i o preen an accurae calibraion procedure for very hor mauriie ha require a minimal compuaional co In Fouque, Papanicolau and Sircar he auhor developed an eay way of idenifying he group parameer ha are needed for pricing and hedging European-ype ecuriie, namely, he average volailiy ogeher wih he lope and inercep of he implied volailiy curve a a funcion of he logmoneyne-o-mauriy raio We build on hi way of idenifying he parameer, alhough our approach i lighly differen By applying he reul in Alò o he Heon model, we deduce a econd-order approximaion formula for he implied volailiy From he analyi of hi approximaion we derive valuable informaion regarding he iing behavior of he implied volailiy hen, uing he erm-rucure for a-he-money opion, we are able o accuraely calibrae all model parameer including he mean-reverion erm hi paper i organized a follow In Secion we briefly decribe he model framework and inroduce he baic noaion In Secion 3 we preen a econd-order approximaion o he Black-Schole price baed on he reul of Alò A calibraion i ypically performed on implied volailiie, in Secion 4 we compue a econd-order approximaion o hi magniude In Secion 5 we preen numerical example in which he parameer are calibraed for he uncorrelaed and he correlaed cae, and heir accuracy i eed on imulaed daa In Secion 6 our reul are compared wih hoe obained wih anoher calibraion procedure derived by a differen mehod and concluion are drawn Preinarie and Framework We conider he following ochaic volailiy model for he ock price under a rik neural probabiliy P choen by he marke: ds rs d + σ S ρdw + ρ db, [, ], where r i he conan inananeou inere rae, W and B are independen andard Brownian moion defined on a complee probabiliy pace Ω, F, P, and ρ [, ] We will aume ha he volailiy proce σ follow a Heon model ee Heon 993, wih dynamic governed by dσ κ σ d + ν σ dw, where κ,, ν are poiive conan aifying he condiion κ ν We denoe by F W and F B he filraion generaed by W and B, repecively, and we define F : F W F B We denoe X : ln S and E : E F Wih hi noaion, he price a ime of a European call wih rike K i given by V e r E [e X K + ] 3

3 In wha follow we will make ue of he following noaion: v : E σd will denoe he fuure average volailiy M : E σ d Noice ha v M σ d BS, x, σ will denoe he Black-Schole price a ime of a European call opion wih conan volailiy σ, curren log-ock price x, ime o mauriy, rike price K and inere rae r We know ha in hi cae BS, x, σ e x Φd + Ke r Φd, 4 where Φ denoe he cumulaive probabiliy funcion of he andard normal law and wih x : ln K r d ± : In order o implify noaion, we define and x x σ ± σ, H, x, σ 3 xxx xxbs, x, σ e x σ π exp d + d + σ, K, x, σ xxxx 4 xxx 3 + xxbs, x, σ e x σ π exp d + d + σd + σ 3 Second-order Approximaion o he Black-Schole Price Our calibraion mehod i buil upon he following wo reul, which provide a econd-order approximaion o he Black-Schole price Boh reul are proved in Alò heorem Decompoiion formula Under condiion of model - we have V BS, X, v + ρ E e r H, X, v σ d M, W + 8 E e r K, X, v d M, M In order o implify noaion, le α κ ν, β α, and γ 5 α heorem Second-order approximaion For he model - and for all [, ] we have he following reul, where C, σ repreen a poiive conan, non-decreaing a a funcion of : If α 5, hen V BS, X ; v ρ H, X, v E σ d M, W 8 K, X, v E d M, M { C, σ ν ρ 3 + ν 3 ρ + ν 4 5/} 3

