Implied non-recombining trees and calibration for the volatility smile

Transcription

1 Impld non-rcombnng trs and calbraton for th volatlty sml hrs haralambous * Ncos hrstofds Eln D. onstantnd and pros H. Martzoukos May 007 JEL classfcaton: 6 63 G3 G5 Kywords: Volatlty ml Impld Non-rcombnng Trs albraton Non- onvx onstrand Optmzaton. Acknowldgmnts: Th authors ar gratful for fnancal support from a rsarch grant on ontngnt lams from th Unvrsty of ypus. Unvrsty of yprus. ntr for Quanttatv Fnanc Impral ollg London. *orrspondng Author haralambous. Profssor Dpartmnt of Busnss Admnstraton Unvrsty of yprus P.O. Box 0537 Y 678 Ncosa yprus Tl.: Fax: mal: bachrs@ucy.ac.cy

2 Abstract In ths papr w captur th mpld dstrbuton from opton markt data usng a non-rcombnng bnary tr allowng th local volatlty to b a functon of th undrlyng asst and of tm. Th problm undr consdraton s a non-convx optmzaton problm wth lnar constrants. W laborat on th ntal guss for th volatlty trm structur and us nonlnar constrand optmzaton to mnmz th last squars rror functon on markt prcs. Th proposd modl can accommodat Europan optons wth sngl maturts and wth mnor modfcatons optons wth multpl maturts. It can provd a markt-consstnt tr for opton rplcaton wth transacton costs oftn ths rqurs a non-rcombnng tr and can hlp prcng of xotc and Ovr Th ountr OT optons. W tst our modl usng optons data of th FTE 00 ndx obtand from LIFFE. Th rsults strongly support our modllng approach.

3 I. Introducton albratng a tr othrws known as constructng an mpld tr mans fndng th stock prc and/or assocatd probablty at ach nod n such a way that th tr rproducs th currnt markt prcs for a st of bnchmark nstrumnts. Th man bnft of calbratng a modl to a st of obsrvd opton prcs s that th calbratd modl s consstnt wth today s markt prcs. Th calbratd modl can thn b usd to prc othr mor complx or lss lqud scurts such as OT optons whos prcs may not b avalabl n th markt. Th bnomal tr s th most wdly usd tool n th fnancal prcng ndustry. Th classc ox-ross-rubnstn RR 979 bnomal tr s a dscrtzaton of th Black-chols B 973 modl snc t s basd on th assumpton of th B modl that th undrlyng asst volvs accordng to a gomtrc Brownan moton wth a constant volatlty factor. Ths howvr contradcts th obsrvd mpld volatlty whch suggsts that volatlty dpnds on both th strk and maturty of an opton a rlatonshp commonly known as th volatlty sml. Ths problm has motvatd th rcnt ltratur on sml consstnt no-arbtrag modls. onsstncy s achvd by xtractng an mpld voluton for th stock prc from markt prcs of lqud standard optons on th undrlyng asst. Thr ar two classs of mthodologs wthn ths approach. ml consstnt dtrmnstc volatlty modls Rubnstn 994 Drman and Kan 994 Dupr 994 Barl and Kakc 995 Rubnstn and Jackwrth 996 Jackwrth 997 tc.; and stochastc volatlty sml consstnt modls whch allow for smlconsstnt opton prcng undr th no-arbtrag voluton of th volatlty surfac Drman and Kan 998 Ldot and anta-lara 998 Brttn-Jons and Nubrgr 000 tc.. Th lattr class of modls s mor gnral and t nsts th formr class of modls kadopoulos 00. Thr also xst nonparamtrc mthods lk tutzr 996 who uss th maxmum ntropy concpt to drv th rsk nutral dstrbuton from th hstorcal dstrbuton 3

4 of th asst prc and At-ahala and Lo 998 who propos a nonparamtrc stmaton procdur for stat-prc dnsts usng obsrvd opton prcs. ml consstnt dtrmnstc volatlty modls ar basd on th assumpton that th local volatlty of th undrlyng asst s a known functon of tm and of th path and lvl of th undrlyng asst prc. Howvr thy do not spcfy local volatlty n advanc but drv t ndognously from th Europan opton prcs. Thrfor thy prsrv th prcng by no-arbtrag proprty of th B modl and th markts ar complt snc th opton s pay-off can b synthszd from xstng assts. Rubnstn 994 fnds th mpld rsk-nutral trmnal-nod probablty dstrbuton whch s n th last-squars sns closst to th lognormal subct to som constrants. Th probablts must add up to on and b non-ngatv. Morovr thy ar calculatd so that th prsnt valu of th undrlyng assts and all th Europan optons calculatd wth ths probablts fall btwn thr rspctv bd-ask prcs. Ths mthodology allows for an arbtrary trmnal-nod probablty dstrbuton but assums that path probablts ladng to th sam ndng nod ar qual. Rubnstn s 994 mthodology suffrs from th fact that optons xprng at arly tm stps cannot b usd for th constructon of th tr. Thus optons wth maturty othr than th maturty of th optons usd durng th constructon of th tr ar not consstnt wth markt prcs. Jackwrth 997 ntroducd gnralzd bnomal trs as an xtnson of Rubnstn 994. Hs modl allows for an arbtrary trmnal-nod probablty dstrbuton but also allows path probablts ladng to th sam nod to tak dffrnt valus. Drman and Kan 994 and Dupr 994 constructd rcombnng bnomal trs usng a larg st of opton prcs. For ach nod thy nd a corrspondng opton prc wth strk prc qual to th nod s stock prc and xprng at th tm assocatd wth that nod. nc thy hav fwr opton prcs than rqurd thy nd to ntrpolat and xtrapolat from 4

