Evaluation of the Stochastic Modelling on Options

Transcription

1 Zhjuan Mao, Zhan Lang, Jnguo Lan, Hongkun Zhang / Inernaonal Journal of Engneerng Research and Applcaons (IJERA) ISSN: Vol., Issue 3, May-Jun 0, pp Evaluaon of he Sochasc Modellng on Opons Zhjuan Mao, Zhan Lang, Jnguo Lan 3, Hongkun Zhang 4 School of Fnance, Shangha Unversy of Fnance and Economcs, PRC. Deparmen of Appled Mahemacs, Shangha Unversy of Fnance and Economcs, PRC. The auhor s parally suppored by he Leadng Academc Dscplne Program and CFKS and TIP (MEC ) 3 Deparmen of Mahemacs and Sascs, UMass Amhers,USA. 4 Deparmen of Mahemacs and Sascs, UMass Amhers, USA. The auhor s parally suppored by NSF (DMS ) Absrac Modern opon prcng echnques are ofen consdered among he mos mahemacally complex of all appled areas of fnancal engneerng. In parcular hese echnques derve her mpeus from four mlesones of opon prcng models: Bacheler model, Samuelson model, Black-Scholes-Meron model and Levy model. In hs paper we evaluae all relaed opon prcng models based on hese mlesones, by comparng he correspondng sochasc dfferenal equaons and opon prcng formulas. In addon we also nclude some smulaons o make he comparsons more ransparen. Key Words: European opon, Prcng model, Brownan moon, Levy processes, Varance Gamma process I. INTRODUCTION Fnancal Engneerng s a cross-dscplnary feld whch reles on fnancal mahemacs, numercal mehods and compuer smulaons o make radng, hedgng and nvesmen decsons. Wh he rapd developmen of fnancal dervaves lke opons, fuures and swaps, has aroused more and more neres especally he opc on prcng models. Modern opon prcng echnques, wh roos n sochasc calculus, are ofen consdered among he mos mahemacally complex of all appled areas of fnancal engneerng. In parcular hese modern echnques derve her mpeus from four mlesone of opon prcng models. Dang back o 900, Bacheler frs proposed o use he Brownan moon o model he dynamcs of sock prce n hs Ph.D dsseraon, and he also derved a closed formula for prcng he sandard European opons. He was consdered o be a poneer n he sudy of fnancal engneerng and fnancal mahemacs, alhough he process he used can generae shares ha allowed boh negave secury prces and opon prces ha exceeded he prce of he underlyng asse. In 959, Osborne refned Bacheler model by usng he sochasc exponenal of he Brownan moon o model sock prce []. Laer, Samuelson [] exended o sudy he opon prcng usng he geomerc Brownan moon n a more sysemac manner, and he came wh he dea o use he dscoun rae n prcng. Due o hs grea conrbuons, Samuelson was awarded he Nobel Memoral Prze n Economc Scences n 970. A breakhrough was made by Black, Scholes and Meron n 973, ha he opon prce s explcly conneced o a hedgng sraegy whch depends on he volaly of he sock prce as well as oher observable quanes. The mos sgnfcan dea of Black-Scholes-Meron (BS) model was o realze ha he expeced reurn of he opon prce should be he rsk-free rae and ha by holdng a ceran amoun of sock, now referred o as he dela; he opon poson could be dynamcally compleely hedged. Laer, Scholes and Meron receved he Nobel Prze n 997 for her key dscovery. Shorly afer he nvenon of BS model, aemps have been made o relax he srngen assumpons and o mprove he performance of he model. Meron proposed a jump dffuson model [3], under whch he log reurn has boh a dffuson componen and a jump componen. And hs dscovery made hm conclude ha he exponenal Brownan moon should be replaced by symmerc -sable Levy process o beer f marke daa. Ths model s ofen called Meron Jump dffuson (MJD) model. He also clamed ha he log prce process s a superposon of a Brownan moon and an ndependen compound Posson process wh lognormally dsrbued jumps. Emprcal sudes reveal ha he sock prce dsrbuon usually has some properes conradcng o he radonal Black-Scholes assumpons, such as "fa al'', "self-smlary'', "long-range dependence'' see [4]. Mandelbro and Taylor [5, 6] proposed ha he sock marke should ake on he characer of fraconal Brownan moon, hen Peers [7] nroduced Fraconal Brownan moon o model he dynamcs of sock prce. Afer hs, many scholars have made ousandng conrbuons on hs opc. In 000, Duncan e al. [8], Hu and Oksendal [9] developed he fraconal Black- Scholes formula by a sochasc negraon wh he fraconal Brownan moon and Wck-produc a nal me. In 463 P a g e