4 If α [, 5, hen V BS, X ; v ρ H, X, v E 8 K, X, v E d M, M σ d M, W { } +γ C, σ ν ρ 3 + ν 3 ρ + ν 4 γ 5/ γ γ If α [ 3,, hen V BS, X ; v ρ H, X, v E 8 K, X, v E d M, M C, σ σ d M, W { +β [ ν β ρ 3 β + νρ β] β + ν 4 γ 5/ γ γ +γ } Alhough he preciion of he bound in he la heorem ha been proved for he horerm hor mauriie, our numerical example how ha hey alo hold for he long-erm ee Example 46 in Alò he following reul, which are eay o check, are ued hroughou he paper In order o faciliae furher reference, we ae hem here a a lemma Lemma 3 he following reul hold: E σ d + σ κ e κ dm νσ 3 E σ d M, W e κr dr dw ν κ σ e κ dw ν κ {κ + σ + e κ σ κ e κ σ } 4 E d M, M { ν σ κ + κ e κ κ e κ σ κ e + e κ σ + e κ e κ } κ κ By ubiuing he above expreion ino he approximaion formula provided by heorem, we will be able o obain explici econd-order approximaion for he implied volailiy hi i wha we do in he nex ecion 4

5 4 Second-order Approximaion o he Implied Volailiy he marke price a ime of a European call opion wih rike K and mauriy i an obervable magniude, which will be referred o a BS ob BS ob, K, he implied volailiy i hen defined a he value of he volailiy parameer ha make he Black-Schole price equal o he oberved marke price From expreion 4 we have ha he implied volailiy i he value IV ha make: BS, x, IV BS ob For impliciy, and wihou lo of generaliy, we conider A i i done in Alò, from he expreion in heorem we deduce a econd-order approximaion o he implied volailiy, ĨV, which we will wrie aĩv v + I + I, wih and I : I : ρ d + E σ d M, W v v d + 8v v d + v v E d M, M he key elemen ha will allow u o calibrae he model parameer i he iing behavior of he implied volailiy, boh cloe and far away from mauriy, ogeher wih he erm rucure for large mauriie We herefore ar hi ecion by udying he iing behavior of ĨV Given ha v + σ κ e κ, i i clear ha v σ when, and v when A all he reul in hi ecion refer o he model -, we will only make an explici reference in he fir lemma From hen on, reference o he model will be implicily aumed 4 Limiing Behavior of I In hi ecion we look a he i behavior of I We ar by compuing he i of I a he ime o mauriy goe o Lemma 4 Aume he model - hen, I ρν x ln K 4σ Proof Replacing d + by i expreion we have: I ρ d + E σ d M, W v v ρ v r v E σ d M, W ρx ln K v 3 E σ d M, W 5

6 A we have E σ d M, W ν E σ e κr dr d r ν ρν v r v ρνx ln K v 3 Applying L Hôpial rule, i follow ha I ρν v r v ρνx ln K 4v 3 r e κr E σ d dr, e κr E σ d dr r e κr E σ d dr e κ E σ d e κ E σ d he fir i in he righ hand ide i clearly, and he econd one i equal o ρνx ln K 4v 3 E σ κe κ E σ e κ d If we now le, he econd erm of he la equaion converge o, while he fir one converge o ρν 4σ x ln K, 5 a we waned o prove In he following lemma we explore he behavior of I when he opion i far away from mauriy Lemma 5 A increae, he following reul hold: I ρν κ Proof Uing he previou compuaion, we have: I ρ d + v v ρν v + r v r E σ d M, W e κ E σ e κ d ρνx ln K 4v 3 E σ ρνx ln K 4v 3 κe κ E σ e κ d I i eay o ee ha, a, e κ E σ e κ d converge o κ and E σ converge o herefore, he econd and hird erm in he la expreion vanih and he fir one converge o: ρν r κ ρν κ r, and he proof i now complee 6

7 4 Limiing Behavior of I We now look a he behavior of I, he econd-order erm of he implied volailiy approximaion We ar by compuing i i a he ime o mauriy goe o zero Lemma 6 A we ge cloe o mauriy, we have: I ν 4σ 3 x ln K Proof We need o compue: [ I d + 8v v d + ] v v E d M, M 6 On one hand, we have [ d + 8v v d + ] v v 8v 3 d + v d+ [ x ln K 8v 3 v + r v ] v 4 On he oher hand, E d M, M Oberve ha ν κ Eσ e κ d E d M, M C, where C Cν, κ, σ, i a conan I follow ha he i in 6 i equivalen o: ν 8v 5 3 E σ e dr κr d x ln K Aplying L Hôpial rule, I ν v 5 ν v 5 ν 4 v 5 ν 4v 5 ν 4σ 3 which conclude he proof E σ e κr dr e κ d x ln K r E σ e κr e κ ddr E σ e κ d x ln K x ln K Eσ κe κ Eσe κd x ln K x ln K, 7 he nex lemma deal wih he behavior of I a he ime o mauriy goe o infiniy 7