5 gvn opton prcs. Thr trs ar snstv to th ntrpolaton and xtrapolaton mthod and rqur adustmnts to avod arbtrag volatons. Barl and akc 995 ntroducd a numbr of modfcatons whch amd to lmnat ngatv probablts and mprov th gnral stablty of Drman s and Kan s 994 modl. Although thr modfd mthod fts th sml accuratly ngatv probablts may stll occur wth ncrass n th volatlty sml and ntrst rat. As thy stat ths s bcaus of thr strct rqurmnt that contnuous dffuson b modld as a bnomal procss and on a rcombnng tr. Ths problm can b rfrrd to as a problm of ntrdpndncs btwn nods. Possbl mthods that can b usd to rduc th problm of ntrdpndncs ar th calbraton of trnomal or multnomal trs or nonrcombnng trs. Ths xtra dgrs of frdom allow for mor flxblty n th stmaton of th dstrbuton of th undrlyng asst. Trnomal trs provd a much bttr approxmaton to th contnuous tm procss than th bnomal trs for th sam numbr of stps. Howvr th xtra dgrs of frdom addtonal numbr of nods rqur a largr numbr of smultanous quatons to b solvd. Drman Kan and hrss 996 proposd mpld trnomal trs. In thr modl thy us th addtonal paramtrs to convnntly choos th stat spac of all nod prcs n th tr and lt only th transton probablts b constrand by markt optons prcs. hrss 996 gnralzd thr mthod for Amrcan styl optons. In ths papr w propos a mthod for calbratng a non-rcombnng bnary tr basd on optmzaton. pcfcally w mnmz th dscrpancy btwn th obsrvd markt prcs and th thortcal valus wth rspct to th undrlyng asst at ach nod subct to constrants that mantan rsk nutralty and prvnt arbtrag opportunts. Our modl s bult on a non-rcombnng tr so as to allow th local volatlty to b a Othr work w ar awar of that uss a non-rcombnng tr s of Talas 005 whr for th calbraton h uss gntc algorthms. 5

6 functon of th undrlyng asst and of tm and to nabl ach nod of th tr to act as an ndpndnt varabl. Effctvly th problm undr consdraton s a non-convx optmzaton problm wth lnar constrants. W laborat on th ntal guss for th volatlty trm structur and usng mthods from nonlnar constrand optmzaton w mnmz th last squars rror functon. pcfcally w adopt a pnalty mthod and for th optmzaton w us a Quas-Nwton algorthm. Bcaus of th combnatoral natur of th tr and th larg numbr of constrants th sarch for an optmum soluton as wll as th choc of an algorthm that prforms wll bcoms a vry challngng problm. Our modl was cratd as a rspons for th nd of a nonrcombnng mpld tr. Th man bnft of th modl s ts analytcal structur whch nabls us to us ffcnt mthods for nonlnar optmzaton. Although th mthod uss a larg numbr of varabls du to th fact that w us ffcnt mthods for optmzaton th modl s not computatonally ntnsv. Also th proposd mthodology can b asly modfd to captur th obsrvd bd/ask sprads n th markt. Ths s vry usful snc th rportd closng prcs may not always b accurat or may b naccurat du to varous markt frctons. In addton calbraton of th non-rcombnng tr can b usd for opton rplcaton wth transacton costs as n Edrsngh t al. 993 and othr rlatd mthodologs that rqur non-rcombnng trs. In contrast to Rubnstn 994 th proposd mthodology can b asly modfd to account for Europan contracts wth dffrnt maturts. Our mthod dos not nd any ntrpolaton or xtrapolaton across strks and tm to fnd hypothtcal optons as opposd to Drman and Kan 994. Fnally th xtra dgrs of frdom and th analytcal structur of th modl would allow us to mpos smoothnss constrants on th dstrbuton of th undrlyng asst f rqurd. W tst our modl usng optons data on th FTE 00 ndx for th yar 003 obtand from LIFFE. Th rsults strongly support our modllng 6

7 approach. Prcng rsults ar smooth wthout th prsnc of an ovr-fttng problm and th drvd mpld dstrbutons ar ralstc. Also th computatonal burdn s not a maor ssu. Th papr contnus as follows: In scton II w dscrb th proposd mthodology and th ntalzaton of th non-rcombnng tr. In scton III w dscuss th mposd rsk nutralty and no-arbtrag constrants. In scton IV w dscrb th optmzaton algorthm. In scton V w tst th modl usng FTE 00 optons data. onclusons ar n scton VI. In Appndx A w prov th fasblty of th ntalzd tr n Appndx B w prov th fasblty of th ntalzd tr takng nto account that th rsk-fr rat dvdnd yld and tm stp ar tm dpndnt and n Appndx w adust th formulas for tm dpndnt rsk fr rat dvdnd yld and stp sz. II. Th proposd mthodology and ntalzaton of th non-rcombnng tr Our goal s to dvlop an arbtrag-fr rsk nutral modl that fts th sml s prfrnc-fr and can b usd to valu optons from asly obsrvabl data. In ordr to allow mor dgrs of frdom w us a nonrcombnng tr. In th followng scton w prsnt th proposd mthodology and dscrb th ntalzaton of th tr. Fgur shows a non-rcombnng tr wth four stps. [Insrt fgur hr] Th pont on th tr dnots: : th tm dmnson... n : th asst tm spcfc dmnson... s th valu of th undrlyng asst at nod. Fgur shows a typcal trplt n a non-rcombnng tr. [Insrt fgur hr] 7

