Research Article Pricing Currency Option in a Mixed Fractional Brownian Motion with Jumps Environment

Transcription

1 Hidawi Publishig Corporaio Mahemaical Problems i Egieerig, Aricle ID 85810, 13 pages hp://dx.doi.org/ /014/85810 Research Aricle Pricig Currecy Opio i a Mixed Fracioal Browia Moio wih Jumps Evirome Foad Shokrollahi ad Adem KJlJçma Deparme of Mahemaics ad Isiue for Mahemaical Research, Uiversiy Pura Malaysia (UPM), Serdag, Selagor, Malaysia Correspodece should be addressed o Adem Kılıçma; akilic@upm.edu.my Received 17 December 013; Acceped 10 February 014; Published 9 April 014 Academic Edior: Abdo Aagaa Copyrigh 014 F. Shokrollahi ad A. Kılıçma. This is a ope access aricle disribued uder he Creaive Commos Aribuio Licese, which permis uresriced use, disribuio, ad reproducio i ay medium, provided he origial work is properly cied. A ew framework for pricig he Europea currecy opio is developed i he case where he spo exchage rae fellows a mixed fracioal Browia moio wih jumps. The jump mixed fracioal parial differeial equaio is obaied. Some Greeks ad properies volailiy are discussed. Fially he umerical simulaios illusrae ha our model is flexible ad easy o impleme. 1. Iroducio A currecy opio is a corac, which gives he ower he righ bu o he obligaio o purchase or sell he idicaed amou of foreig currecy a a specified price wihi a specified period of ime (America Opio) or o a fixed dae (Europea Opio). Sice he currecy opio ca be used as a ool for ivesme ad hedgig, i is oe of he bes ools for corporaios or idividuals o hedge agais adverse movemes i exchage raes. Thus i he prese work he heoreical models for pricig currecy opios have bee carried ou. Opio pricig was iroduced by Black-Scholes i However, Dravid e al. [1, Ho e al. [, Tof ad Reier [3, ad Kwok ad Wog [4 also worked o ha direcio. Dua ad Wei [5 idicaed ha opio pricig by Black-Scholes modelwhichisbasedobrowiamoiocaoillusrae clearly wo pheomea from sock markes: firs asymmeric lepokuricfeauresadsecodhevolailiysmile.ia work by Garma ad Kohlhage (Hereafer G-K) [6 was exeded he Black-Scholes model i order o make valuaio Europea currecy opio, havig wo fudameal feaures: (1) esimaig he marke volailiy of a uderlyig asse geerally as a fucio of price ad ime wihou direc referece o he specific ivesor characerisics such as expeced yield, risk aversio measures, or uiliy fucios; () selfreplicaig sraegy or hedgig. However, some researchers (see [7) preseed some evidece of he mispriced currecy opiosbyheg-kmodel.thesigificacausesofwhy his model is o suiable for sock markes are ha he currecies are differe from he socks i mai respecs ad geomeric Browia moio cao resolve he coduc of currecy reur; see [8. Sice he, i order o overcome hese problems, may sysems for pricig currecy opios were proposed by usig amedmes of he G-K model, such as, Roseberg [9, Sarwar ad Krehbiel [10,ad Bolle ad Rasiel [11. However all hese researches cosider ha he logarihmic reurs of he exchage rae were idepede ad ideically disribued ormal radom variables. Neverheless, he experimeal ivesigaio o asse reur shows ha discoiuiies or jumps are believed o be a idispesable eleme of fiacial sock pricig; see [1 17. Mero, i [18, proposed a jump diffusio process wih Poisso jump o mach he abormal flucuaios of sock price. Based o his heory, Kou [19, Co ad Takov [0 also cosidered he problems of pricig opios uder a jump diffusio evirome i a larger seig. Moreover, he empirical sudies also demosraed ha he disribuios of he logarihmic reurs i he asse marke geerally reveal excess kurosis