2 Zhjuan Mao, Zhan Lang, Jnguo Lan, Hongkun Zhang / Inernaonal Journal of Engneerng Research and Applcaons (IJERA) ISSN: Vol., Issue 3, May-Jun 0, pp , Cpran Necula [0] obaned an explc fraconal Black-Scholes formula for he prce of an opon by usng Fourer ransform. Durng he pas decades, much aenon has been drawn o he consan volaly model. I was shown n [, ] ha sock reurns were heeroscedasc wh consderable evdences. The consan elascy volaly process was frs proposed by Cox and Ross [3, 4]. And hey appled hs dffuson process o opon prcng. Besdes he mpled volaly smle, s dscovered ha when he prce of he underlyng asse decreases, he smle shfs o lower prces; when he prce ncreases, he smle shfs o hgher prces. In order o have a coheren esmae of volaly rsk, he SABR model was developed n [5]. More mporanly, he SABR model capures he correc dynamcs of he smle, and hus yelds sable hedges. In [6], Heson also nroduced a sochasc volaly model and derved a closed-form formula for a European call opon. Ths model s called he -dmensonal BS model. I s crcal o correcly handle marke smles and skewness for hedgng. In 996, Chen developed a 3-dmensonal BS model o descrbe he evoluon of neres rae [7]. Comparng wh Heson model, can be vewed as a "hree-facor model" snce descrbes neres rae as drven by hree sources of marke rsk, sochasc mean and sochasc volaly. Consderng Levy models wh jumps, many mprovemens have been made based on MJD model. Baes found ha sochasc volaly alone could no explan he "volaly smle" well. He furher exends he Heson model o combne sochasc volaly and jump dffuson process (SVJD) under sysemac jumps and volaly rsk [8]. In 00, Kou [9] proposed a double exponenal jump dffuson model, whch can capure he lepokurc and fa al properes beer han MJD model. Madan and Senea proposed Varance Gamma (VG) model n 990, n whch he log reurn of sock follows a varance gamma process [0]. And he asymmerc VG model was suded n []. Then he Normal Inverse Gaussan (NIG) model was proposed by Barndorff-Nelsen [] n 998. Laer Eberlen and Prause [3] exended NIG model o he generalzed hyperbolc class. In 00, Carr, Geman, Madan and Yor developed CGMY model based on VG model [4]. Ths paper manly gves an overvew and evaluaons of he developmen of major opon prcng models, as well as her mprovemens and exensons. Comparng o oher survey papers, Sundaresan [5] manly concerned wh dervave prcng heory, erm srucure heory and asse prcng bu whou prcng models; In [6], Hobson focused on sochasc mehods of dervave prcng and descrpons of some barrer opons. Whle n hs paper, we no only revew he major heory of opon prcng bu also gve a dealed evaluaon along he developmen of European opon prcng models, whch can be of grea help o researchers on opon prcng. The paper s organzed as follows. In secon, we begn wh he revew and evaluaon of he four mporan mlesone models on European opons: he Bacheler Model, whch s he prooype of opon prcng heory; he Samuelson-Osborne Model, whch can be seen as he md-sep o BS model; hen he Black-Scholes-Meron (BS) model, followed by he MJD model. The exensons of BS model are lsed n Secon 3. Besdes from he model drven by he Fraconal Brownan moon, we also evaluaed CEV, SABR, Heson and Chen model. The Kou model, SVJD model as well as he VG, CGMY and NIG model are oulned n secon 4 as hey have grealy mproved he MJD model. We also nclude some smulaons on comparson of BS model wh Bacheler Model as well as Heson model. II. MILESTONES OF OPTION PRICING MODELS We frs brefly revew he concep of opon: A call (or pu) opon s a conrac ha gves an nvesor he rgh bu no he oblgaon o buy (or sell) a sock or oher underlyng asses a a specfed prce (he srke prce) whn a ceran me (he maury). In hs paper we manly consder he European opons, whch can only be exercsed a s maury, and for smplcy we assume he underlyng asses are socks. For a European call opon wh he srke prce K and he maury T, f he prce of s underlyng asse s S(T), hen he payoff of hs opon s gven by C ( S( T), T) ( S( T) K) () Consder he sock prce process {S(): 0}, equpped wh he ncreasng sequence of σ-algebra { F } ha conans all he hsorcal nformaon a me. Besdes s real world expecaon E[S()], s also useful o consder he rskneural expecaon E R [S()] under he rsk-neural probably P R, whch s absoluely connuous wh respec o P, he orgnal probably. Then he prce of a European call opon a me can be expressed as he expecaon of dscouned payoffs a maury T under ceran measure: C( S( ), ) E[ D( T)( S( T) K) F ] () D () where E() denoes eher he real world or rsk-neural expecaon n dfferen models and D() s a dscoun process. From now on, we assume =0 for smplcy. Snce he prce of pu opon can be derved from he pu-call pary [7], we only consder call opons n hs paper. Nex we ls all general assumpons for he followng models: A) The opon s a European call opon wh srke prce K and maury me T; A) The sock prce evolves as a connuous Markov process, homogeneous boh n me and space; 464 P a g e