8 Lt Mkt k k... N dnot th markt prcs of N Europan calls wth strks K k and sngl maturts T. Also lt Mod x k k... N dnot th thortcal prcs of th N calls obtand usng th modl. x dnots a vctor contanng th varabls of th modl whch ar th valus of th undrlyng asst at ach nod of th tr xcludng ts currnt valu. Th dal soluton s to fnd th valus of th undrlyng asst th modl varabls at ach nod of th tr such that a prfct match s achvd btwn th opton markt prcs and thos prdctd by th tr. Howvr du to markt mprfctons and othr factors prfct matchng may not always b possbl. Thrfor w mnmz th dscrpancy btwn th obsrvd markt prcs and th thortcal valus producd by th modl subct to constrants that prvnt arbtrag opportunts. W hav to solv a non-convx constrand mnmzaton problm wth rspct to th valus of th undrlyng asst at ach nod: N x k k mn w f x k k Mod Mkt whr f dnots a sutabl obctv functon on th rror btwn th obsrvd and markt prcs. W can also allow for a wght factor wk. In ths papr w us th last squars rror functon whch s dfnd as th sum of squar dffrncs btwn markt prcs and thortcal prcs producd by th tr. Th mthod can b adustd asly for any othr obctv functon. Th phlosophy of th ntalzaton of th non-rcombnng tr s th sam as that of th constructon of th standard RR bnomal tr but w adust th formulas so that th tr dos not ncssarly rcombn. W dnot wth u and d th up and down factors by whch th undrlyng asst prc can mov n th sngl tm stp t gvn that w ar at nod. t u and d factors ar gvn by th followng formulas 3 : Wghts can b rlatd for xampl to th tradng volum of th optons. 3 For smplcty w mak th assumpton that th rsk fr rat th dvdnd yld and th stp sz do not chang across tm. Formulas adustd for tm dpndnc can b found n Appndx B and. 8

9 T t n u σ t 3a... n... d σ t u 3b whr T s th opton s tm to maturty and σ s th volatlty trm structur at tm stp. W ntalz th tr usng th followng volatlty trm structur: λ t σ σ R λ n- 4 whr λ s a constant paramtr and σ s a proprly chosn ntal valu for th volatlty. If λ s postv thn volatlty ncrass as w approach maturty and f λ s ngatv thn volatlty dcrass as w approach maturty 4. In ordr to prsrv th rsk nutralty at vry tm stp and hnc obtan a fasbl ntal tr w choos λ to blong n th followng ntrval for proof s Appndx A: r f δ t λ log T σ By choosng λ from th abov ntrval w allow th ntal volatlty to ncras or dcras across tm. W mak svral conscutv draws from ntrval 5 untl w fnd th valu of λ that gvs th optmal tr 5. W dnot wth th currnt valu of th undrlyng asst. Th odd nods of th tr ar ntalzd usng th followng quaton: d... n a Th vn nods of th tr ar ntalzd usng th followng quaton: 5 4 Othr non-monotonc functons could also b usd for σ but what w hav trd provd adquat for our purposs. 5 Optmal tr s th on that gvs th lowst-valu obctv functon subct to th ntal constrants. 9

10 u... n b W want to pont out that quatons 3 to 6 ar usd only for ntalzaton. Onc th optmzaton procss starts ach valu of th undrlyng asst xcpt from acts as an ndpndnt varabl n th systm. Upward transton probablts gv th probablty of movng from nod to nod whras downward transton probablts gv th probablty of movng from nod to nod for... n and.... For th upward transton probablts p btwn th varous nods of th tr w us th rsk-nutral probablty formula: p r f δ t... n... 7 whr r f dnots th annually contnuously compoundd rsklss rat of ntrst and δ dnots th annually contnuously compoundd dvdnd yld. Thr rspctv downward probablty s qual to on mnus th upward probablty. Th call opton valu at th last tm stp s gvn by: n max{ n K 0} n... 8 Howvr th functon max s non dffrntabl at n K. To ovrcom ths problm w propos th followng smoothng approxmaton to n : 0 for n / K z / α n n for n / K z / 9a K K n z for z / < n / K < z / z K... whr z s a small postv constant for xampl 0.0 s Fg. 3. n [Insrt fgur 3 hr] 0

11 Th valu of th call at ntrmdat nods s gvn by th followng quaton: p p r f t 9b n III. Rsk nutralty and no-arbtrag constrants In ths scton w dscrb th rsk nutralty and no-arbtrag constrants. In ordr for th transton probablts p dfnd n Eq.7 to b wll spcfd thy should tak valus btwn zro and on. Ths mpls th followng rsk-nutralty constrants: r δ t f 0a... n... r f δ t 0b Rsk nutralty constrants n th non-rcombnng tr prvnt nods and to cross for... n and... s Fg.. Optons puts and calls hav uppr and lowr bounds that do not dpnd on any partcular assumptons on th factors that affct opton prcs. If th opton prc s abov th uppr bound or blow th lowr bound thr ar proftabl opportunts for arbtragurs. To avod such opportunts w nclud th no-arbtrag constrants. pcfcally a Europan call wth dvdnds should l btwn th followng bounds: max δt rf T K 0 Mod Also vry valu of th undrlyng asst on th tr should b gratr or qual to zro. Thus w also mpos th followng constrant: 0... n...