2 Mahemaical Problems i Egieerig wih more probabiliy mass aroud he origi ad i he ails adlessiheflakshawhawouldoccurforormally disribued daa [0.I ca be said ha he properies of fiacial reur series are oormal, oidepede, ad oliear, self-similar, wih heavy ails, i boh auocorrelaios ad cross-correlaios, ad volailiy cluserig [1 3. Sice fracioal Browia moio has wo subsaial feaures such as self-similariy ad log-rage depedece, hus usig i is more applicable o capure behavior from fiacial asse [3 37. Alhough, he fracioal Browia moio is eiher a Markov process or a semimarigale, we cao apply he commo sochasic calculus o aalyze i. Foruaely, Xiao e al. [38 employedhewickproduc raher ha he pahwise produc o defie a fracioal sochasic iegral whose mea is ideed zero. This propery was very coveie for boh heoreical developmes ad pracical applicaios. Furher, i [38, i was saed ha if oe uses he Wick-Io-Skorohod iegral, he ca obai a arbirage-free model. Björk ad Hul [39 showed recely ha uilizig fracioal Browia moio i fiace does o make much ecoomic sese because, while Wick iegraio leads o o arbirage, he defiiio of he correspodig self-fiacig radig sraegies is quie resricive, see for example [40, ha is he simple buyig-ad-hold sraegy is o self-fiacig. Therefore, he fracioal marke based o Wick iegrals is cosidered which is a beauiful mahemaical cosrucio bu wih resriced applicabiliy i fiace. From he above aalysis oe ca coclude ha he classical Io calculus could o help i fracioal Browia moio ad o describe a appropriae sochasic iegral wih respec o fracioal Browia moio is very sric. Ideed, he pricipal obsacle of applyig fracioal Browia moio i fiace is ha i is o semimarigale. To capure his problem, o ake io accou he log memory propery, ad o ge flucuaios form fiacial markes, i is suiable o apply he mixed fracioal Browia moio o ake flucuaios from fiacial asse (see [41 44). The mixed fracioal Browia moio is a family of Gaussia processes ha is a liear combiaio of Browia moio ad fracioal Browia moio. I is cosidered as oe of special class of log memory processes wih Hurs parameer H>1/.I ecoomics he firs work of usig mixed fracioal Browia moio could be foud i [45. Moreover, Cheridio [45 had proved ha, for H (3/4, 1), he mixed model wih depede Browia moio ad fracioal Browia moio is equivale o oe wih Browia moio ad hece i was arbirage-free. Rece furher applicaios have bee acceped i [46. Throughou he prese work, we follow he idea i [45; ha, is we will assume ha H (3/4, 1). Moreover, some empirical sudies have proved ha his hypohesis is useful ad applicable; see [ Furher, o capure jumps or discoiuiies, flucuaios adoakeioaccouhelogmemorypropery,combiaio of Poisso jumps ad mixed fracioal Browia moio is iroduced i his paper. The jump mixed fracioal Browia moio is based o he assumpio ha exchage rae reurs are geeraed by a wo-par sochasic process: (1) small, coiuous price movemes are geeraed by a mixed fracioal Browia moio ad () large, ifreque price jumps are geeraed by a Poisso process (see [5). This woparprocessisiuiivelyappealig,asiiscosisewih a impressive marke i which major iformaio arrives ifrequely ad radomly. I addiio, his process may provide a descripio for empirically observed disribuios of exchage rae chages ha are skewed, lepokuric, log memory ad have faer ails ha comparable ormal disribuios ad for he appare osaioariy of variace. Alhough, various models have bee applied for pricig currecy opios, uilizig jump fracioal Browia moio o pricig currecy opios has o bee ivesigaed. The we illusrae how o price Europea currecy opios usig he G-K ype model derived i a jump mixed fracioal Browia evirome. The comparaive resuls of our model ad oher available valuaio models idicae ha our model is easy o impleme. The res of his paper is orgaized as follows. I Secio, we briefly sae he defiiios ad properies relaed o mfbm ha will be used i forhcomig secios ad we prove some resuls regardig he quasi-codiioal expecaio. We prese a aalyical pricig formula for Europea currecy opio i a mixed fracioal Browia moio wih jumps evirome, i Secio 3. Secio 4 deals wih he jump mixed fracioal parial differeial equaio ad discussed some Greeks of his jump mixed model. I Secio 5, we idicae how o use our model o price currecy opios by umerical simulaios. The compariso of our jump mixed fracioal Browia moio model ad radiioal models is preseed. Secio 6 draws he cocludig remarks.. Prelimiaries I his secio we recall some defiiios ad resuls which we eed for he res of paper. This oucome ca be foud i [45, 46, 53, 54. Defiiio 1. A mixed fracioal Browia moio of parameers α, β, ad H is a liear combiaio of fracioal Browia moio wih Hurs parameer H ad Browia moio, defied o he probabiliy (Ω, F, P) for ay R + by M H =αb +βb H, (1) where B is a Browia moio, B H is a idepede fracioal Browia moio wih Hurs parameer H (0, 1), ad α ad β are wo real cosas such ha (α, β) = (0, 1). Now we lis some properies i [5, 54 byhefollowig proposiios. Proposiio. The mixed fracioal Browia moio M H saisfies he followig properies: (i) M H is a ceered Gaussia process ad o a Markovia oe, for H (0, 1) \ 1/; (ii) M H 0 =0P-almos surely;

3 Mahemaical Problems i Egieerig 3 (iii) he covariaio fucio of M H (α, β) ad MH (a, b)for ay, s R + is give by Cov (M H,MH s )=α ( s) + β (H +s H s H ), () where deoe he miimum of wo umbers; (iv) he icremes of M H (α, β) are saioary ad mixedself similar for ay h>0, M H h (α, β) MH (αh 1/,βh H ), (3) where meas o some law ; (v) he icremes of M H are posiively correlaed if 1/ < H < 1, ucorrelaed if H = 1/, ad egaive correlaed if 0<H<1/; (vi) he icremes of M H are log rage depede if ad oly if H>1/; (vii) for all R +,oehas E[(M H (α, β)) = { 0, = l + 1, {(l)! { l l! (α +β H ) l, = l. Now, le (Ω, F, P) be a probabiliy field such ha B is Browia moio wih respec o P ad B H is a idepede fracioal Browia moio wih respec o P. Now we prese some resuls regardig he quasi-codiioal expecaio ha we will eed for he res of he paper (see [16). Lemma 3. For every 0<<Tad σ Coe has (4) E [e σ(b T+B H T ) =e σ(b T+B H T )+(σ /)(T )+(σ /)(T H H), (5) where E deoes he quasi-codiioal expecaio wih respec o risk-eural measure. Proof. See [44. Lemma 4. Le f be a fucio such ha E [f(b T,B H T ) <. The, for every 0< Tad σ C,oehas E [f (σb T +σb H T ) = R 1 π [σ (T ) +σ (T H H ) (x σb exp [ σb H ) [σ (T ) +σ (T H H ) f (x) dx. [ (6) Le f(x) = 1 A ; oe ca easily obai he followig corollary. Proof. See [44. Corollary 5. Le A B(R).The E [1 A (σb T +σb H T ) = R (x σb ((exp [ σb H ) [σ (T ) +σ (T H H ) ) [ ( π [σ (T ) +σ (T H H 1 )) ) 1 A (x) dx. Le θ, w R.Now,oecosidersheprocess Z =θb +w(bh ) =θb +θ +wb H +w H, 0 T. From he Girsaov heorem, here exiss a measure P such ha Z is a ew mixed fracioal Browia moio. Oe will deoe E [ he quasi-codiioal expecaio wih respec o P.Cosider X = exp ( θb θ wbh w H ). (9) Proof. See [44. Lemma 6. Le f be a fucio such ha E [f(θb +wb H ).Theforevery T,oehas (7) (8) E [f (θb T +wb H T ) = 1 X E [f (θb T +wb H T )X T. (10) Proof. See [44. Theorem 7. Thepriceaeveryime [0,Tof a bouded F H T -measurable claim F L is give by F =e r(t ) E [F, (11) where r represes he cosas risk-free ieres rae. Proof. See [ Pricig Currecy Opios i Mixed Fracioal Browia Moio wih Jumps The aim of his secio is o derive he pricig formula for Europea currecy opios. The mixed fracioal Browia moio wih jump model by combiig he mixed fracioal Browia moio ad jump process (see [43, 5)is obaied ad some properies are aalyzed. To derive he ew currecy opio pricig formula i a jump mixed fracioal marke. The followig hypohesis will be provided: (i) here is o rasacio coss or axes ad all securiies are perfecly divisible;