3 Zhjuan Mao, Zhan Lang, Jnguo Lan, Hongkun Zhang / Inernaonal Journal of Engneerng Research and Applcaons (IJERA) ISSN: Vol., Issue 3, May-Jun 0, pp A3) There are no ransacon coss n buyng or sellng socks, opons,.e. he marke s frconless; A4) No dvdends are pad on he underlyng sock durng he opon lfe; A5) The marke does no adm arbrage. Noe ha he assumpon A3-A4) can be dropped easly, by replacng he sock prce by he effecve sock prce n all he models below.. Bacheler model In [8] Bacheler frs proposed a mahemac model o prce sock prce, assumng ha he dscoun rae s zero and he dynamcs of sock prce sasfes he followng SDE: = S(0)σdB() (3) where S() denoes he spo prce of he underlyng secury a me ; B() s he sandard Brownan moon and σ s he volaly of sock prce. Consderng a European call opon defned as above, s opon a nal me sasfes: C( S(0), 0) E[( S( T) K) ], where E() denoes he expecaon wh respec o he real world. Alhough he soluon was no precsely derved n hs early work, based on () and (3) one can oban: C( S(0), 0) ( S(0) K) F ( d) S(0) T ( d), where S(0) K d, FN ( ) and N () denoe he cumulave dsrbuon funcon and he probably densy funcon S( T) T of sandard normal varable, respecvely. Bacheler proved ha he one dmensonal dsrbuon of hs sock process sasfes he relaon known as he Chapman-Kolmogorov equaon and he Gaussan propery wh lnearly ncreasng varance [9]. Bu he hypohess of absolue Brownan moon (3) leads o negave sock prce wh posve probably. And he fac ha gnores any dscounng conradcs he realy. Despe of hese resrcons, he Bacheler model s wdely celebraed as a landmark n he hsory of prcng heory. I has grea nfluence on he whole developmen of sochasc calculus and fnancal engneerng.. Samuelson/Osborne model In 959, Osborne refned Bacheler model by usng he sochasc exponenal of Brownan moon o model he sock prce process []. He jusfed hs approach n psychologcal erms based on he Weber-Fechner law, whch saes ha humans perceve he nensy of smul on a log scale raher han a lnear scale. Based on hs argumen, Osborne nferred ha he logreurn log( S( ) / S( )) should follow a normal dsrbuon wh mean zero and varance, for any small 0. Laer Samuelson appled hs process o prce opons n a more sysemac way []. Besdes he general hypoheses, he assumed ha he opon derved from a sock has a consan expeced reurn rae. And s greaer han ha of he underlyng secury. The evoluon of sock prce n Samuelson model sasfes he followng SDE: (4) d db() S () where denoes he expeced reurn rae of S(), s same as defned above. Agan Samuelson dd no gve an explc soluon for he opon prce, bu we can derve from () and (4): ( ) C(S(0), 0)=e T T T E[( S( T) K) ] S(0)e FN( d) Ke FN( d) where d [log( S( T) / K) ( ) T]/ T, d d T and, beng he expeced reurn rae of opon value and sock prce, respecvely. Alhough hs prcng formula s very close o Black-Scholes formula, dd no receve much aenon, manly because he wo expeced reurn values are boh unknown quanes from he marke. As hey depend heavly upon he unque rsk characerscs of he underlyng sock and he opon, dfferen nvesors mgh propose dfferen values n erms of her own level of rsk averson. Ths uncerany makes Samuelson model dffcul for buyers and sellers wh dfferen rsk aversons o agree upon a unform prce. On he oher hand, may gve ceran nvesors a more accurae forecasng based on personal keen nsgh no he marke..3 Black-Scholes-Meron (BS) model In 973, Black and Scholes derved he Black-Scholes formula, whch posed a momenous sgnfcance n he hsory of opon prcng heory. The ousandng feaure of hs model s nroducng he rsk neural measure and clamng ha he dscouned value of dervave secury under hs measure s ndeed a marngale [30]. N N 465 P a g e

4 Zhjuan Mao, Zhan Lang, Jnguo Lan, Hongkun Zhang / Inernaonal Journal of Engneerng Research and Applcaons (IJERA) ISSN: Vol., Issue 3, May-Jun 0, pp The model assumes ha he sock prce follows he SDE (4) defned n he Samuelson model. To prce a European call opon, hey furher derved ha s prce process C(S(), ) sasfes he followng PDE: C C C S rs rc 0. S S Thus one obans he so-called Black-Scholes formula rt CBS ( S(0), 0) S(0) FN ( d) Ke FN ( d) (5) Where d [log( S(0) / K) ( r ) T]/ T, d d T. Noe ha here s only one unknown parameer n he above formula: he volaly wh all oher parameers observable from he marke. Besdes hs, from he dervaon of hs formula [30], BS model has a key advanage of posng an explc hedgng sraegy for replcaon. These make Black-Scholes formula very