12 IV. Th optmzaton algorthm Th obctv of th problm s to mnmz th last squars rror functon of th dscrpancy btwn th obsrvd markt prcs and th thortcal valus producd by th modl. Thus w hav th followng optmzaton problm: N mn Mod x k Mkt k 3 x k whr Mod k and Mkt k dnot th modl and markt prc rspctvly of th k th call k... N subct to th constrants: r f δ t g 0... n... 4a r f δ t g 0... n... 4b g k Mod k 0 k... N 4c 3 r T 4 Mod δτ f v g k k max K k 0 0 k... N 4d v g n... 4 nc th problm undr consdraton s a non-convx optmzaton problm wth lnar constrants w adopt an xtror pnalty mthod Facco and Mcornck 968 to convrt th nonlnar constrand problm nto a nonlnar unconstrand problm. Th Extror Pnalty Obctv functon that w us s th followng: P x α N k x k k Mod Mkt [ mn g 0 ] [ mn g 0 ] α n α N [ mn g k 0] [ mn g k ] k α n [ mn g 0 ] 5 5 Th scond thrd and fourth trms n P x α gv a postv contrbuton f and only f x s nfasbl. Undr mld condtons t can b provd that mnmzng th abov pnalty functon for strctly ncrasng

13 squnc α tndng to nfnty th optmum pont x α of P tnds to x* a soluton of th constrand problm. For th optmzaton w us a Quas-Nwton algorthm. pcfcally w us th BFG formula 6 Fltchr 987. For th procdur of Ln arch n th algorthm w us th haralambous 99 mthod. To achv th bst fasbl soluton.. th soluton that gvs us a fasbl tr wth th smallst rror functon w forc th algorthm to draw conscutvly valus of λ from th spcfd ntrval 5 untl th obctv functon s smallr than.e-4 and also th pnalty trm quals zro.. w hav a fasbl soluton. Implmntaton For th mplmntaton of th optmzaton mthod w nd to calculat th partal drvatvs of Mod k 7 wth rspct to th valu of th undrlyng asst at ach nod for k k... N.. w want to fnd... n... 8 and k... N. For notatonal smplcty n th followng w assum that w hav only on call opton. For th computaton of w mplmnt th followng stps: W dfn th trplt vctor s Fg.: l [ ] 6 st stp: omput th partal drvatvs of th rsk nutral transton p probablts p and p for... n and.... W summarz th drvatvs n vctor form 7. 6 Th BFG formula was dscovrd n 970 ndpndntly by Broydn Fltchr Goldfarb and hanno. 7 From now on w wll us nstad of Mod. 8 W do not calculat k snc s a known fxd paramtr and thus dos not tak part n th optmzaton. 3

14 4 p p p p p p t r f l δ 7 nd stp: omput th partal drvatvs for... n and... and for n. W summarz th drvatvs n vctor form 8. t r t t r t f f l p p δ δ 8 whr Rato Dlta t δ 9 3 rd stp: omput th partal drvatvs n n α for... n. Thy ar gvn by th followng formula: < < / / / / 0 z K n z K for z K n z z K n for z K n for n n α 0

15 4 th stp: omput th partal drvatvs for 3. { of th probablts on th path that tak us from nod to nod k k x rft } k / / for for vn odd For xampl 46 p p p p r f t 4 p3 53 r f t V. Applcaton usng FTE 00 optons data W us th daly closng prcs of FTE 00 call optons of January 003 to Dcmbr 003 as rportd by LIFFE 9. For th rsk-fr rat r f w us nonlnar cubc spln ntrpolaton for matchng ach opton contract wth a contnuous ntrst rat that corrsponds to th opton s maturty by utlzng th -month to -month LIBOR offr rats collctd from Datastram. Our ntal sampl for th months prod conssts of 9905 obsrvatons. W adopt th followng fltrng ruls: 9 FTE 00 optons ar tradd wth xprs n March Jun ptmbr and Dcmbr. Addtonal sral contracts ar ntroducd so that optons trad wth xprs n ach of th narst 3 months. FTE 00 optons xpr on th thrd Frday of th xpry month. FTE 00 optons postons ar markd-to-markt daly basd on th daly sttlmnt prc whch s dtrmnd by LIFFE and confrmd by th larng Hous. 5

16 Elmnat calls for whch th call prc s gratr than th valu of th undrlyng asst.. Mkt >. No obsrvatons ar lmnatd from ths rul. Elmnat calls f th call prc s lss than ts lowr bound.. Mkt < δt rf T K. Ths rul lmnats 306 obsrvatons. Elmnat calls wth tm to maturty lss than 6 days.. T < 6. Ths rul lmnats 309 obsrvatons. v Elmnat calls f thr closng prc s lss than 0.5 ndx ponts. Ths rul lmnats 373 obsrvatons. v Elmnat calls for whch th tradng volum s zro snc w want hghly lqud optons for calbraton. Ths rul lmnats 6686 obsrvatons. Th fnal sampl conssts of 4537 obsrvatons. In th mplmntaton for σ w us th at-th-mony mpld volatlty gvn by LIFFE and for tm to maturty T w us th calndar days to maturty. Also snc th undrlyng asst of th optons on FTE 00 s a futurs contract w mak th standard assumpton that th dvdnd yld quals th rsk fr rat. Th modl s appld vry day wth n 6 and also wth n 7. For ach mplmntaton th optons usd hav th sam undrlyng asst and th sam tm to maturty. Th vdnc for th bhavour of th futurs volatlty n th ltratur s not clar. Accordng to amulson 965 th volatlty of futurs prc changs should ncras as th dlvry dat nars. Howvr Bssmbndr t al. 996 fnd that th amulson hypothss s not supportd for optons on fnancals futurs. In ordr to choos th valu of λ that gvs th bst fasbl soluton w mak conscutv draws from ntrval 5 whch allows for both postv and ngatv valus of λ. Th frst valu of λ s that of ts lowr bound. Howvr snc dvdnd yld quals rsk fr rat nstad of r δ w st.e-8. Th nxt valu of λ quals th old plus an appropratly f chosn stp sz. 6