4 4 Mahemaical Problems i Egieerig (ii) securiy radig is coiuous; (iii) he shor-erm domesic ieres rae r d ad foreig ieres rae r f are kow ad cosa hough ime; (iv) here are o risk-free arbirage opporuiies. Now cosider a jump fracioal Browia moio Black- Scholes currecy marke ha has wo ivesmes: (a) a moey marke accou: dπ =r d Π d, (1) where r d represe he domesic ieres rae; (b) a sock whose price saisfies he followig equaio: where d 1 =(l (S e J( i) K )(r d r f λμ J() ) (T ) + 1 σ (T ) + 1 σ (T H H )) ( σ (T ) +σ (T H H 1 )), d =d 1 σ (T ) +σ (T H H ), (16) ds =S (μ λμ J() )d+σs db +σs db H +S (e J() 1)dN, 0< T, S 0 =S>0, (13) where S deoe he spo exchage rae a ime of oe ui of he foreig currecy measured i he domesic currecy; he drif μ ad volailiy σ aresupposedo be cosas; B H is a fracioal Browia moio wih Hurs parameer H > 3/4; N is a poisso process wih rae λ; J() is jump size perce a ime which is a sequece of idepede ideically disribued; ad (e J() 1) N(μ J(),δ ()). Moreover, all hree sources of radomess, he fracioal Browia moio B H,hepoissoprocessN,adhejumpsize e J() 1, are supposed o be idepede. By usig he Io formula, he soluio for sochasic differeial equaio (13)is ε deoe he expecaio operaor over he disribuio of ej( i),adn( ) is he cumulaive ormal disribuio. I addiio, we ca calculae price of a pu currecy opio which is made by he followig corollary. Corollary 9. The price a every [0,Tof a Europea pu currecy opio wih srike price K ad mauriy T is give by P(,S )= =0 e λ(t ) λ (T )! ε [Ke r d(t ) N( d ) S e J( i) e (r f+λμ J() )(T ) N( d 1 ), (17) N S =S e J( i) exp [(μ λμ J() )+σb H +σb (14) where 1 σ 1 σ H. Le C(, S ) be he price of a Europea call currecy opio a ime wih srike price K ha maures a ime T. The we obai he pricig formula for currecy opio by he followig heorem. Theorem 8. Thepriceaevery [0,Tof a Europea call currecy opio wih srike price K ha maures a ime T is give by C(,S )= =0 e λ(t ) λ (T )! ε [S e J(i) e (r f+λμ )(T ) J() N(d 1 ) Ke r d(t ) N(d ), (15) d 1 =(l (S e J( i) K )(r d r f λμ J() ) (T ) + 1 σ (T ) + 1 σ (T H H )) ( σ (T ) +σ (T H H 1 )), d =d 1 σ (T ) +σ (T H H ). 4. Propery of Pricig Formula (18) Assume ha V is he value of a whole porfolio of differe opio. The value of whole porfolio saisfies i he jump mixed fracioal Black-Scholes parial differeial equaio ha prese i his heorem.