easy o use and raher aracve. However, emprcal sudes fnd ha he marke prce s generally hgher han compued from he formula. One of he possbles s ha he exsence of rsk neural measure needs effcen marke hypoheses, whch s defnely no so n realy. The assumpon of frconless marke, such as no coss or fees for ransacons, connuous me radng, may also lead o lqudy rsk and gap rsk. Apar from ha, volaly surface from marke daa mples ha he volaly s no consan. One of he mos serous lmaons s he underesmaon of exreme moves. I s found ha he reurn dsrbuon has hgher peak and heaver als comparng wh normal dsrbuon, whch dsagree wh he "deal condons". In general, hs formula s popular wh praconers for s smplcy and easeness o use. To beer undersand he advanage of Black-Scholes formula, we now compare he Bacheler model and BS model n fgure. Le he parameers be (S(0), K,, T, r)=(0,, 0.4,, 0). We smulae he pahs of S() under he Bacheler model and BS model, respecvely. The blue pah denoes he pah n he Bacheler model, whereas he red one refers o ha n BS model. As shown n fgure, he pahs obaned from hese wo models f very well a he begnnng of he perod. However, he dfference s enlarged when he me nerval ges longer. Ths mples ha he Bacheler model can be seen as a lnearzaon of BS model wh shor maury and vanshng drf erm..4 Levy model---mjd model Levy Process s defned as a connuous-me sochasc process {X=X(): 0} sars a he orgn and possesses he properes of ndependen ncremens, saonary ncremens and sochasc connuy. Besdes Markovan and saonary dsrbuon properes assumpon on sock prce, s also mporan o model he suaon when S() has he properes ha large jumps for exreme marke movemens and small jumps and dffuson for nsananeous radng [3]. Noe ha crcal assumpons n he BS model nclude radng akes place connuously n me and he dynamcs of he sock has a connuous sample pah wh probably one. However, enough evdences show ha he exsence of jumps n response o ceran announcemens. In 976, Meron [3] proposed he followng jump dffuson model, where he sock prce s he soluon of he followng SDE: N() ( ) d db( ) d Y. S () where N() s a Posson process, denoes he mean number of jumps per un me; Y s a random varable characerzng he percenage change of he sock prce when jump occurs, and EY. The assumpon ha Y s o guaranee he fac ha he sock prce could no jump from a posve o a negave value or o zero. The sock prce a me can be obaned under he rsk-neural measure: N() S( ) S(0)exp[( r ) B( )] ( Y ). (6) 466 P a g e

5 Zhjuan Mao, Zhan Lang, Jnguo Lan, Hongkun Zhang / Inernaonal Journal of Engneerng Research and Applcaons (IJERA) ISSN: Vol., Issue 3, May-Jun 0, pp Here Y are ndependen and have dencally lognormal dsrbuons under he rsk-neural measure,.e. log( Y ) follow normal dsrbuons N ( m, ). Thus we can oban R E Y exp( m ) n (6). For a European call opon, s prce process C(S(), ) sasfes he followng PDE: C C C R S ( r ) S rc [ E C( SY, ) (, )] 0. C S S S By condonng on he number of jumps N(T), we oban T n e ( T) C( S(0), 0) CBS ( Sn(0), 0), n! n0 where C ( S (0), 0) s defned as he Black-Scholes formula (5) wh / BS n m n n T and n Sn(0) S(0)exp[ nm e T T ]. The MJD model has grealy mproved BS model by nroducng jump process o model he sock prce. I can beer f marke daa especally when here are many oulers or ceran announcemens. Besdes, hs model leads o he lepokurc feaure and he mpled volaly smle. The hypohess of lognormal dsrbuon makes he good propery of he dsrbuon of he sum of jump random varables and Brownan moon. Thus a closed-form soluon can be derved. On he oher hand, hs model has s own drawback. For example, he hypohess abou jump process s oo src and he fa al feaure s no obvous under hs model. We wll revew some mprovemens of hs model n secon 4. III. IMPROVEMENTS AND EXTENSIONS ON BLACK-SCHOLES-MERTON MODEL As emprcal sudes fnd ha he prce of sock somemes does no follow he soluon of he Black-Scholes formula, many models have been consruced o make relaed mprovemens. 