17 For brvty w prsnt rsults only for th frst tradng day of ach month of th yar 003 and only for n 6 Tabl. Tradng Day s th tradng day of ach contract Expry s th xpraton month of ach contract Asst s th valu of th undrlyng asst at th spcfd tradng day N s th numbr of contracts usd for th calbraton th contracts that on th sam tradng day hav th sam undrlyng asst and th sam xpraton day Error s th valu of th obctv functon Pnalty s th valu of th pnalty trm. Idally w want th rror functon and th pnalty trm to tnd to zro. Maturty s th calndar days tll th maturty of th contract and lambda s th valu of λ that gvs th bst fasbl soluton. Also w prsnt rsults only whn th numbr of opton contracts s gratr than 3 snc wth fwr optons th dstrbuton of th undrlyng asst takn wll not b rlabl 0. [Insrt Tabl hr] Th rsults obtand support our modlng approach. As w can s n Tabl n all cass th soluton strctly satsfs th constrants snc th pnalty trm quals zro. Also w s that n 67 out of 69 cass.. n 97.% of th cass th rror functon tnds to zro wth an avrag valu of.34e-08. In th othr cass whr th rror functon s gratr than.e-4 th avrag rror s 0.0. mlar rsults wr found for n 7. Evn though th problm rqurs a constrand non-convx optmzaton n n varabls th us of ffcnt optmzaton algorthms prvnts th calbraton of th modl from bcomng computatonally too ntnsv. On avrag th computatonal tm n mnuts rqurd for ach calbraton had a man mdan for n 6 and for n 7. Th computr usd for th calbraton of th modl had th followng spcfcatons: a Pntum 4 3. GHz PU Mmory GB RAM and Wndows XP Profssonal opratng systm. Th cods wr wrttn n Matlab R006a. Th computatonal tm ndd would hav dcrasd f th cods wr wrttn n th / languag. 0 In Tabl w not that for th sam contract sam undrlyng asst sam xpraton th numbr of contracts usd n th modl changs across months. That s bcaus som contracts wr rmovd bcaus of th fltrng ruls. 7

18 Whn modls provd an xact ft thr s always th concrn of ovrfttng. W chckd th modl for ovr-fttng by prcng optons wth strks n-btwn thos usd for th optmzaton calbraton. Thn w mad plots of th call prcs markt prcs and stmatd from th modl vrsus monynss. Ovr-fttng was also chckd usng a rstrctd sampl consstng only of optons wth monynss btwn 0.8 and. snc ths optons ar xpctd to b mor lqud and mor accuratly prcd. For brvty w xhbt only th plots for optmzatons don n th frst tradng day of Jun mddl of th yar for th two sampls usng a tr wth n 6. As w s for both sampls th stmatd call valus ncras smoothly wth ncrasng monynss wthout any vdnc of ovr-fttng s Fg.4. mlar rsults wr obtand whn a tr wth n 7 was usd for th calbraton procdur. [Insrt fgur 4 hr] As a furthr chck for ovr-fttng w us only part of th nformaton to calbrat th tr and th othr part to chck th modl usng n pcfcally w lav out conscutvly on of th N optons at ach tm and w calbrat our modl wth th rmanng optons. In ordr to prsrv th optons monynss rang stabl and avod problms of xtrapolaton w do not rmov th optons wth th hghst and lowst monynss. Ovr-fttng s chckd lk bfor usng th full and th rstrctd sampl of optons. For th calbraton only cass consstng of N > 8 wr usd. Rsults for th man and mdan absolut rrors ar gvn n Tabl. W s that th rror gvn an avrag contract sz of 90 for th full and 74.4 for th rstrctd sampl s small and rathr stabl. Ths sub-sampl has a total of 3696 obsrvatons for th yar 003. Also w compar our modl wth rspct to ovr-fttng wth th Black-chols modl usng th Whaly 98 approach. Accordng to ths approach w fnd th volatlty that mnmzs th sum of squar dffrncs of th Black-chols opton prcs wth thr corrspondng markt prcs usng nonlnar mnmzaton. Rsults show that th man mdan absolut rror usng ths approach s for th full sampl and for th rstrctd sampl whch ar much hghr than th rrors obtand usng our modl for n