5 Mahemaical Problems i Egieerig 5 Theorem 10. The price of a currecy opio wih a bouded payoff f(s ) is give by V(S,),whereV(S,)is he soluio of he PDE: + 1 σ S V S +Hσ H 1 S V S +(r d r f λμ J() )S S r dv +λe[v(e J() S,) V(S,)=0. (19) Now, we discuss he properies of jump mixed fracioal Browia moio such as Greeks which are basic ools i risk maageme ad radig opios wihou he kowledge of Greeks ca resul i high loses. Theorem 11. Suppose C = C(, S ) be he price of Europea call currecy opio he Greeks are give by Δ= C S = e λ(t ) λ (T ) ε! [ =0 e J( i) e (r f+λμ J() )(T ) N(d 1 ). = C K = e λ(t ) λ (T )! ρ rd ρ rf =0 ε [ e r d(t ) N(d ). = C r d = =0 e λ(t ) λ (T )! ε [K (T ) e r d(t ) N(d ). = C r f = =0 ε [ S (T ) e λ(t ) λ (T )! e J( i) e (r f+λμ J() )(T ) N(d 1 ). Θ= C = (e λ(t ) λ +1 (T ) =0 e λ(t ) λ (T ) 1 ) (!) 1 ε [S e J(i) e (r f+λμ )(T ) J() N(d 1 ) Ke rd(t ) e λ(t ) λ (T ) N(d )+! =0 [(r f +λμ J() )S e J(i) e (r f+λμ )(T ) J() N(d 1 ) ε Γ= C S r d Ke r d(t ) N(d ) 1 σ +Hσ H 1 σ (T ) +σ (T H H ) S = e J( i) e (r f+λμ J() )(T ) N (d 1 ). e λ(t ) λ (T )! =0 [ ε ej(i) e (r f+λμ J ())(T ) N (d 1 ), S [ σ (T ) +σ (T H H ) σ = C σ = e λ(t ) λ (T )! [ [ =0 S ε e J( i) e (r f+λμ J() )(T ) σ (T ) +σ(th H ) σ (T ) +σ (T H H ) N (d 1 ). (0) The followig heorem represes he ifluece of Hurs parameer H i jump mixed fracioal Browia moio. Theorem 1. The ifluece of he Hurs parameer ca be wrie as C H = e λ(t ) λ (T ) ε! =0 [ [ S σ (T H l T H l ) σ (T ) +σ (T H H ) e J(i) e (r f+λμ )(T ) J() N (d 1 ). Suppose he forward price of a exchage rae is as f S() e J( i) e (r d r f λμ J() )(T ), (1) d ± = l (f/k) ± (σ /) (T H H )±(σ /) (T ). σ T H H +T () Remark 13. The relaioship of call-pu pariy saisfies as C(,S ) P(,S )=S e (r f+λμ J() )(T ) Ke r d(t ). (3)

6 6 Mahemaical Problems i Egieerig Table 1: The valuaio of chose parameers used i hese models. Model ype r d r f σ K H J λ μ J δ G-K PFBM PMFBM JMFBM JMFBM Thisisjusmorecomplicaedwayowrieherivial equaio S =S + S. Remark 14. The relaioship of pu-call pariy ca be wrie as C (, S ) 5. Simulaio Sudies P (, S ) =e (r f+λμ J())(T ). (4) Thissecioidicaehowoperformaceourjumpmixed fracioal Browia moio model ad o illusrae he effecs of jump parameers of our pricig model. For hese aims, we repor o wo ses of umerical experimes. I he firs se, we compare heoreical prices of some assumpive opios amog of he followig models: he G-K, he pure fracioal Browia moio (hereafer PFBM), he pure mixed fracioal Browia moio (hereafer PMFBM), ad our jump mixed fracioal Browia moio (hereafer JMFBM). These ess will o be based o empirical daa, bu hey will cosis of some simulaios of differe pricig models wih some chose parameers of some simulaios of differe pricig models wih some chose parameers ha will o be basedoempiricaldaa.thesecodseshowsheifluece of differe parameers of JMFBM. The code lie is wrie i MATLAB Compariso of Opio Prices. Now, for descripio of discrepacies amog hese mehods: he G-K, he PFBM, he PMFBM, ad our JMFBM mehods, we prese he heoreical prices of some assumpive opios applyig several models. All parameers which are eeded for compuig he assumpive currecy call opios prese by he G-K, PFBM, PMFBM, ad JMFBM mehods are prese i Table 1, respecively. The fourh ad fifh lie, which has low ad high jump parameers, respecively, provides he parameers for calculaig he prices by he JMFBM mehod. Tables ad 3 prese he prices compued by differe models, where S deoes he price of exchage rae; P G-K deoes he prices calculaed by he G-K mehod; P P-F shows he prices calculaed by he PFBM; P P-MF idicaes he prices simulaed correspodig o PMFBM; ad P J-MF shows he prices calculaed by he JMFBM. By comparig colums P G-K, P P-F, P P-MF,adP,a J-MF i Tables ad 3 for he small mauriy cases, we ca coclude ha he opio prices give by four assessme mehods almos are he same. The mai reaso i his case is ha jump parameers are very small. Furhermore, he prices obaied by MFBM ad JMFBM wih small parameers i high ad low expiraio are close, sice he jump parameers are very small. Also, Tables ad 3 show ha he parameers i colum P,a J-MF are greaer ha prices which is obaied by he G-K, PFBM, ad MFBM. Nex, he prices of assumpive opios are ivesigaed i large jump parameers case. Wheever he expiraio ime icreases, he discrepacy amog values calculaed by JMFBM model ad oher models icreases i large jump parameers case. Tables ad 3 show his resul by comparig colums P G-K, P P-F, P P-MF,adP,b J-MF for he small ad large ime o mauriy cases. The fidigs from he colums P G-K, P P-F, P,b J-MF,adP P-MF wih S [80, 140 idicae ha he amou of discrepacy proporio i prices is larger for ou-of-hemoey opios i he large expiraio ime case ad his discrepacy proporio declies wih he icrease he exchage rae. 5.. The Ifluece of Parameers. I wha follows, he value of call currecy opios is displayed by usig JMFBM for differe parameers. We jus ivesigae he ou-of-hemoey case. I fac, he oher cases i-he-moey ad ahe-moey ca also be cosideres by applyig he same mehod.now,wecosiderhevaluesofourjumpmixed fracioal Browia moio currecy opios for various Hurs parameers H ad he ivesigae he values for differe jump parameers. Figure 1 shows he values of call currecy opio agais is parameers, H, λ, adμ J.The defaulig parameers are S = 90, K = 100, r d = 0.010, r f = 0.30, σ = , = 0, T =, H = 0.76, J = , λ = 7.3, μ J = , adδ = No woder ha Figure 1 shows ha (1) he opio value is a decreasig fucio of μ J, δ ad () icreasig parameer of H ad λ comes alog wih a icrease of he opio value. Now,wecomparehehreepricesderivedfromheG- K, PFBM, ad JMFBM models for ou-of-he-moey. These simulaio parameers are seleced: S = 89.8, r d = 0.010, r f = 0.030, σ = , H = 0.76, J = 0.008, λ =.3, μ j = , ad = 0.1 wih differe exercise price K [90, 10 ad expiraio T [0.11, 10. For ou-of-hemoey Figure idicaes he opio price discrepacies by heg-k,pmfbm,adjmfbmmehods,respecively.from he figures i is cocluded ha he prices by JMFBM are beer fied o he G-K model. Therefore, we ca see ha our JMFBM seems reasoable.