3. Fraconal Brownan Moon model For 0<H<, le B ( ), H R be he fracal Brownan moon, whch has zero means and covarance gven by: H H H CovH (, s) E[ BH ( ) BH ( s)] ( s s ) Obvously, BH () concdes wh he sandard Brownan moon B() when H /, and has homogeneous ncremens and connuous pahs. Under he fraconal Brownan moon framework, he dynamcs of sock prce can be modeled by he followng SDE: d dbh ( ). S () The correspondng fraconal Black-Scholes formula was developed n [8, 9]. For a European call opon, s prce process C(S(), ) sasfes he followng PDE: H C C C H S rs rc 0. S S rt Solvng he above PDE obans he prcng formula: C( S(0), 0) S(0) FN( d) Ke FN( d), where S(0) H H H d (log rt T ) / ( T ), d d T. K Comparng wh BS model, hs formula s smlar o Black-Scholes formula excep ha here s one exra parameer--- Hurs ndex. If H /, BH ( ) s a sandard Brownan moon, hen he ncremens of he process are ndependen. If H /, he ncremens of he process are posvely correlaed and he process s perssen. Whle f H /, he ncremens of he process are negavely correlaed and n hs case he process s an-perssen. Of course by usng Fraconal Brownan moon o model he sock prce, one needs o esmae he Hurs ndex, whch ndeed consrans he feasbly of hs model. 3. Consan Elascy Volaly (CEV) model The consan volaly hypohess n BS model ofen leads o resuls whch are nconssen wh he marke daa. To mprove he dscrepancy, he consan elascy of varance dffuson process was proposed n [3] o model he heeroscedascy and he leverage effec n reurns of common socks. The CEV model can well capure he 467 P a g e

6 Zhjuan Mao, Zhan Lang, Jnguo Lan, Hongkun Zhang / Inernaonal Journal of Engneerng Research and Applcaons (IJERA) ISSN: Vol., Issue 3, May-Jun 0, pp heeroscedascy of he volaly of sock reurns. Recenly, hs model has been furher nvesgaed n [9, 3], and he correspondng SDE s: d S( ) db( ) (7) S () where 0, s a parameer ha conrols he relaonshp beween volaly and prce. The varance of log(s()) s, whch mples ha he elascy s. ( S, ) S( ) and hus d( S, ) / ds ( S, ) / S The equlbrum prce of a call prce for was nvesgaed n [3], whle he case for In summary, he CEV soluon consss of a par of nfne summaons of gamma densy and survvor funcons: rt C( S(0), 0) S(0) P Ke P, was done n [33]. where P P( S(0), T;, ) denoes he probably of he call opon exprng n-he-money. Is dervaon ress on he rsk-neural prcng heory, see [3, 33] for furher nformaon. Aemps were made n early 980s, as a hree-sage procedure o esmae, was proposed n [33]. However, he resuls were no sasfacory. A breakhrough was made by usng he non-cenral Ch-squared dsrbuon, whch faclaes he compuaons sgnfcanly wh suable sascal sofware, see [33] for furher reference. If, he elascy s zero, so he sock prce s lognormally dsrbued, whch s jus he case of BS model; f, s observed from equy markes ha he volaly of a sock ncreases when s prce falls. Conversely, for, one ofen observes he so-called nverse leverage effec whereby he volaly of he prce of a commody ends o ncrease when s prce ncreases. In emprcal sudy, we manly focus on he case of as he relaonshp beween he sock prce and s reurn volaly usually s negave. 3.3 SABR model The SABR model,.e. sochasc alpha, bea, rho model, s purposed o capure he volaly smle n dervave markes. Ths model has been wdely used by praconers n fnancal ndusry. And was frs nroduced n [5]. Snce SABR model can be used o descrbe a sngle forward, a forward swap rae or a forward sock prce, we jus denoe he underlyng asse by S(). The sochasc process S() sasfes he SDE (7), bu also follows a sochasc process: d ( ) S( ) db( ), d ( ) ( ) db( ) S () where db( ) B( ) d,, and are consan. () s a volaly-lke parameer, whereas s hough o be he volaly of he volaly. Ths model was nroduced o model he dscouned sock prce process under he rskneural measure and hus he drf erm d vanshes under hs ransformaon [5]. Consderng a European call opon, he mpled volaly s obaned by usng seres expanson, see [5] for furher reference. To oban he opon prce, we can replace he volaly n Black-Scholes formula by he value obaned from hs model. Emprcal sudes found ha alhough he formula s complcaed, he approxmae soluon s que accurae f parameers are relavely small. Noe ha f 0 he volaly s no longer sochasc and he SABR model concdes wh he CEV model (7). Consequenly, we can also replace he volaly n CEV model by he valude derved from SABR model and hen oban he opon prce. Theorecally, s dffcul o oban he closed-form soluon or effcen esmae, bu s feasble o smulae wh compuer programmng. The parameer [0,] deermnes he relaonshp beween he underlyng asse prcng and a he money volaly. ndcaes ha he nvesor beleves ha f he marke s o move up or down n an orderly fashon, he a he money volaly level would no be sgnfcanly affeced; 0 ndcaes ha f he marke s o move up or down, hen a he money volaly would move n he oppose drecon. 3.4 Heson model In [6], Heson nroduced a sochasc volaly model and derved a closed-form soluon for he prce of a European call opon. The sock prce process {S()} sasfes: d ( ) db ( ), where () follows a CIR process, S () d ( ) ( ( )) d ( ) db ( ), 468 P a g e