19 [Insrt Tabl hr] nc mpld volatlty changs wth strk and tm to maturty volatlty sml th ndx should hav a non-lognormal dstrbuton whch mpls that th log-rturns wll dvat from normalty. In ordr to s how ralstc s th dstrbuton obtand from our modl for yar 003 w calculat th statstcs of th -month log-rturns obtand from our modl and compar thm wth th hstorcal -month log-rturns for th yar 003 and th yars pcfcally for ach calbraton wth n 6 and n 7 for whch th optons maturty was btwn 8 and 3 calndar days w calculat th frst four momnts man varanc skwnss and kurtoss. Thn n ordr to gt a flng for th rprsntatv statstcs of -month logrturns w provd for ach of thos momnts th man and th mdan. Th statstcs for n 6 ar summarzd n Tabl 3. mlar statstcs wr found for n 7. Lu t al. 005 dscuss th drvatons of hstorcal and mpld ral and rsk-nutral dstrbutons for th FTE 00 ndx. Thy dmonstrat that th ndd adustmnts to gt th mpld ral varanc skwnss and kurtoss from th mpld rsk-nutral ons ar mnmal. Thus knowng that our mpld rsk-nutral momnts byond th man ar vry clos to th mpld ral ons w can thn compar thm wth th hstorcal ons wthout xpctng th two dstrbutons to b dntcal. As w would xpct th man of th mpld rsk-nutral dstrbuton of log-rturns dffrs from that of th hstorcal dstrbuton. Also as w s both th mpld rsknutral and th hstorcal dstrbuton dvat from normalty snc thy xhbt ngatv skwnss and mostly xcss kurtoss. Ths s an ndcaton that th mpld dstrbuton s ralstc. [Insrt Tabl 3 hr] In ordr to gv furthr vdnc for th mpld dstrbutons obtand by our modl rprsntatv mpld dstrbutons hstograms for th - month log-rturns n Jun 003 ar shown n Fgurs 5a full sampl and 5b rstrctd sampl for n 6 and n 7. To mak th hstograms of th mpld 9

20 dstrbutons w mak us of th Parson systm of dstrbutons 3 as appld n Matlab 4. Usng th frst four momnts of th data t s asy to fnd n th Parson systm th dstrbuton that matchs ths momnts and to gnrat a random sampl n ordr to produc a hstogram corrspondng to th mpld dstrbuton. From th fgurs t s obvous that th mpld dstrbutons hav ngatv skwnss and postv kurtoss whch s consstnt wth hstorcal data. Ths fgurs ar rprsntatv of th vast maorty of cass 5. Anothr ntrstng thng w obsrv s that dstrbutons for n 6 and n 7 ar practcally ndstngushabl for both sampls. [Insrt Fgurs 5a 5b hr] VI. onclusons In most optons markts th mpld Black chols volatlts vary wth both strk and xpraton a rlatonshp commonly known as th volatlty sml. In ths papr w captur th mpld dstrbuton from opton markt data usng a non-rcombnng bnary tr allowng th local volatlty to b a functon of th undrlyng asst and of tm. Th problm undr consdraton s a non-convx optmzaton problm wth lnar constrants. W laborat on th ntal guss for th volatlty trm structur and us nonlnar constrand optmzaton to mnmz th last squars rror functon on markt prcs. pcfcally w adopt a pnalty mthod and th optmzaton s mplmntd usng a Quas-Nwton algorthm. Approprat constrants allow us to mantan rsk nutralty and to prvnt arbtrag opportunts. Th proposd modl can accommodat Europan optons wth sngl maturts and wth mnor modfcatons optons wth multpl maturts. Also ths mthod s flxbl snc t appls to arbtrary undrlyng asst dstrbutons whch mpls arbtrary local volatlty dstrbutons. Markt mpld nformaton mbodd n th constructd tr 3 In th Parson systm thr s a famly of dstrbutons that ncluds a unqu dstrbuton corrspondng to vry vald combnaton of man standard dvaton skwnss and kurtoss. 4 opyrght 005 Th MathWorks Inc. 5 In rar xcptons only w hav mpld dstrbutons clos to normal or vn lptokurtc. 0

21 can hlp th prcng and hdgng of xotc optons and of OT optons on th sam undrlyng procss. W tst our modl usng FTE 00 optons data. Th rsults obtand strongly support our modllng approach. Prcng rsults ar smooth wthout th prsnc of an ovr-fttng problm and th drvd mpld dstrbutons ar ralstc. Also th computatonal burdn s not a maor ssu.

22 APPENDIX A: Fasblty of th ntalzd non-rcombnng tr W ntalz th tr usng th followng volatlty trm structur: λ t σ σ R λ whr n Th fasblty of th ntal tr dpnds on th rght choc of th local volatlty trm structur; hnc to obtan a fasbl ntal tr w must fnd an ntrval wth th approprat valus of λ. In ordr to prsrv th rsk nutralty at vry tm stp th followng constrants must b satsfd: r f δ t A a r f δ t A b Also u σ t A a d σ t A b ubsttutng A a and A b to A a and A b rspctvly w gt th followng nqualts: σ rf δ t A 3a σ rf δ t A 3b Thus w hav that σ rf δ t A 4 λ t For λ 0 σ σ s strctly ncrasng. nc A 4 holds for vry ths mans that mn σ rf δ t or σ r f δ t A 5 Th mnmum valu of σ s for σ thus A 5 s ndpndnt of λ. Thrfor f λ s postv thr s no uppr bound for λ.

