Barrier Options and a Reflection Principle of the Fractional Brownian Motion
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1 rrier Opions nd Reflecion Principle of he Frcionl rownin Moion Ciprin ecul DOFI Acdem of conomic udies uchres Romni mil: Firs drf: epember Absrc he purpose of his pper is o obin he price of he brrier opions in frcionl rownin moion environmen in he specil cse of zero ineres re. As consequence we derive reflecion principle for he frcionl rownin moion.
2 . Inroducion If 0 he frcionl rownin moion fm wih urs prmeer is he coninuous Gussin process 0 wih men [ ] 0 nd whose covrince is given b: C [ ] + s s s If hen coincides wih he sndrd rownin moion. he frcionl rownin moion is self-similr process mening h for n α 0 α hs he sme lw s α. he consn deermines he sign of he covrince of he fuure nd ps incremens. his covrince is posiive when zero when nd negive when. Anoher proper of he frcionl rownin moion is h for i hs long-rnge dependence. he self-similri nd long-rnge dependence properies mke he frcionl rownin moion suible ool in differen pplicions like mhemicl finnce. ince for he frcionl rownin moion is neiher Mrkov process nor semimringle we cnno use he usul sochsic clculus o nlze i. Worse sill fer phwise inegrion heor for frcionl rownin moion ws developed in 995 Decreusefond nd Usunel 999 i ws proven h he mrke mhemicl models driven b could hve rbirge Rogers 997. he frcionl rownin moion ws no longer considered fi for mhemicl modeling in finnce. owever fer he developmen of new kind of inegrl bsed on he Wick produc Duncn u nd Psik-Duncn 000 u nd Oksendl 000 clled frcionl Io inegrl i ws proved u nd Oksendl 000 h he corresponding Io pe frcionl lck-chools mrke hs no rbirge. here re some oher ws of defining he frcionl Io inegrl. ee for emple Alos Mze nd ulr 00 Alos nd ulr 00 Perez-Abreu nd udor 00 or ender 00. A review of he resuls concerning he frcionl Io inegrl cn be found in ecul 00 or ecul 003. In he pper of u nd Oksendl 000 formul for he price of uropen opion 0 is derived nd he formul is eended for ever [ 0 ] in he pper of ecul 00. ender 003 generlized i for he cse of non consn bu deerminisic volili using no he qusi-condiionl epecion bu liner frcionl D. he purpose of his pper ws o derive he price of brrier opions in frcionl rownin moion environmen. u so fr we onl mnged o obin he price of brrier opion for he cse of zero ineres re. hese resuls re no ver ppeling for he finnce bu he cn be used o obin for he frcionl rownin moion somehing h cn be clled he reflecion principle.
3 . A Reflecion Principle for he Frcionl rownin Moion Consider frcionl lck-choles mrke wih mone mrke ccoun nd sock. Under he risk-neurl mesure P we hve h he sock price sisfies he equion: r d + d d. where r represen he consn riskless ineres re. We will denoe b [] he qusi-condiionl epecion u nd Oksendl 000 wih respec o he risk-neurl mesure. We know h he price of ecul 00: F F - mesurble coningen clim F is given b e [ F] We consider he coningen clims: inr cll nd pu wih srike :. r C P Gp cll nd pu wih srike : GC GP emm. We hve h: r r C e d P e d.3 GC d GP d where d d ln + r ln + r + nd nd is he cumulive probbili of he sndrd norml disribuion. 3
4 4 Proof: ee he proof of heorem 4. in ecul 00. We will mke he following noions: inf : m 0 inf :.4 M 0 sup : Firs we will nlze he cse of no drif in. i.e 0 r We know h + ep.5 heorem. Consider h 0 r.. If nd hen m.6 where ln nd ln. If nd hen b b m.7 where b ln nd b ln 3. If nd hen M.8