7 Zhjuan Mao, Zhan Lang, Jnguo Lan, Hongkun Zhang / Inernaonal Journal of Engneerng Research and Applcaons (IJERA) ISSN: Vol., Issue 3, May-Jun 0, pp wh db( ) B( ) d and,,,, are consan. No arbrage assumpon mples ha he opon value should depend on S( ), ( ) and. Heson was able o oban he followng resul: C( S( ), ( ), ) S( ) P Ke P r( T ) where P P ( S( ), ( ), ) represen he probably of he call opon exprng n-he-money, whch can be obaned j j from he Fourer Transform. Comparng wh BS model, one aracve feaure of hs model s he updang srucure of s volaly. And based on solvng wo paral dfferenal equaons abou Pj, j,, a closed-form soluon for opon value s derved under hs sochasc volaly hypohess. In Heson model, skewness s generaed by he correlaon parameer, whle kuross s generaed by he volaly of volaly parameer, whch can explan he dscrepancy beween he BS model and marke daa. Snce n BS model, log(s(t)/s(0)) follows a normal dsrbuon, whch s symmerc abou s mean. Here we use MATLAB o smulae he pahs of log(s(t)/s(0)) under Heson model. Le ( r,,,, ) =(0.04, 0., 0.3, 0., 0.5) and he nal value of volaly (0) 0.4. We do he smulaon for 000 mes and hen plo he hsogram of he dsrbuon of log(s(t)/s(0)). As shown n fgure, he dsrbuon s posvely skewed comparng wh sand normal dsrbuon. Ths mples ha a posve correlaon resuls n hgher varance when he spo asse rses. And herefore, he Heson model explans he fa al phenomenon observed n emprcal sudes. 3.5 Chen/Three-facor model As demonsraed n [34], nearly all he varaon n bond reurns can be explaned by hree facors: he level of neres rae, he slope of he yeld curve and he curvaure of he yeld curve. Based on hs argumen, n 996, Chen developed a hree-facor model descrbng he evoluon of bond, whch s now called Chen model. I descrbes neres rae drven by hree sources of marke rsk---he neres rae, he sochasc mean and sochasc volaly. In [7, 35], he dynamcs of he nsananeous neres rae are proposed o follow he SDEs: dr( ) ( ( ) r( )) d ( ) db ( ), (8) d ( ) ( ( )) d ( ) db ( ), (9) d ( ) ( ( )) d ( ) db ( ). (0) Here B ( ),,, 3 are assumed o be ndependen, and,,,,,, are all consan. Consderng a defaul free dscoun bond whch promses o pay one un a me T, he bond prce sasfes a ceran PDE (see [35] for furher reference). Under correspondng boundary condons, he soluons can be obaned usng numercal mehods. In Chen model, also known as he hree-facor model, boh mean and volaly of he shor neres rae are sochasc. Ths propery fs he marke daa much beer and provdes addonal nsghs and explanaory powers. Alhough s dffcul o oban a closed-form formula, we can oban he soluon by smulaon. Bu how o evaluae he rsk prces of he hree facors from marke daa poses a grea challenge, whch can be an neresng opc of fuure research n he feld of fnancal engneerng. Noe ha he neres rae follows a sochasc process n Chen model, hus hs model can no be appled o opon prcng. Insead, we conjecure he followng model by replacng (8) wh ds( ) ( ( ) S( )) d ( ) db ( ) () P a g e

8 Zhjuan Mao, Zhan Lang, Jnguo Lan, Hongkun Zhang / Inernaonal Journal of Engneerng Research and Applcaons (IJERA) ISSN: Vol., Issue 3, May-Jun 0, pp Consderng a European call opon on hs sock, s prce should be obaned by he opon prcng formula () under rsk-neural measure wh D() e r. I would be very neresng o oban eher a closed-form soluon or numercal soluons for he opon prce n he mos general case. IV. IMPROVEMENTS AND EXTENSIONS ON LEVY MODELS In hs secon, we wll nroduce four models: he SVJD model, whch can be seen as he combnaon of Heson model and MJD model; he Kou model, a jump dffuson model; Varance Gamma (VG) and CGMY model, fnally s he Normal Inverse Gaussan (NIG) model. 4. SVJD model In 996, Baes [8] proposed a model ha he evoluon of he underlyng asse should follow: N() ( ) d ( ) db( ) d Y, d ( ) ( ( )) d ( ) db( ). S () Here s assumed ha db( ) B( ) d, EY, N() s a Posson process wh nensy of, and Y s he random percenage of prce change condonal on jumps occurrng. And log( Y ) follows he normal dsrbuon same as MJD model. The prce of a European call opon on hs underlyng asse was derved: r( T ) C( S( ), ) S( ) P Ke P where P and P are defned same as Heson model and can be obaned by Fourer Transform. If he volaly process degeneraes no consan, hs model concdes wh MJD model. The sochasc volaly propery makes hs model capure he feaure of smle volaly beer han MJD model. In addon, f he jump parameers are se o zero, he SVJD model urns no he Heson model. Thus hs model ouperforms Heson model especally when here exs oulers n he marke. In general, more parameers make SVJD model more flexble. 