23 λ t For λ < 0 σ σ s strctly dcrasng. nc A 4 holds for vry ths mans that mn σ rf δ t> σ n rf δ t > λ n t r f δ σ t But n t T thus λ log T r f δ σ t A 6 If w allow λ to tak both ngatv and postv valus thn λ should blong n th ntrval r f δ t λ log T σ A 7 3

24 APPENDIX B: Fasblty of th ntalzd non-rcombnng tr assumng tm dpndnt rf δ and t W dnot wth r f and δ th rsk fr rat and dvdnd yld rspctvly btwn two conscutv tm stps.. btwn tm stp and... n. [Insrt Fgur A] W ntalz th tr usng th followng volatlty trm structur: λ t σ σ λ R whr n Th fasblty of th ntal tr dpnds on th rght choc of th local volatlty trm structur; hnc to obtan a fasbl ntal tr w must fnd an ntrval wth th approprat valus of λ. In ordr to prsrv th rsk nutralty at vry tm stp th followng constrants must b satsfd: r f δ t A a r f δ t A b Also u σ t A a d σ t A b ubsttutng A a and A b to A a and A b rspctvly w gt th followng nqualts: σ rf δ t A 3a σ rf δ t A 3b Thus w hav that σ rf δ t A 4 For λ 0 σ σ λ t Lt ξ max δ t M r f s strctly ncrasng. 4

25 Thn A 4 holds for vry f mn σ ξ or M σ ξ M A 5 Th mnmum valu of σ s for σ thus A 5 s ndpndnt of λ. Thrfor f λ s postv thr s no uppr bound for λ. For λ < 0 σ σ λ t Lt ξ mn δ t m r f Thn A 4 holds for vry f mn σ ξ σ n ξ m m s strctly dcrasng. σ λ n t ξ m But n t T thus ξ m λ log A 6 T σ If w allow λ to tak both ngatv and postv valus thn λ should blong n th ntrval ξ m λ log T σ A 7 5

26 Appndx : Formulas adustd for tm dpndnt rf δ and t W dnot wth r f and δ th rsk fr rat and dvdnd yld rspctvly btwn two conscutv tm stps.. btwn tm stp and... n and wth r f and δ w dnot th rsk fr rat and dvdnd yld rspctvly from today tll th maturty of th opton.. from to n. If w allow r f δ and ar rplacd wth th followng: t to b tm dpndnt th quatons of th man txt u σ t 3a d σ t u... n... 3b ξ m λ log T σ whr 5 ξ m mn δ r f t p r f δ t... n... 7 p p t 9b r f n r f δ t... n... 0a r f δ t... n... 0b max ' ' δ T rf T K 0 Mod 6

27 7 0 g t r f δ n 4a 0 t r f g δ n 4b 0 3 k k g Mod N k... 4c 0 0 max ' ' 4 T r T Mod f k K k k g δ N k... 4d p p p p p p t r f l δ 7 t r t t r t f f l p p δ δ 8 Rato Dlta t δ 9 } { x h t h r h f k k nod to from nod us tak path that probablts on th th of odd for vn for k / /

28 Rfrncs At-ahala Y. Lo A. W Nonparamtrc stmaton of stat-prc dnsts mplct n fnancal asst prcs. Journal of Fnanc 53 pp Barl. akc N Growng a smlng tr. Rsk 8 pp Bssmbndr H. oughnour J. F. gun P. J Is thr a trm structur of futurs volatlts? R-valuatng th amulson Hypothss. Workng Papr. Black F. and chols M Prcng of optons and corporat lablts. Journal of Poltcal Economy 8 pp Brttn-Jons M. and Nubrgr A Opton prcs mpld prc procsss and stochastc volatlty. Journal of Fnanc 55 pp haralambous. 99. onugat gradnt algorthm for ffcnt tranng of artfcal nural ntworks. IEE Procdngs-G 39 pp hrss N Transatlantc trs. Rsk 9 pp ox J. Ross. and Rubnstn M Opton prcng: A smplfd approach. Journal of Fnancal Economcs 7 pp Drman E. Kan I Rdng on a sml. Rsk 7 pp Drman E. Kan I. and hrss N Impld trnomal trs of th volatlty sml. Journal of Drvatvs 3 pp. 7-. Drman E. Kan I tochastc mpld trs: Arbtrag prcng wth stochastc trm and strk structur of volatlty. Intrnatonal Journal of Thortcal and Appld Fnanc pp Dupr B Prcng wth a sml. Rsk 7 pp Edrsngh Nak and Uppal 993. Optmal rplcaton of optons wth transacton costs and tradng rstrctons. Journal of Fnancal and Quanttatv Analyss 8 pp Facco A. V. Mcormck G. P Nonlnar Programmng: quntal Unconstrand Mnmzaton Tchnqus. John Wly and ons Inc. Fltchr R Practcal mthods of optmzaton. hchstr Wly. 8

29 Jackwrth J. and Rubnstn M Rcovrng probablty dstrbutons from opton prcs. Journal of Fnanc 5 pp Jackwrth J Gnralzd bnomal trs. Journal of Drvatvs 5 pp Ldot O. anta-lara P Rlatv prcng of optons wth stochastc volatlty. Workng Papr Unv. of alforna Los Angls. Lu X. hacklton B. M. Taylor J.. Xu X losd transformatons from rsk-nutral to ral-world dstrbutons. Workng Papr Unv. of Lancastr. Rubnstn M Impld bnomal trs. Journal of Fnanc 49 pp amulson P Proof that proprly antcpatd prcs fluctuat randomly. Industral Managmnt Rvw 6 pp kadopoulos G. 00. Volatlty sml consstnt opton modls: A survy. Intrnatonal Journal of Thortcal and Appld Fnanc 4 pp tutzr M A smpl nonparamtrc approach to drvatv scurty valuaton. Journal of Fnanc 5 pp Talas K Impld bnomal trs and gntc algorthms. Doctoral Dssrtaton Impral ollg London. Whaly R. 98. Valuaton of Amrcan call optons on dvdnd-payng stocks. Journal of Fnancal Economcs 0 pp

30 FIGURE Fgur : Non-rcombnng tr wth 4 stps. Fgur : A typcal trplt n a non-rcombnng tr. 30

31 Fgur 3: moothng of th opton pay-off functon at maturty. 3

32 Fgur 4: Plots of th call prcs markt and stmatd for th FTE 00 ndx for th st tradng day of Jun 003. dnots th valu of th undrlyng asst and T th calndar days to maturty. 3

33 8 Impld Probablty Dstrbuton Hstogram Jun Full ampl n Frquncy Month log-rturn FTE00 8 Impld Probablty Dstrbuton Hstogram Jun Full ampl n Frquncy Month log-rturn FTE00 Fgur 5a: Impld probablty dstrbutons hstograms obtand for th - month log-rturn of Jun 003 usng th full sampl for n 6 and n 7. 33

34 Impld Probablty Dstrbuton Hstogram Jun Rstrctd ampl n Frquncy Month log-rturn FTE00 Impld Probablty Dstrbuton Hstogram Jun Rstrctd ampl n Frquncy Month log-rturn FTE00 Fgur 5b: Impld probablty dstrbutons hstograms obtand for th -month log-rturn of Jun 003 usng th rstrctd sampl for n 6 and n 7. 34

35 35 Fgur A: A typcal trplt n th ntalzaton of th non-rcombnng tr assumng f r δ and t to b tm dpndnt. t σ t r f δ t σ r f δ t

36 TradngDay Expry Asst N Error Pnalty Maturty lambda 0/0/003 Jan E /0/003 Fb E /0/003 Mar E /0/003 Jun E /0/003 Dc /03/003 Fb E /03/003 Mar E /03/003 Apr E /03/003 May E /03/003 Jun E /03/003 p E /03/003 Dc E /03/003 Mar E /03/003 Apr E /03/003 May E /03/003 Jun E /03/003 p E /0/003 Apr E /0/003 May E /0/003 Jun E /0/003 Jul E /0/003 p E /0/003 Mar E Tabl : Rsults for th applcaton of th modl on th st tradng day of ach month of th yar 003: Tradng Day s th tradng day of ach contract Expry s th xpraton month of ach contract Asst s th valu of th undrlyng asst at th spcfd tradng day N s th numbr of contracts usd for th calbraton Error s th valu of th obctv functon Pnalty s th valu of th pnalty trm Maturty s th calndar days tll th maturty of th contract and lambda s th valu of λ that gvs th bst fasbl soluton 36

37 TradngDay Expry Asst N Error Pnalty Maturty lambda 05/0/003 May E /0/003 Jun E /0/003 Jul E /0/003 p E /0/003 Mar E /0/003 Jun E /0/003 Jul E /0/003 Aug E /0/003 p E /0/003 Dc E /0/003 Jun E /0/003 Jul /0/003 Aug E /0/003 p E /0/003 Oct E /0/003 Dc E /0/003 Mar E /0/003 Aug E /0/003 p E /0/003 Oct E /0/003 Nov E /0/003 Dc E /0/003 Mar E Tabl contnud: Rsults for th applcaton of th modl on th st tradng day of ach month of th yar 003: Tradng Day s th tradng day of ach contract Expry s th xpraton month of ach contract Asst s th valu of th undrlyng asst at th spcfd tradng day N s th numbr of contracts usd for th calbraton Error s th valu of th obctv functon Pnalty s th valu of th pnalty trm Maturty s th calndar days tll th maturty of th contract and lambda s th valu of λ that gvs th bst fasbl soluton. 37

38 TradngDay Expry Asst N Error Pnalty Maturty lambda 08/0/003 Jun E /0/003 p E /0/003 Oct E /0/003 Nov E /0/003 Dc E /0/003 Mar E /0/003 Oct E /0/003 Nov E /0/003 Dc E /0/003 Jan E /0/003 Mar E /0/003 Jun E /03/003 Nov E /03/003 Dc E /03/003 Jan E /03/003 Fb E /03/003 Mar E /03/003 Jun E /0/003 Dc E /0/003 Jan E /0/003 Fb E /0/003 Mar E /0/003 Jun E Tabl contnud: Rsults for th applcaton of th modl on th st tradng day of ach month of th yar 003: Tradng Day s th tradng day of ach contract Expry s th xpraton month of ach contract Asst s th valu of th undrlyng asst at th spcfd tradng day N s th numbr of contracts usd for th calbraton Error s th valu of th obctv functon Pnalty s th valu of th pnalty trm Maturty s th calndar days tll th maturty of th contract and lambda s th valu of λ that gvs th bst fasbl soluton. 38

39 Absolut Errors Modl Full ampl Rstrctd ampl n 6 Man Mdan n 7 Man Mdan n 8 Man Mdan Obsrvatons Tabl : Man and mdan absolut rrors usng our modl for n6 7 8 and data from th full and th rstrctd sampl. Impld 003 n 6 Man Varanc kwnss Kurtoss Obsrvatons Man Mdan Hstorcal Man Varanc kwnss Kurtoss Obsrvatons Tabl 3: Impld rsk-nutral and hstorcal statstcs of th dstrbuton of th FTE 00 -month log-rturns 39