7 Mahemaical Problems i Egieerig 7 S Table : Resuls by differe pricig models. Low ime o mauriy, T= 0.5, =0 P G-K P P F P P MF P,a J MF The maximum umber of ieraio is 100. a The parameers for calculaios are i he hird lie of Table 1. b The parameers for calculaios are i he forh lie of Table 1. P,b J MF S Table 3: Resuls by differe pricig models. Low ime o mauriy, T=, =0 P G-K P P F P P MF P,a J MF The maximum umber of ieraio is 100. a The parameers for calculaios are i he hird lie of Table 1. b The parameers for calculaios are i he forh lie of Table 1. P,b J MF 6. Coclusios This paper compouds he jump process ad mixed fracioal Browia moio. Furhermore, some paricular feaures of currecy pricig formula are also idicaed. Furhermore, experimeal resuls obaied i Secio 5 idicae ha whe he expiraio dae is grea sufficie he JMFBM model is esseial o use. However, our jump mixed fracioal Browia moio model is simple o employ ad has he especial properies o describe he uusual movemes of fiacial markes. I coclusio, sice mixed fracioal Browia moio is a suiable mahemaical model of profoudly correlaed radom processes ad jumps are udeiable compoe i fiacial markes, jump mixed fracioal Browia model will be a more appropriae mehod for pricig currecy opios. Appedix Proof of Theorem 8. I a risk aural world, from Theorem 7, a Europea call currecy opio wih mauriy T ad srike price K cabedisplayedas C(,S )= E P H [e r d(t ) (S T K) + F H =e r d(t ) E P H [S T 1 ST >K F H Ke r d(t ) E P H [1 ST >KF H, where 1 ST >K is a idicaor fucio. Le B + B H = μ λμ J() +r f r d +B σ +B H. (A.1) (A.)

8 8 Mahemaical Problems i Egieerig Opio value 6 Opio value Hurs parameer λ Opio value 6 Opio value μ j δ Figure 1: Value of he currecy call opio. The we obai Usig he idepedece of N T ad J( i ) ad he heory of Poisso disribuio wih iesiy λ(t ),we have N T S T =S exp [(μ λμ J() ) (T ) +σ( B H T B H ) +σ ( B T B ) (A.3) S T = =0 P(N =)S T = e λ(t ) λ (T ) S T!. =0 (A.5) Le [ 1 σ (T ) 1 σ (T H H ). By usig he fracioal Girsaov heorem, we obai ha B H B is a ew mixed fracioal Browia moio uder P H. Hece, seig S T =S exp [(μ λμ J() ) (T ) +σ( B H T B H ) +σ ( B T B ) [ 1 σ (T ) 1 σ (T H H ). (A.4) d = l [ K S (r ej( i) d r f λμ J() ) (T ) +σ( B H + B )+ 1 σ (T H H )+ 1 σ (T ) (A.6)

9 Mahemaical Problems i Egieerig 9 Differece Srike price JMFBM versus G-K Pure MFBM versus G-K Time o mauriy (year) Figure : Relaive differece amog he G-K model, PMFBM, ad JMFBM for ou-of-he-moey. ad usig Theorem 7 we derive ha E P H [1 ST >K F H = E P H [1 d <σ( B H T + B T )< FH = d 1 π [σ (T ) +σ (T H H ) (x σ B H exp [ σ B ) [σ (T ) +σ (T H H ) dx [ (σ B H + B d )/ σ (T )+σ (T H H ) 1 = π e z / dz = N (d ), (A.7) from he fracioal Girsaov formula we kow ha here exiss a probabiliy measure P H i which σb +σ(b H ) is a ew MFBM. Le X = exp (σ B H +σ B 1 σ H 1 σ ) ; (A.10) he S T =S e (r d r f )(T ) X T from Lemma 6 ad Theorem 7,we obai ha E P H [S T 1 ST >K F H where =S e (r d r f )(T ) E P H [X T 1 d <σ B H +σ B <+ (σ B H +σ B ) F H =S e (r d r f )(T ) E P H [1 d 1 <σ(b H T ) +σb T <+ (σ(bh T ) +σb T ) FH =S e (r d r f )(T ) (σ(b H ) +σb d 1 )/ σ (T )+σ (T H H ) 1 π e z / dz =S e (r d r f )(T ) N(d 1 ), (A.11) where d = σ B H +σ B d σ (T ) +σ (T H H ) d 1 = l ( K S ) (r ej( i) d r f μ J() ) (T ) +σ(b H ) +σb 1 σ (T ) = (l (S e J( i) K )+(r d r f λμ J() ) (T ) 1 σ (T H H ) 1 σ (T ) ) (A.8) d 1 = 1 σ (T H H ), σ(b H ) +σb d 1 σ (T ) +σ (T H H ) (A.1) ( σ (T ) +σ (T H H 1 )). Now, we cosider E P H [S T 1 ST >K F H ;seig σb +σ(bh ) =σ( B σ)+σ( B H σ H ) (A.9) K = (l ( S ) (r ej( i) d r f μ J() ) (T ) + 1 σ (T ) + 1 σ (T H H )) ( σ (T ) +σ (T H H 1 )).