8 Lemma 7 A increae, he following reul hold: I ν r 8κ 4 Proof We need o compue or, equivalenly, Given ha 8v 3 [ d + 8v v d + ] v v E x ln K v 3 d M, M, + r v v 4 E d M, M E d M, M C, he only i ha i no zero a i r 8v v 4 4 E d M, M ν r 8v κ Eσ 4 e κ d Applying L Hôpial rule, he above i i equivalen o which i he ame a v 4 ν r 4v κ v 4 e κ Eσ 4 e κ d e κ Eσe κ d, ν 4v κ r v 4 + σ 4 e κ e κ e κ d I i raighforward o ee ha he inegral converge o κ We herefore have ha and he proof i complee I ν r 8κ 4 Remark 8 When he opion i cloe o mauriy, he above reul allow u o wrie he econd-order approximaion o he implied volailiy a, ÎV σ ρν 4σ x ln K + ν 4σ 3 x ln K 8 Remark 9 When he opion i far away from mauriy, he econd-order approximaion o he implied volailiy become ÎV σ + ρν κ r + ν r 8κ 9 4 8

9 43 Derivaive of ĨV when he opion i a-he-money In order o calibrae he model parameer we will ue he implied volailiy erm rucure In paricular, from he available daa we will need o eimae he implied volailiy inercep and lope, and hen compue he parameer value ha beer fi uch magniude In hi ecion we herefore derive expreion for he derivaive of ĨV a and a When he opion i a-he-money, hen x x and d + v he expreion for ĨV hu become: ĨV v + ρ 4v E σ d M, W 8v 4 + v E d M, M From hi expreion we can prove he following reul Lemma he derivaive of he approximaion o he implied volailiy a i equal o: ĨV 3σ ρν 6κσ ν 4σ Proof Uing he reul from Lemma 3 i i eay o check ha ĨV σ he derivaive of ĨV wih repec o, a, i herefore equal o: ĨV σ v σ ρ + 4v E 8v 4 + v σ d M, W E d M, M For he fir erm, we have v σ E σ d σ E σ d E σ E σ d E σ d E σ E σ d σ σ E σ d E σ E σ d + σ κ e { E σ 4σ + σ e κ + κ σ e κ } κ σ 4σ 9

10 he econd erm of equaion i equal o: ρ 4v E σ d M, W ρν 4σ E σ e κr dr d ρν 4σ ρν 8σ ρνσ 8 Finally, he i in he hird erm of expreion give u: 8v 4 + v E d M, M 8v r E σ e κr d dr E σ e κ d 4 + v E d M, M ν 8v 4 + v E σ e dr κr d ν 8v v E σ e dr κr d ν 8v 3 Applying now L Hôpial, he above i i equal o: ν v 3 ν v 3 ν 4v 3 which complee he proof 3 E σ e dr κr d E σ e κr dr r e κ d E σ e κr e κ d dr E σ e κ d ν 4v 3 E σ κe κ ν, 4σ Eσ e κ d Remark When he opion i a-he-money, he above reul allow u o wrie he aylor expanion of ĨV near a: ĨV σ + 3σ ρν 6κσ ν 4σ We now compue he derivaive of ĨV when he opion i far away from mauriy In order o prove he nex lemma, noe ha ĨV + νρ ν 3κ, which follow eaily from