5 4. If nd hen b b.9 M Proof. Consider down-nd-ou binr cll DOC wih srike price brrier nd muri. he poff of his opion is. he price of his m coningen clim DOC is nonzero if nd zero if. Consider now porfolio h consiss in long posiion of one binr cll wih srike nd muri nd shor posiion of gp pus wih srike price nd muri. I cn be seen from emm. h if i.e. he price of his porfolio is zero. o if he brrier is hi previous o he muri he vlue of his porfolio is equl o h of he opion. If he brrier is no hi he porfolio nd he opion will hve he sme poff muri since he gp pu is ou of he mone. ince he frcionl lck-choles does no hve rbirge nd he down-ndou binr cll nd he porfolio hve he sme poff heir vlue will be he sme for. o DOC C GP. Consider coningen clim h ps one uni if he sock price does no hi he brrier before down-nd-ou bond. he poff of his coningen clim is. As in he previous cse we look for porfolio h hs he sme vlue m s he coningen clim nd s consequence of he no-rbirge proper of he frcionl lck-choles he will hve he sme vlue for ever. In his cse we cn chose porfolio consising in long posiion of one binr cll wih srike nd muri nd shor posiion of gp pus wih srike price nd muri. 3. In his cse we consider n up-nd-ou binr pu nd porfolio h consiss in long posiion of one binr pu wih srike nd muri nd shor posiion of gp clls wih srike price nd muri. 5
6 4. In his cse we consider n up-nd-ou bond nd porfolio h consiss in long posiion of one binr pu wih srike nd muri nd shor posiion of gp clls wih srike price nd muri. heorem. Consider h r 0.. If nd hen c c.0 m where ln + c nd c ln. If nd hen c c. M Proof. Consider down-nd-ou cll wih srike price brrier nd muri. he poff of his opion is. he vlue of his opion m is given b: M M One cn see h his coningen clim hs he sme poff s porfolio h consiss in long posiion of one cll wih srike nd muri nd shor posiion of pus wih srike price nd muri. Using heorem 4. in ecul 00 he vlue of his porfolio is: c c 6
7 7. In his cse we consider n up-nd-ou pu nd porfolio h consiss in long posiion of one pu wih srike nd muri nd shor posiion of clls wih srike price nd muri. We know h here is probbili mesures P such h + is fm under P. he reflecion principle of he rownin moion gives he common disribuion of he rownin moion nd is minimum or mimum. he ne corollr gives similr resul for he frcionl rownin moion. Corollr.. If 0 nd hen + m + ep.. If 0 nd hen M + ep.3 he shorcoming of he resul is h he qusi-condiionl epecion i is under P no under he probbili mesure P is fm under P no under P.
8 Remrks For finnce n imporn resul would be formul for he price of brrier opions in he cse in which he ineres re is no zero. u he eension o he cse of non-zero drif in. seems ver difficul. References Alos. O. Mze nd D. ulr 00 ochsic clculus wih respec o Gussin processes Annls of Probbili Alos. nd D. ulr 00 ochsic clculus wih respec o he frcionl rownin moion Preprin Universi of rcelon ender C. 00 he Frcionl Io Inegrl Chnge of Mesure nd Absence of Arbirge Preprin ender C. 003 plici oluions of Clss of iner Frcionl Ds nd Applicions o Finnce Preprin Decreusefond nd A.. Usunel 999 ochsic nlsis of he frcionl rownin moion Poenil Anlsis Duncn.. Y. u nd. Psik-Duncn 000 ochsic clculus for frcionl rownin moion I. heor IAM J. Conrol Opim u Y. nd. Oksendl 000 Frcionl whie noise clculus nd pplicion o Finnce Preprin Universi of Oslo u Y.. Oksendl nd A. ulem 000 Opiml consumpion nd porfolio in lck-choles mrke driven b frcionl rownin moion Preprin 3/000 Universi of Oslo in.j. 995 ochsic nlsis of frcionl rownin moion frcionl noises nd pplicions IAM Review ecul C. 00 Opion Pricing in Frcionl rownin Moion nvironmen working pper DOFI Acdem of conomic udies uchres ecul C. 003 ochsic Clculus of he Frcionl rownin Moion nd Applicions in Mhemicl Finnce Mc Disserion Pper Fcul of Mhemics Universi of uchres Oksendl 003 Frcionl rownin moion in finnce Preprin 8/003 Universi of Oslo 8
9 Perez-Abreu V. nd C. udor 00 A rnsfer Principle for Muliple ochsic Frcionl Inegrls Preprin CIMA Rogers.C.G. 997 Arbirge wih frcionl rownin moion Mhemicl Finnce udor C. 00 On he Wiener inegrl wih respec o he frcionl rownin moion ol. Me. M. oc
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