10 10 Mahemaical Problems i Egieerig From he aalysis above ad Theorem. from [55 iis calculaed ha he price of Europea call currecy opio ca be wrie as C(,S )= =0 e λ(t ) λ (T )! ε [S e J(i) e (r f+λμ )(T ) J() N(d 1 ) Ke r d(t ) N(d ), ad for he price of pu currecy opio P(,S )= =0 e λ(t ) λ (T )! ε [Ke r d(t ) N( d ) S e J( i) e (r f+λμ J() )(T ) N( d 1 ). (A.13) (A.14) The erm of [V(e J() S,) V(S,)dN describes he differece i he opio value whe a jump occurs. The chage i he porfolio value ca be expressed as follows: dπ =dv(,s ) = + 1 σ S V S +Hσ H 1 S V S +(μ λμ J() )S +σs db +σs db H +[V(e J() S,) V(S,)dN Δ[S (μ λμ J() )d+σs db +σs db H +S (e J() 1)dN, dπ = [ + 1 σ S V S +Hσ H 1 S V S +(μ λμ J() )S Δ(μ λμ J() )S d +(σs ΔσS )db +(σs ΔσS )db H +[V(e J() S,) V(S,) ΔS (e J() 1)dN. (A.18) Proof of Theorem 10. Le V(, S ) be he price of he currecy derivaives a ime ad le Π be he porfolio value. The we have Π =V(S,) ΔS. (A.15) Porfoliovaluechagesiaveryshorperiodofime.Hece dπ =dv(s,) ΔS sice S =Sexp [(μ λμ J() )+σb H 1 σ H + N J( i ). +σb 1 σ (A.16) By seig Δ = / o elimiae he sochasic oise, he dπ = ( + 1 σ S V S +Hσ H 1 S V S +λe[v(e J() S,) V(S,) (e J() 1)S ) d. (A.19) The reur of a amou Π ivesed i bak accou is equal o r d Π d a ime d. For absece of arbirage hese values mus be he same [43. By applyig he Io formula for jump diffusio process [55 we have dv (, S )= + 1 σ S V S +Hσ H 1 S V S Proof of Theorem 11. Firs, we derive a geeral formula. Le y be oe of he ifluece facors. By seig +(μ λμ J() )S +σs db +σs db H +[V(e J() S,) V(S,)dN. (A.17) C 1 (, S )=S e J(i) e (r f+λμ )(T ) J() N(d 1 ) Ke r d(t ) N(d ), (A.0)

11 Mahemaical Problems i Egieerig 11 he we have Bu C 1 = (S ej(i) e (r f+λμ )(T ) J() ) N(d 1 ) +S e J(i) e (r f+λμ J() )(T ) N (d 1 ) Ke r d(t ) N(d ) Ke r N (d d(t ) ). (A.1) +S e J( i) e (r f+λμ J() )(T ) N (d 1 ) σ (T ) +σ (T H H ). Subsiuigi (A.3) we ge he desired. (A.3) Proof of Theorem 1. We firs differeiae C(, S ) wih respec o H;hewehave C H = e λ(t ) λ (T )! =0 ε N (d ) =N (d ) d π e d = 1 / d = 1 π exp [ (d 1 (σ (T ) π e d = 1 +σ (T H H )) 1/ ) / d. / exp [d 1 σ (T ) +σ (T H H ) exp [ σ (T ) +σ (T H H ) d = [ [ =0 [ [ S e J( i) e (r f+λμ J() )(T ) N (d 1 ) σ (T ) +σ (T H H ) H e λ(t ) λ (T ) S! ε e J( i) e (r f+λμ J() )(T ) N (d 1 ) σ (T H l T H l ). σ (T ) +σ (T H H ) (A.4) π e d = 1 / exp [l ( S ej( i) ) K +(r d r f λμ J() ) (T ) d Coflic of Ieress The auhors declare ha here is o coflic of ieress regardig he publicaio of his paper. =N (d 1 ) S ej( i) K exp [(r d r f λμ J() ) (T ) d. Thewehave C 1 = (S ej(i) e (r f+λμ J())(T ) ) N(d 1 ) Ke r d(t ) N(d ) (A.) Ackowledgme The auhors express heir sicere haks o he referees for he careful ad deailed readig of he paper ad very helpful suggesios. The auhors graefully ackowledge ha his research was parially suppored by he Uiversiy Pura Malaysia uder he GP-IBT Gra Scheme havig projec o. GP-IBT/013/ Refereces [1 A. Dravid, M. Richardso, ad T. Su, Pricig foreig idex coige claims: a applicaio o Nikkei idex warras, Joural of Derivaives,vol.1,o.1,pp.33 51,1993. [ T. S. Ho, R. C. Sapleo, ad M. G. Subrahmayam, Correlaio risk, crossmarke derivaive producs ad porfolio performace, Joural of Europea Fiacial Maageme, vol. 1,o.,pp ,1995.