11 Lemma A become large ie, a he ime o mauriy increae, he following reul hold: [ ĨV + νρ ] ν 3κ σ κ + νρσ 4κ σ ν 5 + 4κ 3 κ 3 Proof Noe ha [ ĨV { v + 8v + νρ ] ν 3κ σ d M, W ρ 4v E 4 + v E d M, M νρ 4κ + ν 3κ } 3 Applying L Hôpial, he fir wo erm in he i above give u: v E σ d / σ κ { E } σ d E σ v / + σ e κ σ e κ v κ We now conider he nex wo erm in he righ-hand ide of expreion 3 Uing Lemma 3 i follow ha { } ρ E σ d M, W 4v νρ 4κ { νρ κ + σ 4κ v + [ σ κ σ ] e κ νρ } 4κ νρ σ 4κ Applying Lemma 3 again, he la wo erm in he righ-hand ide of 3 give u: { 8v 4 + } v E d M, M + ν 3κ { } E d M, M 3v + ν 3κ 8v 3 E d M, M { [ ν 3v κ + σ κ κ + ] } + ν κ 3κ ν 8v 3 κ ν 3 κ 3 σ 5 + 4κ Puing hee i ogeher, we ge ha [ ĨV + νρ ] ν 3κ which i he deired reul σ κ + νρσ 4κ ν σ 5 + 4κ 3 κ 3,

12 Remark 3 he above reul allow u o conclude ha, for an a-he-money European call opion which i far away from mauriy, he following approximaion hold: ĨV + νρ ν σ 3κ + κ + νρσ 4κ σ ν 5 + 4κ 3 κ Calibraion In hi ecion we e he accuracy of our reul by calibraing he model from imulaed daa Le S, σ, κ 3, ν 3, ρ, r, and 9 From he model equaion, we fir generae opion price for mall and large mauriie and we hen inver hem in order o ge he correponding a-he-money implied volailiie When we ay hor mauriie we roughly refer o o 5 year, while large mauriie mean 3 o 4 year Before moving on o numerical example, we offer a bird -eye view of he calibraion procedure he fir ep i o plo he implied volailiie for hor mauriie a a funcion of and o fi a curve o he daa From expreion we obain σ a he inercep of he fied equaion he econd ep, which alo follow from, coni of making κ σ + ρνσ ν 4σ 8 4σ equal o he lope of he regreion curve he hird ep i o plo he implied volailiie for hor mauriie a a funcion of he log-moneyne x ln K and fi a curve o he daa he produc νρ i obained, according o 8, from he coefficien of he linear erm in he regreion equaion he fourh ep coni of ploing he implied volailiie a a funcion of / for large value of and fiing a curve o he daa Uing expreion 4, we hen make and + νρ ν 3κ σ κ + νρσ 4κ σ ν 5 + 4κ 3 κ 3 equal o he inercep and he lope, repecively, of he regreion curve he final ep i o olve he yem of equaion generaed in he econd and fourh ep, uing he informaion gahered in he fir and hird ep We conider wo cae, he uncorrelaed one ρ and he correlaed one ρ 5 he uncorrelaed cae ρ In Figure we plo he differen implied volailiie obained from he imulaed price again he correponding mauriie expreed in year By running a regreion, we ge ha he inercep i and he lope of he linear erm i 676 When he opion i a-he-money, he implied volailiy a i σ Uing a ha o deignae calibraed magniude, we hu wrie σ From expreion i follow ha or, equivalenly, 3σ ρν 6κσ ν 4σ 676, ρν 6κ4 ν 8448 We now look a he implied volailiy a a funcion of he log-moneyne x ln K I follow from expreion 5 and 7 ha, for hor mauriie, he implied volailiy end o σ νρ 4σ x ln K + ν 4σ 3 x ln K