4. Kou model In he MJD model, he hypohess of normal jump dffuson mples symmerc dsrbuon of log( Y ), whch s no conssen wh sudes n behavoral fnance [9]. As a resul, a new model was proposed by Kou o prce opons by consderng he psychology elemen of nvesors whn he frame of radonal Brownan moon and effcen marke hypohess. The sochasc process of he asse prce S() sasfes he SDE: N() d db() d Y. S () I s assumed ha log( Y ) are ndependen and dencally dsrbued wh asymmerc double exponenal dsrbuon. The probably densy funcon s as below: y y f ( y) pe q e, y0 y0 where p, q>0, p+q= represen he probables of upward and downward jumps respecvely. Thus he prce of a European call opon can be obaned by he formula () under he rsk-neural measure wh D( ) e r. To oban an explc form, one needs o undersand he dsrbuon of he sum of he double exponenal random varables and normal random varables. Kou [9] nroduced he Hh funcon, a specal funcon of mahemacal physcals and derved an analycal formula. Comparng wh he MJD model, alhough boh models can explan he followng wo emprcal phenomena: he asymmerc lepokurc feaure and mpled volaly smle; Kou model defnely has some good feaures. I can also be used o prce pah-dependen opons, and embedded no a raonal expecaons equlbrum framework. However, Kou model also has ceran lmaons: alhough he closed-form soluon can be obaned, needs compuer programmng for s complexy. Mos mporanly, he jump process makes he marke neffcen and hus he hedgng sraegy s mpraccable. 4.3 Varance Gamma/CGMY model Under he assumpons of BS model, a seres of connuous me models have been developed o allow boh fne or nfne jumps durng fne me nerval [, 4, 36]. The sock prce process can be expressed as a subordnaed Brownan moon wh a me changng process. In VG model, he sock prce process S() sasfes: log S( ) G B( G ), 470 P a g e

9 Zhjuan Mao, Zhan Lang, Jnguo Lan, Hongkun Zhang / Inernaonal Journal of Engneerng Research and Applcaons (IJERA) ISSN: Vol., Issue 3, May-Jun 0, pp where G s a Gamma dsrbued process wh mean and varance wh be a posve consan. The absence of arbrage mples ha for a European call opon, s prce can be obaned from formula () under rsk-neural measure wh D( ) e r. More precsely, was shown n [30] ha: where d ( ) T / T / r T / T / N N C( S(0), 0) S(0) e ( ( ) / ) F ( d ) Ke ( / ) F ( d ) log( S(0) / K) r (/ )log[( ( ) / ) / ( / )] [ ] T and d d T. T In 00, Carr, Geman, Madan and Yor developed CGMY model based on VG model []. Le X ( ) log S( ). Then s characersc funcon s: VG ( u, G ) E[exp( ux ( ))] ( ) u u / Consderng he nfnely dvsbly propery of VG process, he process X() can be expressed as X (,,, ) X (,,, ) X (,,, ), p p p q q q p q p p q q where ( ) /, ( ) /,,. Ths mples ha he Levy densy for 4 4 he VG process s [36] p exp( x ) p p for x 0 p x kvg ( x). q exp( x ) q q for x 0 q x In [4], he auhors developed he CGMY model---a ransformaon of VG model. They manly change he parameers n he above Levy densy: exp( G x ) C for x 0 Y x kcgmy ( x), exp( M x ) C for x 0 Y x where C>0, G0, M 0 and Y<. In CGMY model, he sock prce can be gven by: S( ) S(0)exp[( ) X ( ; C, G, M, Y)], where s a convexy correcon defned by exp( ) ( ;, C, G, M, Y). The analycal soluon n he CGMY framework could no be obaned. However, by Fas Fourer Transformaon, we can use numercal mehods o smulae he pahs of sock prce and hen evaluae he prce of opon. Thus he characersc funcon plays a key role. Also noce ha VG model s a specal case of CGMY model when Y 0, C, G, M. q p Comparng wh prevous models, boh VG model and CGMY model can capure he long aledness for daly reurns, alhough he reurns under hese models over long perods approach normaly. In he MJD model, here are only fne jumps n a fne me nerval, whle n he VG and CGMY model here can be nfne number of jumps. In MJD model, closed-form soluon has been derved bu n hese models he opon prce can only be obaned by Fas Fourer Transformaon or numercal mehods. 4.4 Normal Inverse Gaussan (NIG) model Emprcal sudes found ha he log reurn of fnancal asses can ofen be fed well by Normal Inverse Gaussan (NIG) dsrbuons. And hs process was frs nroduced o model he log reurn of sock prces by Barndorff and Nelsen n []. We can relae he NIG process o a me-changng Brownan moon by nroducng an ndependen nverse Gaussan process. Le T dynamcs of sock prce follows: CGMY CGMY be he frs me ha a Brownan moon wh drf reaches he posve level. And he log S( ) T B( T ), 47 P a g e