12 1 Mahemaical Problems i Egieerig [3 K. B. Tof ad E. S. Reier, Currecy-raslaed foreig equiy opios: he America case, Advaces i Fuures ad Opios Research,vol.9,aricle13,pp.33 64,1997. [4 Y.K.KwokadH.Y.Wog, Currecy-raslaedforeigequiy opios wih pah depede feaures ad heir muli-asse exesios, Ieraioal Joural of Theoreical ad Applied Fiace,vol.3,o.,pp.57 78,000. [5 J. Dua ad J. Wei, Pricig foreig-currecy ad crosscurrecy opios uder GARCH, Joural of Derivaives, vol. 7, o. 1, pp , [6 M. B. Garma ad S. W. Kohlhage, Foreig currecy opio values, Joural of Ieraioal Moey ad Fiace, vol.,o. 3, pp , [7 R. Cookso, Models of imperfecio, Risk, vol. 9, o. 5, pp , 199. [8 N. Ekvall, L. P. Jeergre, ad B. Näslud, Currecy opio pricig wih mea reversio ad ucovered ieres pariy: a revisio of he Garma-Kohlhage model, Europea Joural of Operaioal Research,vol.100,o.1,pp.41 59,1997. [9 J.V.Roseberg, Pricigmulivariaecoigeclaimsusig esimaed risk-eural desiy fucios, Joural of Ieraioal Moey ad Fiace, vol. 17, o., pp. 9 47, [10 G. Sarwar ad T. Krehbiel, Empirical performace of aleraive pricig models of currecy opios, Joural of Fuures Markes,vol.0,o.3,pp.65 91,000. [11 N. P. B. Bolle ad E. Rasiel, The performace of aleraive valuaio models i he OTC currecy opios marke, Joural of Ieraioal Moey ad Fiace, vol.,o.1,pp.33 64, 003. [1 P. Jorio, O jump processes i he foreig exchage ad sock markes, Review of Fiacial Sudies,vol.1,o.4,pp , [13 D. S. Baes, Dollar jump fears, : disribuioal abormaliies implici i currecy fuures opios, Joural of Ieraioal Moey ad Fiace,vol.15,o.1,pp.65 93,1996. [14 D. S. Baes, Jumps ad sochasic volailiy: exchage rae processes implici i deusche mark opios, Review of Fiacial Sudies,vol.9,o.1,pp ,1996. [15 M. Cherov, A. R. Galla, E. Ghysels, ad G. Tauche, Aleraive models for sock price dyamics, Joural of Ecoomerics, vol. 116, o. 1-, pp. 5 57, 003. [16 J. Pa, The jump-risk premia implici i opios: evidece from a iegraed ime-series sudy, Joural of Fiacial Ecoomics, vol.63,o.1,pp.3 50,00. [17 B. Eraker, Do sock prices ad volailiy jump? Recocilig evidece from spo ad opio prices, Joural of Fiace, vol. 59,o.3,pp ,004. [18 R. C. Mero, Opio pricig whe uderlyig sock reurs are discoiuous, Joural of Fiacial Ecoomics, vol. 3, o. 1-, pp , [19 S. G. Kou, A jump-diffusio model for opio pricig, Maageme Sciece,vol.48,o.8,pp ,00. [0 R. Co ad P. Takov, No-parameric calibraio of jumpdiffusio opio pricig models, Joural of Compuaioal Fiace,vol.7,o.3,pp.1 49,004. [1 A. W. Lo, Log-erm memory i sock marke prices, Ecoomerical,vol.59,o.5,pp ,1991. [ R. N. Maega ad H. E. Saley, A Iroducio o Ecoophysics, Cambridge Uiversiy Press, Cambridge, UK, 007. [3 B. Huag ad C. W. Yag, The fracal srucure i muliaioal sock reurs, Applied Ecoomics Leers,vol.,pp.67 71, [4 D. O. Cajueiro ad B. M. Tabak, Log-rage depedece ad mulifracaliy i he erm srucure of LIBOR ieres raes, Physica A,vol.373,pp ,007. [5 S. H. Kag ad S.-M. Yoo, Log memory properies i reur ad volailiy: evidece from he Korea sock marke, Physica A,vol.385,o.,pp ,007. [6 S. H. Kag ad S.-M. Yoo, Log memory feaures i he high frequecy daa of he Korea sock marke, Physica A, vol. 387, o.1,pp ,008. [7 S. H. Kag, C. Cheog, ad S.-M. Yoo, Log memory volailiy i Chiese sock markes, Physica A, vol. 389, o. 7, pp ,010. [8 Y. Wag, Y. Wei, ad C. Wu, Cross-correlaios bewee Chiese A-share ad B-share markes, Physica A, vol.389,o. 3, pp , 010. [9 Z. Dig, C. W. J. Grager, ad R. F. Egle, A log memory propery of sock marke reurs ad a ew model, Joural of Empirical Fiace,vol.1,o.1,pp ,1993. [30 Y. Liu, P. Gopikrisha, P. Cizeau, M. Meyer, C.-K. Peg, ad H. E. Saley, The saisical properies of he volailiy of price flucuaios, Physical Review E, vol. 60, o., pp , [31 B. Podobik ad H. E. Saley, Dereded cross-correlaio aalysis: a ew mehod for aalyzig wo osaioary ime series, Physical Review Leers,vol.100,o.8,AricleID08410, 008. [3 B. Podobik, D. Horvaic, A. M. Peerse, ad H. E. Saley, Cross-correlaios bewee volume chage ad price chage, Proceedigs of he Naioal Academy of Scieces of he Uied Saes of America,vol.106,o.5,pp ,009. [33 A. Carboe, G. Caselli, ad H. E. Saley, Time-depede hurs expoe i fiacial ime series, Physica A,vol.344,o. 1-, pp , 004. [34 X.-T. Wag, E.-H. Zhu, M.-M. Tag, ad H.-G. Ya, Scalig ad log-rage depedece i opio pricig II: pricig Europea opio wih rasacio coss uder he mixed Browiafracioal Browia model, Physica A, vol. 389, o.3,pp , 010. [35 X.-T. Wag, Scalig ad log rage depedece i opio pricig, IV: pricig Europea opios wih rasacio coss uder he mulifracioal Black-Scholes model, Physica A,vol. 389, o. 4, pp , 010. [36 X.-T. Wag, M. Wu, Z.-M. Zhou, ad W.-S. Jig, Pricig Europea opio wih rasacio coss uder he fracioal log memory sochasic volailiy model, Physica A, vol. 391, o. 4, pp , 01. [37W.Xiao,W.Zhag,W.Xu,adX.Zhag, Thevaluaioof equiy warras i a fracioal Browia evirome, Physica A,vol.391,o.4,pp ,01. [38 W. Xiao, W. Zhag, ad W. Xu, Parameer esimaio for fracioal Orsei-Uhlebeck processes a discree observaio, Applied Mahemaical Modellig, vol.35,o.9,pp , 011. [39 T. Björk ad H. Hul, A oe o Wick producs ad he fracioal Black-Scholes model, Fiace ad Sochasics,vol.9, o., pp , 005. [40 R. J. Ellio ad J. va der Hoek, A geeral fracioal whie oise heory ad applicaios o fiace, Mahemaical Fiace,vol. 13, o., pp , 003.