13 ,6 y,73x +,676x +, R², Implied Volailiy,8,4,,,4,6,8,, ime o Mauriy Figure : Implied Volailiy when he a-he-money opion i cloe o mauriy remaining figure, ime o mauriy i expreed in year In hi and he In Figure we plo he implied volailiie again he log-moneyne By running a regreion, we ge ha he coefficien of he fir order erm i zero 4 7, which give νρ Noe ha he regreion coefficien of he econd order erm i 443, from where one could be emped o conclude ha ν 4 x 443 x However, calibraion of he econd-order coefficien from he above equaion can be unable, for i depend on he correc eimaion of he implied volailiy for rike ha are ou-of-he-money We herefore conider 964 only a a rough approximaion o he value of ν We will ee below ha a more accurae calibraion of hi parameer require he ue of he erm-rucure for long mauriie,88 Implied volailiy,8,7 y,443x 4E 7x +,7,64,3,,,,,3 Log moneyne Figure : Implied Volailiy a a funcion of log-moneyne for hor mauriie he nex ep i o gaher addiional informaion from he implied volailiy We know from 4 ha, for large value of, he implied volailiy i approximaely equal o: + νρ ν σ 3κ + κ + νρσ 4κ σ ν 5 + 4κ 3 κ 3 3

14 By performing a regreion on he daa from Figure 3, we ge ha he inercep i + νρ ν 3κ 998, while he lope i σ κ + νρσ 4κ σ ν 5 + 4κ 3 34 κ 3,93,9 Implied volailiy,9,9 y,34x +,998 R²,89,5,,5,,5,3,35 / Figure 3: Implied Volailiy a a funcion of / Puing everyhing ogeher, we have he following yem: 4 κ 6κ4 ν 8448 ν 3κ 998 ν κ 3 κ 3 34 Solving he yem we find ha he calibraed value of he parameer are a in he able below where he error i expreed in abolue value parameer real calibraed error σ % ν % κ % % 5 he correlaed cae ρ Recall ha when he opion i a-he-money, he implied volailiy a i σ We alo have from Lemma ha he derivaive of he volailiy a i κ σ + ρνσ ν 4σ 8 4σ In Figure 4 we plo he implied volailiie obained from he imulaed price again he correponding mauriie, a in he previou example By running a regreion, we ee ha he inercep i, which give σ 4

15 ,6 Implied volailiy,,8,4 y,77x +,647x +, R²,,,4,6,8,, ime o mauriy Figure 4: Implied Volailiy a a funcion of ime o mauriy A he fir-order coefficien i equal o 647, equaion allow u o wrie κ σ + ρνσ ν 647 4σ 8 4σ We now look a he implied volailiy a a funcion of he log-moneyne x ln K From expreion 5 and 7 we know ha, for hor mauriie, he implied volailiy end o σ νρ 4σ x ln K + ν 4σ 3 x ln K,4 Implied volailiy,3,,, y,83x +,7 R²,99,5,,5,5,,5 Log moneyne Figure 5: Implied Volailiy a a funcion of log-moneyne for hor mauriie In Figure 5 we plo he implied volailiie again he log-moneyne Noe ha he plo i rongly linear, which make i difficul o eimae he econd-order coefficien A he coefficien of he fir-order erm i equal o 83, i follow from he la equaion ha νρ 4648 he nex ep i o gaher addiional informaion from he a-he-money implied volailiy We know from 4 ha, for large value of, he implied volailiy i approximaely equal o: + νρ ν σ 3κ + κ + νρσ 4κ σ ν 5 + 4κ 3 κ 3 5

16 By performing a regreion on he daa from Figure 6, we ge ha + νρ ν 3κ 96 σ κ + νρσ 4κ σ ν 5 + 4κ 3 33 κ 3,89 Implied volailiy,88,87,86 y,33x +,96 R²,85,5,,5,,5,3,35 / Figure 6: Implied Volailiy a a funcion of / Puing everyhing ogeher, we have he following yem: 4 κ 464 6κ 4 ν κ ν 3κ κ ν κ 3 κ 3 33 Solving i aking ino accoun ha νρ 4648, we ge ha he calibraed value provide a fairly accurae eimae of he real parameer value, a can be een in he able below he error are expreed in abolue value parameer real calibraed error σ % ν % κ % % ρ % 6 Summary and Concluion In hi paper we have preened a calibraion procedure for very hor mauriie in he conex of he Heon model When applied o imulaed daa, hi mehod allow u o calibrae he full e of Heon parameer σ, ν, κ,, ρ Our reul provide a way o perform a quick and accurae calibraion of a cloed-form approximaion o he price of vanilla opion ha can hen be ued o price exoic derivaive A a way o illurae he mehod accuracy, we compare our reul wih hoe recenly repored by Forde, Jacquier and Lee heir mehod, differen han our, i baed on addlepoin expanion in he complex plane and he properie of holomorphic funcion hey conider a Heon model wih parameer value σ, ν, κ 5, 4 and ρ 4 In he able below we preen he calibraed value of he parameer obained wih our mehod, ogeher wih hoe repored by Forde, Jacquier and Lee 6