10 Zhjuan Mao, Zhan Lang, Jnguo Lan, Hongkun Zhang / Inernaonal Journal of Engneerng Research and Applcaons (IJERA) ISSN: Vol., Issue 3, May-Jun 0, pp where, and B() are defned same as above. Defne X()=log(S()), hen he characersc funcon can be obaned. By Fas Fourer Transform, we can oban he opon prce based on (). Comparng wh he VG model, NIG model dffers n he dfferen dsrbuon of he me changng process. I s somewha dffcul o say one model ouperforms he oher as he good fng also depends on daa. In general, Levy process opens a new way o mprove he BS model, because wh jump process hese models can always capure he long aledness and heaver peak. However, due o he saonary of ncremens of Levy process, he sock prce reurns for a fxed me horzon should have he same law. I s herefore mpossble o ncorporae any knd of new marke nformaon no he model, whch dsagrees wh he realy. V. CONCLUSION Fnancal Engneerng s a muldscplnary feld nvolvng he heory of fnance, he mehods of engneerng, he ools of mahemacs and he pracce of programmng. And prcng models acs lke he lnk beween hese opcs. Begnnng from he work of Bacheler, he heory of prcng opons and oher fnancal dervaves had been grealy mproved. The process of he sock prce can be descrbed by he arhmec Brownan moon n he Bacheler model and Geomerc Brownan moon n Samuelson/Orsborne model BS model ec. Recenly he Levy process wh Jump process such as MJD, SVJD, Kou model, VG, NIG and CGMY model are developed. Apar from ha, he Fraconal Brownan moon s proposed. On he oher hand, he volaly s mproved from a consan o a sochasc process n he Heson model, SABR model, Chen model and SVJD model. In applcaons, how o choose an approprae model s a bg challenge. Snce he key crera s o compare he errors beween he daa generaed beween he model and obaned from he marke. A good model should f marke and provde addonal nsghs and explanaory powers. From hs pon of vew, he hree-facor model has more advanages. There are evdences n [34] ha he hree-facor model, whch nvolves he underlyng asse process ogeher wh he sochasc mean and sochasc volaly processes, can f bond reurns much beer. Alhough more parameers make a model more flexble, he soluon may no have closed-form. Thus one needs smulaon o ge he soluon, and somemes s a raher dffcul ask o esmae hese parameers. Ths requres effcen compuer programmngs and accurae algorhm mehods. Consequenly, calbrang carres more and more wegh n Fnancal Engneerng, whch also poses a challenge problem n numercal compuaons. Manly here are hree numercal mehods o oban he soluon of a SDE model: Fas Fourer Transformaon, Fne Dfferenal mehod and he Mone Carlo. In he fuure sudy of Fnancal engneerng, one mporan queson should be relaed o valdae hese models wh marke daa, and meanwhle ge beer conrol on predcng he fnancal marke. Noe ha all he models we revew n hs paper can be used o calculae he premer of general dervaves wh underlyng asses ha are no resrced o socks. For smplcy we only consder dervaves wh underlyng asses o be socks. In addon, when coss of ransacons or dvdend of socks are consdered, he models can sll be appled by usng he effecve sock prces. Fnally all models lsed n he paper are under he assumpon of ndependen ncremen for he sock prce, excep he Fraconal Brownan moon model. Bu emprcal sudes convnce us ha here exss dependency consecuve perods. So s anoher challengng queson o se up proper models by consderng he dependency of he sock prce process. REFERENCES [] Osborne, M.F.M., Brownan moon n he sock marke, Operaons Research, 7() (959), [] Samuelson, Paul A., Raonal Theory of Warran Prcng, Indusral Managemen Revew, 6() (965), [3] Meron, Rober C., Opon Prcng When underlyng Sock Reurns are Dsconnuous, Journal of Fnancal Economcs, 3 (976), [4] L M. and M. Wang, Comparson of Black-Scholes Formula wh Fraconal Black-Scholes Formula n he Foregn Exchange opon Marke wh Changng Volaly, Asa-Pacfc Fnan. Markes, 7 (00), 99-. [5] Mandelbro, B. B., The varaon of ceran speculave prces, Journal of Busness, 36 (963), [6] Moore, B.C., L.K. Tyler and W.D. Marslen-Wlsen, Tme requred for judgemens of numercal nequaly, Naure, 5(509) (967), [7] Peers, Edgar E., Fracal Srucure n he Capal Markes, Fnancal Analyss Journal, 45(4) (989), [8] Duncan, T.E., B. Maslowsk and B.Pask Duncan, Adapve conrol for semlnear sochasc sysems, SIAM J.Conrol Opm., 38 (000), [9] Hu, Y.Z, Oksendal Bern, Fraconal Whe Nose Calculus and Applcaons o Fnance, Infne Dmensonal Analyss, Quanum Probably and Relaed Topcs, 6() (003), -3. [0] Necula, C., Opon Prcng n a Fraconal Brownan Moon Envronmen, Buchares Unversy of Economcs, (008). [] Blaerg, R.C. and N.J. Gonedes, A comparson of he sable and suden dsrbuons as sochasc models for 47 P a g e

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