13 Mahemaical Problems i Egieerig 13 [41 C. El-Nouy, The fracioal mixed fracioal Browia moio, Saisics ad Probabiliy Leers,vol.65, o.,pp , 003. [4 Y. Mishura, Sochasic Calculus for Fracioal Browia Moios ad Relaed Processes, Spriger, 008. [43 L. Su, Pricig currecy opios i he mixed fracioal Browia moio, Physica A, vol. 39, o. 16, pp , 013. [44W.Xiao,W.Zhag,W.Xu,adX.Zhag, Thevaluaioof equiy warras i a fracioal Browia evirome, Physica A,vol.391,o.4,pp ,01. [45 P. Cheridio, Arbirage i fracioal Browia moio models, Fiace ad Sochasics,vol.7,o.4,pp ,003. [46 Y. Mishura, Sochasic Calculus for Fracioal Browia Moios ad RelaedProcesses, Spriger, Berli, Germay, 008. [47 M. C. Mariai, I. Florescu, M. P. Beccar Varela, ad E. Ncheuguim, Sudy of memory effecs i ieraioal marke idices, Physica A, vol. 389, o.8, pp , 010. [48 F. Serialdi, Use ad misuse of some Hurs parameer esimaors applied o saioary ad o-saioary fiacial ime series, Physica A,vol.389,o.14,pp ,010. [49 Y.-P. Rua ad W.-X. Zhou, Log-erm correlaios ad mulifracal aure i he ierrade duraios of a liquid Chiese sock ad is warra, Physica A,vol.390,o.9,pp , 011. [50 Y. Wag, Y. Wei, ad C. Wu, Dereded flucuaio aalysis o spo ad fuures markes of Wes Texas Iermediae crude oil, Physica A,vol.390,o.5,pp ,011. [51 L.-Y. He ad W.-B. Qia, A Moe Carlo simulaio o he performace of he R/S ad V/S mehods saisical revisi ad real world applicaio, Physica A,vol.391,o.14,pp , 01. [5 W.-L. Xiao, W.-G. Zhag, X.-L. Zhag, ad Y.-L. Wag, Pricig currecy opios i a fracioal Browia moio wih jumps, Ecoomic Modellig,vol.7,o.5,pp ,010. [53Y.HuadB.Øksedal, Fracioalwhieoisecalculusad applicaios o fiace, Ifiie Dimesioal Aalysis, Quaum Probabiliy ad Relaed Topics,vol.6,o.1,pp.1 3,003. [54 M. Zili, O he mixed fracioal Browia moio, Joural of Applied Mahemaics ad Sochasic Aalysis,vol.006,Aricle ID 3435, 9 pages, 006. [55 Kazuhisa Masuda, Iroducio o Mero Jump Diffusio Model, Deparme of Ecoomics, The Graduae Ceer, The Ciy Uiversiy of New York.

14 Advaces i Operaios Research Hidawi Publishig Corporaio hp:// Advaces i Decisio Scieces Hidawi Publishig Corporaio hp:// Joural of Applied Mahemaics Algebra Hidawi Publishig Corporaio hp:// Hidawi Publishig Corporaio hp:// Joural of Probabiliy ad Saisics The Scieific World Joural Hidawi Publishig Corporaio hp:// Hidawi Publishig Corporaio hp:// Ieraioal Joural of Differeial Equaios Hidawi Publishig Corporaio hp:// Submi your mauscrips a hp:// Ieraioal Joural of Advaces i Combiaorics Hidawi Publishig Corporaio hp:// Mahemaical Physics Hidawi Publishig Corporaio hp:// Joural of Complex Aalysis Hidawi Publishig Corporaio hp:// Ieraioal Joural of Mahemaics ad Mahemaical Scieces Mahemaical Problems i Egieerig Joural of Mahemaics Hidawi Publishig Corporaio hp:// Hidawi Publishig Corporaio hp:// Hidawi Publishig Corporaio hp:// Discree Mahemaics Joural of Hidawi Publishig Corporaio hp:// Discree Dyamics i Naure ad Sociey Joural of Fucio Spaces Hidawi Publishig Corporaio hp:// Absrac ad Applied Aalysis Hidawi Publishig Corporaio hp:// Hidawi Publishig Corporaio hp:// Ieraioal Joural of Joural of Sochasic Aalysis Opimizaio Hidawi Publishig Corporaio hp:// Hidawi Publishig Corporaio hp://