17 our Forde e al parameer real calibraion error calibraion error σ % ν % % κ % 4 4% % 4 5% ρ % % One of he reaon ha make our calibraion o accurae i he fac ha we make ue of he erm-rucure for large mauriie ie, he region of he volailiy urface ha i far away from mauriy, while Forde, Jacquier and Lee only conider he hor erm We may hu ay ha calibraion i no memoryle, in he ene ha he opion behavior far away from mauriy doe influence calibraion when he opion ge cloe o expiraion he main rai of he calibraion mehodology ha ha been preened in hi paper are impliciy, accuracy, and peed: he procedure i fa o implemen and i require a minimal compuaional co By including he erm-rucure for large mauriie we are able o coniderably improve i accuracy Reference [] E Alò : A decompoiion formula for opion price in he Heon model and applicaion o opion pricing approximaion o appear in Finance and Sochaic [] F Anonelli and S Scarlai 9: Pricing opion under ochaic volailiy: a power erie approach Finance and Sochaic 3 : [3] E Benhamou, E Gobe and M Miri 9: Smar expanion and fa calibraion for jump diffuion Finance and Sochaic 3 4: [4] E Benhamou, E Gobe and M Miri a: Expanion formula for European opion in a local volailiy model Inernaional Journal of heoreical and Applied Finance 3 4: [5] E Benhamou, E Gobe and M Miri b: ime dependen Heon model SIAM Journal on Financial Mahemaic : [6] R DeSaniago, J P Fouque and K Sølna 8: Bond Marke wih Sochaic Volailiy Advance in Economeric : 5-4 [7] N El Karoui, M Jeanblanc-Pique and S Shreve 998: Robune of Black and Shole Formula Mahemaical Finance 8: 93-6 [8] M Forde, A Jacquier and R Lee : he Small-ime Smile and erm Srucure of Implied Volailiy under he Heon Model SIAM Journal of Financial Mahemaic o appear [9] J P Fouque, G Papanicolau and K R Sircar : Derivaive in Finanial marke w ih Sochaic Volailiy Cambridge [] J P Fouque, G Papanicolau, K R Sircar and K Sølna 3: Singular Perurbaion in Opion Pricing SIAM Journal of Applied Mahemaic 63 5: [] J P Fouque, G Papanicolau, K R Sircar and K Sølna : Mulicale Sochaic Volailiy for Equiy, Inere Rae, and Credi Derivaive Cambridge [] P S Hagan, D Kumar, A Leniewki and D E Woodward : Managing mile rik Willmo magazine 5: 84-8 Forde, Jacquier and Lee do no calibrae σ 7

18 [3] S L Heon 993: A cloed-form oluion for opion wih ochaic volailiy wih applicaion o bond and currency opion Review of Financial Sudie 6 : [4] J C Hull and A Whie 987: he pricing of opion on ae wih ochaic volailiie Journal of Finance 4: 8-3 [5] A Janek, Kluge, R Weron and U Wyup : FX mile in he Heon Model Humbold Univeriy, SFB 649 Dicuion Paper -47 [6] E M Sein and J C Sein 99: Sock price diribuion wih ochaic volailiy: An analyic approach he Review of Financial Sudie 4: [7] J Wiggin 987: Opion value under ochaic volailiie Journal of Financial Economic, 9: